THE DECENNIAL PUBLICATIONS OF THE UNIVERSITY OF CHICAGO THE DECENNIAL PUBLICATIONS ISSUED IN COMMEMORATION OF THE COMPLETION OF THE FIRST TEN YEARS OF THE UNIVERSITY'S EXISTENCEAUTHORIZED BY THE BOARD OF TRUSTEES ON THE RECOMMENDATION OF THE PRESIDENT AND SENATE EDITED BY A COMMITTEE APPOINTED BY THE SENATE EDWARD CAPPS STARE WILLARD CUTTING ROLLIN D. SALISBURY JAMES ROWLAND ANGELL WILLIAM I. THOMAS SHAILER MATHEWS CARL DARLING BUCK FREDERIC IVES CARPENTER OSKAR BOLZA JULIUS STIEGLITZ JACQUES LOEB THESE VOLUMES ARE DEDICATED TO THE MEN AND WOMEN OF OUR TIME AND COUNTRY WHO BY WISE AND GENEROUS GIVING HAVE ENCOURAGED THE SEARCH AFTER TRUTH IN ALL DEPARTMENTS OF KNOWLEDGE LECTURES ON THE CALCULUS OF VARIATIONS 7< D. LECTURES ON THE CALCULUS OF VARIATIONS BY OSKAR BOLZA OF THE DEPARTMENT OF MATHEMATICS THE DECENNIAL PUBLICATIONS SECOND SERIES VOLUME XIV CHICAGO THE UNIVERSITY OF CHICAGO PRESS 1904 Copyright 1904 BY THE UNIVERSITY OF CHICAGO September, 1904 PREFACE THE principal steps in the progress of the Calculus of Variations during the last thirty years may be characterized as follows: 1. A critical revision of the foundations and demonstrations of the older theory of the first and second variation according to the modern requirements of rigor, by WEIERSTRASS, ERDMANN, DU BOIS-REYMOND, SCHEEFFER, SCHWARZ, and others. The result of this revision was: a sharper formulation of the problems, rigorous proofs for the first three necessary conditions, and a rigorous proof of the sufficiency of these conditions for what is now called a "weak" extremum. 2. WEIERSTRASS'S extension of the theory of the first and second variation to the case where the curves under consideration are given in parameter-representation. This was an advance 'of great importance for all geometrical applications of the Calculus of Variations; for the older method implied -for igeometrical problems-a rather artificial restriction. 3. WEIERSTRASS'S discovery of the fourth necessary condition and his sufficiency proof for a so-called "strong" extremum, which gave for the first time a complete solution, at least for the simplest type of problems, by means of an entirely new method based upon what is now known as " WEIERSTRASS'S construction." These discoveries mark a turning-point in the history of the Calculus of Variations. Unfortunately they were given by WEIERSTRASS only in his lectures, and thus became known only very slowly to the general mathematical public. Chiefly under the influence of WEIERSTRASS'S theory a vigorous activity in the Calculus of Variations has set in ix X PREFACE during the last few years, which has led-apart from extensions and simplifications of WEIERSTRASS'S theory-to the following two essentially new developments: 4. KNESER'S theory, which is based upon an extension ofcertain theorems on geodesics to extremals in general. This new method furnishes likewise a complete system of sufficient conditions and goes beyond WEIERSTRASS'S theory, inasmuch as it covers also the case of variable end-points. 5. HILBERT'S a priori existence proof for an extremum of a definite integral-a discovery of far-reaching importance, not only for the Calculus of Variations, but also for the theory of differential equations and the theory of functions. To give a detailed account of this development was the object of a series of lectures which I delivered at the Colloquium held in connection with the summer meeting of the American Mathematical Society at Ithaca, N. Y., in August, 1901. And the present volume is, in substance, a reproduction of these lectures, with such additions and modifications as seemed to me desirable in order that the book could serve as a treatise on that part of the Calculus of Variations to which the discussion is here confined, viz., the case in which the function under the integral sign depends upon a plane curve and involves no higher derivatives than the first. With this view I have throughout supplied the detail argumentation and introduced examples in illustration of the general principles. The emphasis lies entirely on the theoretical side: I have endeavored to give clear definitions of the fundamental concepts, sharp formulations of the problems, and rigorous demonstrations. Difficult points, such as the proof of the existence of a "field," the details in HILBERT'S existence proof, etc., have received special attention. For a rigorous treatment of the Calculus of Variations the principal theorems of the modern theory of functions of a real variable are indispensable; these I had therefore to PREFACE xi presuppose, the more so as I deviate from WEIERSTRASS and KNESER in not assuming the function under the integral sign to be analytic. In order, however, to make the book accessible to a larger circle of readers, I have systematically given references to the following standard works: Encyclopaedie der mathematischen Wissenschaften (abbreviated E.), especially the articles on "Allgemeine Functionslehre" (PRINGSHEIM) and "Differential- und Integralrechnung" (Voss); JORDAN, Cours d'Analyse, second edition (abbreviated J.); GENOCCHIPEANO, Differentialrechnung und Grundzige der Integralrechnung, translated by BOHLMANN and SCHEPP (abbreviated P.); occasionally also to DINI, Theorie der Functionen einer veranderlichen reellen Grosse, translated by LUROTH and SCHEPP; STOLZ, Grundzige der Differential- und Integralrechnung. The references are given for each theorem where it occurs for the first time; they may also be found by means of the index at the end of the book. Certain developments have been given in smaller print in order to indicate, not that they are of minor importance, but that they may be passed over at a first reading and taken up only when referred to later on. A few remarks are necessary concerning my attitude toward WEIERSTRASS'S lectures. WEIERSTRASS'S results and methods may at present be considered as generally known, partly through dissertations and other publications of his pupils, partly through KNESER'S Lehrbuch der Variationsrechnung (Braunschweig, 1900), partly through sets of notes ("Ausarbeitungen") of which a great number are in circulation and copies of which are accessible to everyone in the library of the Mathematische Verein at Berlin, and in the Mathematische Lesezimmer at Gottingen. Under these circumstances I have not hesitated to make use of WEIERSTRASS'S lectures just as if they had been published in print. xii PREFACE My principal source of information concerning WEIERSTRASS'S theory has been the course of lectures on the Calculus of Variations of the Summer Semester, 1879, which I had the good fortune to attend as a student in the University of Berlin. Besides, I have had at my disposal sets of notes of the courses of 1877 (by MR. G. SCHULZ) and of 1882 (a copy of the set of notes in the "Lesezimmer" at Gottingen for which I am indebted to PROFESSOR TANNER), a copy of a few pages of the course of 1872 (from notes taken by MR. OTT), and finally a set of notes (for which I am indebted to DR. J. C. FIELDS) of a course of lectures on the Calculus of Variations by PROFESSOR H. A. SCHWARZ (1898-99). I regret very much that I have not been able to make use of the articles on the Calculus of Variations in the Encyclopaedie der mathtematischen Wissenschaften by KNESER, ZERMELO, and HAHN. When these articles appeared, the printing of this volume was practically completed. For the same reason no reference could be made to HANCOCK'S Lectures on the Calculus of Variations. In concluding, I wish to express my thanks to PROFESSOR G. A. BLISS for valuable suggestions and criticisms, and to DR. H. E. JORDAN for his assistance in the revision of the proof-sheets. OSKAR BOLZA. THE UNIVERSITY OF CHICAGO, August 28, 1904. TABLE OF CONTENTS CHAPTER I x y. PAGE THE FIRST VARIATION OF THE INTEGRAL (x, y, y')dx ~ 1. Introduction - - - - - - - - 1 ~ 2. Agreements concerning Notation and Terminology - 5 ~ 3. General Formulation of the Problem - - - 9 ~ 4. Vanishing of the First Variation - - - - - 13 ~ 5. The Fundamental Lemma and Euler's Differential Equation - - - -- - 20 ~ 6. Du Bois-Reymond's and Hilbert's Proofs of Euler's Differential Equation- - - - - - - 22 ~ 7. Miscellaneous Remarks concerning the Integration of Euler's Differential Equation - - - 26 ~ 8. Weierstrass's Lemma and the E-function - - - 33 ~ 9. Discontinuous Solutions - - - - - 36 ~10. Boundary Conditions - - - - - - - 41 CHAPTER II THE SECOND VARIATION OF THE INTEGRAL F(x, y, y')dx ~11. Legendre's Condition - - - - - - 44 ~12. Jacobi's Transformation of the Second Variation - 51 ~ 13. Jacobi's Theorem - - - - - 54 ~14. Jacobi's Criterion - - - - - - - 57 ~15. Geometrical Interpretation of the Conjugate Points - 60 ~16. Necessity of Jacobi's Condition - - - - - 65 CHAPTER III SUFFICIENT CONDITIONS FOR AN EXTREMUM OF THE INTEGRAL F(x, y, y')dx.0O ~17. Sufficient Conditions for a Weak Minimum - - 68 ~18. Insufficiency of the Preceding Three Conditions for a Strong Minimum, and Fourth Necessary Condition 73 xiii xiv TABLE OF CONTENTS ~19. Existence of a Field of Extremals - - - - 78 ~20. Weierstrass's Theorem 84 ~21. Hilbert's Proof of Weierstrass's Theorem - - - 91 ~22. Sufficient Conditions for a Strong Minimum - - 94 ~23. The Case of Variable End-Points 102 CHAPTER IV WEIERSTRASS'S THEORY OF THE PROBLEM IN PARAMETER-REPRESENTATION ~ 24. Formulation of the Problem - 115 ~25. The First Variation - - - 122 ~ 26. Examples - - - - - - - - - 126 ~27. The Second Variation 130 ~28. The Fourth Necessary Condition and Sufficient Conditions - - - - - - - - - 138 ~29. Boundary Conditions - - - - - - - 148 ~30. The Case of Variable End-Points - - - - 153 ~ 31. Weierstrass's Extension of the Meaning of the Definite / il Integral F(x, y, x', y')dt - - - - - 156 Jto CHAPTER V KNESER'S THEORY ~32. Gauss's Theorems on Geodesics - - - - - 164 ~ 33. Kneser's Theorem on Transversals and the Theorem on the Envelope of a Set of Extremals - - - 166 ~34. Construction of a Field - 175 ~35. Kneser's Curvilinear Co-ordinates - - 181 ~36. Sufficient Conditions for a Minimum in the Case of One Movable End-Point - - - - - 187 ~ 37. Various Proofs of Weierstrass's Theorem The Assumption F(t, a) 0 - - - - - - 193 ~38. The Focal Point - - - - - - - 199 CHAPTER VI WEIERSTRASS'S THEORY OF THE ISOPERIMETRIC PROBLEMS ~ 39. Euler's Rule - - - - 206 ~40. The Second Necessary Condition - - - - 213 TABLE OF CONTENTS XV ~41. The Third Necessary Condition and the Conjugate Point 218 ~42. Sufficient Conditions - - - 232 CHAPTER VII HILBERT'S EXISTENCE THEOREM ~43. Introductory Remarks 245 ~44. Theorems concerning the Generalized Integral Je - 247 ~45. Hilbert's Construction - - - - - - 253 ~46. Properties of Hilbert's Curve - - - - 259 ADDENDA - - - -- 265 INDEX - - - - -- 269 CHAPTER I THE FIRST VARIATION ~1. INTRODUCTION THE Calculus of Variations deals with problems of maxima and minima. But while in the ordinary theory of maxima and minima the problem is to determine those values of the independent variables for which a given function of these variables takes a maximum or minimum value, in the Calculus of Variations definite integralsl involving one or more unknown functions are considered, and it is required so to determine these unknown functions that the definite integrals shall take maximum or minimum values. The following example will serve to illustrate the character of the problems with which we are here concerned, and its discussion will at the same time bring out certain points which are important for an exact formulation of the general problem: EXAMPLE I: In a plane there are given two points A, B and a straight line 2. It is required to determine, among all curves which can be drawn in this plane between A and B, the one which, if revolved around the line i, generates the surface of minimum area. We choose the line S for the x-axis of a rectangular system of co-ordinates, and denote the co-ordinates of the points A and B by x0, yo and xl, yj respectively. Then for a curve y =f(x) 1 The problem of the Calculus of Variations has, however, been extended beyond the domain of definite integrals (viz., to functions defined by differential equations) by A. MAYER, Leipziger Berichte, 1878 and 1895. Compare KNESER, Lehrbuch, chap. vii. 1 2 CALCULUS OF VARIATIONS [Chap. I joining the two points A and B, the area in question is given by the definite integral1 J27r r Vl +y'2dx, where y' stands for the derivative f'(x). For different curves the integral will take, in general, different values; and our problem is then analytically: among all functions f (x) which take for x x0 and x = x the prescribed values Y0 and yI respectively, to determine the one which furnishes the smallest value for the integral J. This formulation of the problem implies, however, a number of tacit assumptions, which it is important to state explicitly: a) In the first place, we must add some restrictions concerning the nature of the functions f (x) which we admit to consideration. For, since the definite integral contains the derivative y', it is tacitly supposed that f (x) has a derivative; the function f (x) and its derivative must, moreover, be such that the definite integral has a determinate finite value. Indeed, the problem becomes definite only if we confine ourselves to curves of a certain class, characterized by a well-defined system of conditions concerning continuity, existence of derivative, etc. For instance, we might admit to consideration only functions f (x) with a continuous first derivative; or functions with continuous first and second derivatives; or analytic functions, etc. b) Secondly, by assuming the curves representable in the form y f f(x), where f (x) is a single-valued function of x, we have tacitly introduced an important restriction, viz., that we consider only those curves which are met by every ordinate between x0 and xl at but one point. 1 a being a real positive quantity, I/a will always be understood to represent the positive value of the square root. ~1] FIRST VARIATION 3 We can free ourselves from this restriction by assuming the curve in parameter-representation: x= —(t), y —=(t). The integral which we have to minimize becomes then J- 27r y x2+ y'2dt, where x' = '(t), y' '(t), and where to and 1l are the values of t which correspond to the two end-points. c) It is further to be observed that our definite integral represents the area in question only when y 0 throughout the interval of integration. The problem implies, therefore, the condition that the curves shall lie in a certain region2 of the x, y-plane (viz., the upper half-plane). d) Our formulation of the problem tacitly assumes that there exists a curve which furnishes a minimum for the area. But the existence of such a curve is by no means selfevident. We can only be sure that there exists a lower limit3 for the values of the area; and the decision whether this lower limit is actually reached or not forms part of the solution of the problem. The problem may be modified in various ways. For instance, instead of assuming both end-points fixed, we may assume one or both of them movable on given curves. An essentially different class of problems is represented by the following example: 1Compare chap. iv. Even in this generalized form the analytic problem is not quite so general as the original geometrical problem. For the area in question may exist and be finite, and yet not be representable by the above definite integral. This suggests an extension of the problem of the Calculus of Variations, first considered by WEIERSTRASS. Compare ~~ 31 and 44. 2 A restriction of the same nature, but from other reasons, occurs in the problems of the brachistochrone and of the geodesic; compare ~ 26. 3 Compare E. I A, p. 72, and II A, p. 9; J. I, No. 25; and P., No. 20. 4 CALCULUS OF VARIATIONS [Chap. I EXAMPLE II: Among all closed plane curves of given perimeter to determine the one which contains the maximum area. If we use parameter-representation, the problem is to determine among all curves for which the definite integral Jtj tl/X/'2 + y 2 dt has a given value, the one which maximizes the integral t' J 2 (x y'- x y) dt Here the curves out of which the maximizing curve is to be selected are subject-apart from restrictions of the kind which we have mentioned before-to the new condition of furnishing a given value for a certain definite integral. Problems of this kind are called "isoperimetric problems;" they will be treated in chap. vi. The preceding examples are representatives of the simplest -and, at the same time, most important-type of problems of the Calculus of Variations, in which are considered definite integrals depending upon a plane curve and containing no higher derivatives than the first. To this type we shall almost exclusively confine ourselves. The problem may be generalized in various directions: 1. Higher derivatives may occur under the integral. 2. The integral may depend upon a system of unknown functions, either independent or connected by finite or differential relations. 3. Extension to multiple integrals. For these generalizations we refer the reader to C. JORDAN, Cours d'Analyse, 2e ed., Vol. III, chap. iv; PASCAL-(SCHEPP), Die Variationsrechnung (Leipzig, 1899); and KNESER, Lehrbuch der Variationsrechnung (Braunschweig, 1900), Abschnitt VI, VII, VIII. ~2] FIRST VARIATION 5 ~2. AGREEMENTS1 CONCERNING NOTATION AND TERMINOLOGY a) We consider exclusively real variables. The "interval (a b)" of a variable x-where the notation always implies a<b-is the totality of values x satisfying the inequality a -cx -b. The "vicinity (8) of a point x=- ca, x2:=a:z *. *, x,= ac," is the totality of points x1, x2, *, xn satisfying the inequalities: 1x — al <8, | x,- a,'<2,, x,, — a < 8. The word "domain" will be used in the same sense as the German Bereich, i. e., synonymous with "set of points" (compare E. II A, p. 44). The word "region" will be used: (a) for a "continuum," i. e., a set of points which is "connected" and made up exclusively of "inner" points; in this case the boundary does not belong to the region ("open" region); (b) for a continuum together with its boundary ("closed" region); (c) for a continuum together with part of its boundary. The region may be finite or infinite; it may also comprise the whole n-dimensional space. When we say: a curve lies "in" a region, we mean: each one of its points is a point of the region, not necessarily an inner point. For the definition of "inner" point, "boundary point" (frontiere), and "connected" (d'un setl tenant) we refer to E. II A, p. 44; J. I, Nos. 22, 31; and HURWITZ, Verhandlzngen des ersten internationalen Mathematikercongresses in Zurich, p. 94. b) By a "functiont" is always meant a real single-valued function. The substitution of a particular value x x0 in a function ((x) will be denoted by 1 The reader is advised to proceed directly to ~ 3 and to use ~ 2 only for reference. 6 CALCULUS OF VARIATIONS [Chap. 1 X=X0 similarly 1 -(x, y) = (Xo, yo); also (x) (X()- (x0) Instead we shall also use the simpler notation () +z, (x, ),) [(x)] where it can be done without ambiguity, compare e). We shall say: a function has a certain property IN1 a domzain V of the independent variables, if it has the property in question at all points of the domain A, no matter whether they are interior or boundary points. A function of x, x2, ', xn has a certain property in the vicinity of a point xt — a,, x2- a2'', x, x a-, if there exists a positive quantity 8 such that the function has the property in question in the vicinity (8) of the point a,, a2,', a C,. If L (h) -0, we shall say: (h) is an "infinitesimal" h =0 (for L h 0); such an infinitesimal will in a general way be denoted by (h). Also an independent variable h which in the course of the investigation is made to approach zero, will be called an "infinitesimal." c) Derivatives of functions of one variable will be denoted by accents, in the usual manner:, _() c-lf(x) df () tc. f M dxdx d etc. For brevity we shall use the following terminology2 for various classes of functions which will frequently occur in the sequel. We shall say that a function f(x) which is defined in an interval (x0xl) is 1 Or, with more emphasis, "throughout." 2The letters C, D,are to suggest "continuous," "discontinuous;" the accents the order of the derivative involved. ~2] FIRST VARIATION 7 of class C if f (x) is continuous ] of class C' if f(x) and f' (x) are continuous n (X of class C(' if f(x), f' (x),. * andf 'a(x) are continuous J with the understanding concerning the extremities of the interval that the definition of f (x) can be so extended beyond (xoxl) that the above properties still hold at x0 and xl. If f(x) itself is continuous, and if the interval (xo0l) can be divided into a finite number of subintervals (X0l), (CC2),., (C.n_), such that in each subinterval f(x) is of class C'(C"), whereas f'(x) (f" (x)) is discontinuous at cl, c2,, Cnl, we shall say that f(x) is of class D'(D"). We consider class C'(C") as contained in D'(D"), viz., for n -1. From these definitions it follows that, for a function of + class D', the progressive' and regressive derivatives f'(c,), f'(cv) exist, are finite and equal to the limiting values2 f' (c, + 0), f' (c - 0) respectively. d) Partial derivatives of functions of several variables will be denoted by literal subscripts (KNESER): r1(/, F(x, y p) p) Fyp (X y a p) - ( a-, etc.;F? (0, Q P0) F (x, y,p) P= Also of a function of several variables we shall say that it is of class C(n) in a domain 2 if all its partial derivatives 1E. II A, p. 61; DINI, Grundclagen, etc., ~68: and STOLZ, Grundz ige, etc., Vol. I, p. 31. E. II A, p. 13. 8 CALCULUS OF VARIATIONS [Chap. I up to the nth order inclusive exist and are continuous in' the domain F. e) The letters x, y will always be used for rectangular co-ordinates with the usual orientation of the positive axes, i. e., the positive y-axis to the left of the positive x-axis. It will frequently be convenient to designate points by numbers: 0, 1, 2,; the co-ordinates of these points will then always be denoted by x0, yo;, i x 2, 2; X2 Y respectively; their parameters, if they lie on a curve given in parameterrepresentation, by to, tl, t2, ' A curve2 (arc of curve) y =f(x), x0 C x x, will be said to be of class C, C', etc., if the functionf(x) is of class C, C', etc., in (xoxi). In particular, a curve of class D' is continuous and made up of a finite number of arcs with continuously turning tangents, not parallel to the y-axis. The points of the curve whose abscissae are the points of discontinuity C1, C2, ', n Cn_ off'(x),. will be called c its corners. At a corner the ^ /|v ~ Gcurve has a progressive and a Xjr \regressive tangent, and, + + _ 4 _____, _ __,_,"' tan a =f'(), tan a =f'(c) FIG. 1 (See Fig. 1.) f) The integral J- F(x,, y')dx taken along the curve G ' yy=f(x), Xo0 X x lWhen > contains boundary points, an agreement similar to that in the case of one variable is necessary with respect to these points. 2The" Corresponding definitions for curves in parameter-representation will be given in ~ 24. __ ~3] FIRST VARIATION 9 from the point A (x0, Yo) to the point B (xi, yl), i. e., the integral f'1 ( x, ), f(x)) d will be denoted by Jo (AB) (more briefly J, or J (AB)); or by J,,, if the end-points are designated by numbers: ',L, v. g) The distance between the two points P and Q will be denoted by I PQ 1, the circle with center 0 and radius r by (0, r) (HARKNESS AND MORLEY). The angle which a vector makes with the positive x-axis will be called its amplitude. ~3. GENERAL FORMULATION OF THE PROBLEM1 a) After these preliminary explanations, the simplest problem of the Calculus of Variations may be formulated in the most general way, as follows: There is given: 1. A well-defined infinitude M of curves, representable in the form y =f(x), X O X; the end-points and their abscissae x0, xl may vary from curve to curve. We shall refer to these curves as "admissible curves." 2. A function F (x, y, p) of three independent variables such that for every admissible curve G, the definite integral J = F(x, y, y')d (1) has a determinate finite value. 1Until rather recently a certain vagueness has prevailed with respect to the fundamental concepts of the Calculus of Variations. The most important contributions toward clear definitions and sharp formulations of the problems are due to Du BOIS-REYMOND, "Erlauterungen zu den Anfangsgriinden der Variationsrechnung," Mathematische Annalen, Vol. XV (1879), p. 283; SCHEEFFER, "Ueber die Bedeutung der Begriffe 'Maximum und Minimum' in der Variationsrechnung," ibid., Vol. XXVI (1886), p. 197; WEIERSTRASS, Lectures on the Calculus of Variation, especially those since 1879. Compare also ZERMELO, Untersuchulngen zur Variationsrechnung, Dissertation (Berlin, 1894), p. 24; KNESER, Lehrbuch, ~17, and OSGOOD, "Sufficient Conditions in the Calculus of Variations," Annals of Mathematics (2), Vol. II (1901), p. 105. 10 CALCULUS OF VARIATIONS [Chap. I The set' of values J, thus defined has always a lower limit, K, and an upper limit, G (finite or infinite2). If the lower (upper) limit is finite, and if there exists an admissible curve d such that J = K, (J G = ), the curve ( is said to furnish the absolute minimum (mactximum) for the integral J (with respect to A). For every other admissible curve d we have then Jo, Jt, I (J JO). (2) The word "extremum" 3 will be used for maximum and minimum alike, when it is not necessary to distinguish between them. Hence the problem arises: to determine all admissible curves which, in this sense, minimize or maximize the integral J. b) As in the theory of ordinary maxima and minima, the problem of the absolute extremum, which is the ultimate aim of the Calculus of Variations, is reducible to another problem which can be more easily attacked, viz., the problem of the relative extrenium: An admissible curve ( is said to furnish a relative minuimumn (maximnum) if there exists a "neighborhood I of the culrve (," however small, such that the curve ( furnishes an absolute minimum with respect to the totality ltt of those curves of it which lie in this neighborhood; and by a neighborhood U of the curve 6 we understand any region5 which contains ( in its interior. 1 By "set" we translate the German Punktmenge, the French ensemble, J. I, No. 20. 2The upper limit is +-r, if for every preassigned positive quantity A there exist curves C( for which J(5 > A; see E. II A, p. 9. 3 Du BOIS-REYMoND, Mathematische Annalen, Vol. XV, p. 564. 4 In the use of tLe words " absolute" and "relative" I follow Voss in E. II A, p. 80. Many authors call the isoperimetric problems "problems of relative maxima and minima." 5 For the definition of the termn "region," see p. 5. ~3a] FIRST VARIATION 11 According to STOLZ, the relative minimum (maximum) will be called proper, if there exists a neighborhood 3I such that in (2) the sign > (<) holds for all curves ( different from (; implrope' if, however the neighborhood M may be chosen, there exists some curve C different from (for which the equality sign has to be taken. A curve which furnishes an absolute extremum evidently furnishes a fortiori also a relative extremum. Hence the original problem is reducible' to the problem: to determine all those curves uwhich ft orish a relative minimum,'; and in this form we shall consider the problem in the sequel. We shall henceforth always use the words "minimum," "maximumll in the sense of relative minimum, maximum; and we shall confine ourselves to the case of a minimum, since every curve which minimizes J, at the same time maximizes - J, and vice versa. c) In the abstract formulation given above, the problem would hardly be accessible to the methods of analysis; to make it so, it is necessary to specify some concrete assumptions concerning the admissible curves and the function F. For the present, we shall make the following assumptions: A. The infinitude Al of admissible curves shall be the totality of all curves satisfying the following conditions: 1. They pass through two given points A (x0, yo) and B(x1, y1). 2. They are representable in the form y =f(x), Xo X x,1 f (x) being a single-valued function of x. 3. They are con/tinuous and consist of a finite number of 1 After the relative problem has been solved, it merely remains to pick out among its solutions those which furnish the smallest or largest value for J. Only if the relative problem should have an infinitude of solutions, new difficulties would arise. For a direct treatment of the problem of the absolute extremum compare HILBERT'S existence proof (chap. vii); DARBOUX, Thgorie des surfaces, Vol. III, p. 89; and ZERMELO, Jahresbericht der Deutschen Mathenmatiker-Vereinigung, Vol. XI (1902), p. 184. 12 CALCULUS OF VARIATIONS [Chap. 1 arcs with continuously turning tangents, not parallel to the y-axis; i. e., in the terminology of ~2, c),f(x) is of class D'. 4. They lie in a given region'1 of the x, y-plane. B. The function F(x, y, p) shall be continuous2 and admit continuous partial derivatives of the first, second, and third orders in a domain3 ~ which consists of all points4 (x, y, p) for which (x, y) is a point of A, andp has a finite value. Under these assumptions the definite integral J, taken along any admissible curve ( is always finite and determinate,5 provided we define, in the case of a curve with corners, the integral as the sum of integrals taken between two successive corners. Since we suppose the end-points A and B fixed and the curves representable in the form y f(x), the curves ( all lie between the two lines X= x and x=xl, with the exception of the end-points, which lie on these lines. Hence it follows that we may, in the present case, give the following simpler definition of a minimum: An admissible curve (: y=f(x) minimizes the integral J, if6 there 1 Compare ~2, a). 2I follow here the example of PASCAL, loc. cit., p. 21, and OSGOOD, loc. cit., p, 105. WEIERSTRASS, JORDAN, and KNESER suppose the function F (x, y, p) to be analytic. 3 If we interpret p as a third co-ordinate perpendicular to the x, y-plane, C is the cylinder, infinite in both directions, whose base is the region E. 4" Point" in the sense of the theory of "point-sets." Compare E. II A, p. 44, and J. I, No. 20. 5 If the curve has no corners, this follows at once from elementary theorems on continuous functions (J. I, Nos. 60, 66). If the curve has corners, the integral Jo has no immediate meaning. But the two integrals Xai X F (x, f (, f'W() dz and F (Ix, f (x), f(x)) dx are finite and determinate and equal to each other, and at the same time equal to the sum of integrals mentioned in the text. Compare DINI, loc. cit., ~62; ~187, 2; ~190, 9; and ~ 190, 2. 6 In admitting the equality sign in the inequality (2), I deviate from the conventions generally adopted in the Calculus of Variations and follow STOLZ (Grundziige der Differenzialrechnung, Vol. I, p. 199), whose definition is more consistent with the usual definition of absolute minimum. If the equality sign were omitted, it could not be said that every curve which furnishes an absolute minimum furnishes afortiori also a relative minimum. ~41 FIRST VARIATION 13 exists a positive quantity p such that J Jo for every admissible curve (S: =f(x) which satisfies the inequality IY —Y <P for Xo x e,. (3) This means geometrically that the curve ( lies in the interior' of the strip of the x, y-plane between the two curves =f(x) +p, y f(x) - on the one hand, and the two f4 lines X= 0, x= x on the L fo) other hand. This strip we f( shall call "the neighborhood2 (p) of the curve,," the points A and B being included, the rest of the boundary excluded. FIG. 2 ~4. VANISHING OF THE FIRST VARIATION We now suppose we have found a curve G: y =f(x) which minimizes the integral ~ X1 J=' F(x y, y')dx in the sense explained in the last section. We further suppose, for the present,3 that f' (x) is continuous in (xxl) and that ( lies entirely in the interior of the region t. From the last assumption it follows that we can construct4 a neighborhood (p) of G which lies entirely in the interior of. Except, of course, the points A and B. 2 Compare OSGOOD, loc. cit., p. 107. 3 These restrictions will be dropped in ~~9 and 10. 4 About any point P of C we can construct a circle (P, r) which lies entirely in At, since P is an inner point of E. Let pp be the upper limit of the values of r for which this takes place. Then pp varies continuously as P describes the curve ( (WEIERSTRASS, Werke, Vol. II, p. 204) and reaches therefore a positive minimum value po (compare E. II A, p. 19 and J. I, No. 64, Cor.). If we choose p < po the neighborhood (p) of v will lie in the interior of B. 14 CALCULUS OF VARIATIONS [Chap. I We then replace1 the curve ( by another admissible curve 6(: y=f(X), lying entirely in the neighborhood (p). The increment A y= y - y = f (,) f- (X ) which we shall denote by o, is called the total variation of y. Since ( and ( pass through A and B, we have (to)= =, (X)= 0, (4) and since 6 lies in (p), o )()| < p in (X01). (4a) The corresponding increment of the integral, A J = J — J, is called the total variationof the integral J; it may be written: aJ=JX [F ((x, y+, y'+'+ )-4F(x, y, y')adx Since ( is supposed to minimize J, we shall have provided that p has been chosen sufficiently small. For the next step in the discussion of this inequality two different methods have been proposed: a) Application of Taylor's formula: If we apply Taylor's2 formula to the integrand of A J, we obtain, in the notation of ~2, d), 1 The process of replacing C by 6 is called " a variation of the curve; " the same term is frequently applied to the curve (S itself, which is sometimes also called "the varied curve," or "a neighboring curve." 2 The conditions for the applicability of Taylor's formula are fulfilled, compare E. II A, p. 77, and J. I, No. 253. Fy,, Fyy, etc., are synonymous with F FP, F, etc. The method here used was first given by LAGRANGE. See Oeuvres, Vol. IX, p. 297. Compare also Du BOIS-REYMOND, Mathematische Annalen, Vol. XV (1879), p. 292, and PASCAL-SCEEPP, Die Variationsrechnung, p. 22. Instead of T a ylor's formula with the remainder-term, WEIERSTRASS (Lece tures), KNESER (Lehrbuch der Variationsrechnung, ~8), and C. JORDAN (Cours d'Analyse, Vol.III, No. 350), who suppose F(x, y,p) to be analytic, use Taylor's expansion into an infinite series. Here, however, the question of integration by terms should be considered. ~4] FIRST VARIATION 15 A J -f (F (, + ~ o)'?) da o.. + f (F 2 + 2 Fy, 'W + 'F,, W2) dx o where the arguments of Fy and Fy, are x, y, y', those of Fiyy, 1Fyy,, F, y+0w, y'+-9 o', e being a quantity between 0 and 1. We now consider, with LAGRANGE,, special2 variations of the form o, = e ~, (5) where D is a function of x of class D' which vanishes for x0= and x —xl, and e a constant whose absolute value is taken so small that (4a) is satisfied. Then A J takes the form' r I AJ=e e \(Fy + F) )dx+ (,), (6) where (e) denotes an infinitesimal for LE =0. Hence we infer that we must have xi p.(Eve + Fv - ) dx = o (7) for all functions V of class D' which vanish at x0 and xl; 1 Oeuvres, Vol. IX, p. 298. 2For the purpose of deriving necessary conditions, we may specialize the variations as much as convenient. It will be different when we come to sufficient conditions (compare ~17). 3 Proof: We suppose first that a' (x) is continuous in (xoxl) and denote by tA and /' the maxima of I r (x) I and I,'(x) I in (xoxl), and by q a quantity greater than the maximum of If'(x) I in (xoxl). Having once chosen the function - (x), we can then determine a positive quantity 8 such that the point (x, y) lies in the neighborhood (p) of ( and that - q < y' < q for every x in (xoxl), provided that | e < S. On the other hand, the three functions I Fyl, Iv FY i, II Fy, remain, in this domain, below a finite fixed quantity G. Hence, by the mean-value theorem, |~(7F { 2+2Fy WW'+F Y,'2)dx e2 G (2+2,,'+ 2) (X -X) - If r' (x) is not continuous in (X0Xl), apply the same reasoning to the integrals taken between two successive corners of 8. CALCULUS OF VARIATIONS' [Chap. I for otherwise we could make A J negative as well as positive by giving e once negative and once positive sufficiently small values. b) Differentiation withj respect to e: The same result (7) as well as formula (6) can be obtained by the remark, dueto LAGRANGE,1 that by the substitution of er for a, the integral J becomes a function of e, say J (e), which must have a minimum for e — 0. Hence we must have2 J'(0) 0. If r (x) is of class C' in (xoxl), it follows from our assumptions concerning the function F and the curve ( that aF(x, y (x) + e(x), y'(x) + /' (x)) De is a continuous function of x and e in the domain, x0 x X l, I e e0, e0 being a sufficiently small positive quantity, and therefore the ordinary rule' for the differentiation of a definite integral with respect to a parameter may be applied. Hence we obtain dJ(c) - ~ (FYy+ ~,~') This proves (7) and at the same time (6), since by the definition of the derivative, A J=J (e)-J(0)= - (= ' (0) + (E)) If? (x) is of class D', decompose the integral J in the manner described in ~3, c), and then proceed as above. c) The symbol.: We now make use of the following permanent notation introduced by LAGRANGE' (1760). Let c(x, y, y' y",. -) be a function of x, y and some of the derivatives of y, whose partial derivatives with respect 1 Oeuvres, Vol. X, p. 400. This method has been adopted by LINDELOF-MOIGNO, DIENGER, and OSGOOD. 2 Moreover J"(0) must be - 0. This condition will be discussed in chap. ii. 3 Compare E. iI A, p. 102; J. I, No. 83. 4 euvres, Vol. I, p. 336. Compare also J. III, No. 348. ~4] FIRST VARIATION 17 to y, y', y",' * up to the nth order exist and are continuous in a certain domain. Then if we replace y by y =- yer, and accordingly y' by '-= y' -+ e7', etc., we can expand the function - = (x +, Y+E, y'+, ) according to powers of e and obtain an expansion of the form E E2 E - = + I+4 +... + + (e), where (e) denotes as usual an infinitesimal, and 01 = - y + / + y"A+, + 2 - /,!2, X + 1,~,, v' 2'+ + 2,y, 1'7+.+ The quantities eo1, e2 2, *. are called the first, second, *~ variation of < and are denoted by sb, 382,... respectively. It is easily seen that ak4 - a (8/i-14) Again, if p does not contain e, 8ko may be defined by ak = =0. Similarly, 8kJ is defined as the term of order k, multiplied by k!, in the expansion of xo -F (x, y + 1Eq, y' + E') dx according to powers of e, the possibility of this expansion up to terms of order k being, of course, presupposed. Accordingly J/ dk jJ k de-J 18 CALCULUS OF VARIATIONS [Chap. I It follows immediately1 that Sk= kFdx. So In particular 8J=e (Fv+F,,)q'). (8) 0 We may therefore formulate the result reached above as follows: For an extremilnm it is necessary that the first variation of the integral J shall vanish for all admissible variations of the function y. d) More general type of variations: For many investigations it is necessary to extend the important formula (6) to variations of the following more general type:2 = ) (, ), (5a) where w (x, e) is a function of x and e which vanishes identically for e=0. We suppose that c (x, e) together with the partial derivatives wx, we, a'xE are continuous in the domain X0 X - XI, |E Ie E0, eO being a sufficiently small positive quantity. Moreover, in the case when both end-points are fixed oa (x0, e) =0 and o (x,, e) = for every [ e f eo. If we denote we (x, 0) by (x), formula (6) holds also for variations of type (5a). This can be most easily proved by the method explained under b). For the function x J(e) = F (x, y (x) + wo(x, ), y'(x)+ o (x, e)) dx Xo must have a minimum for = 0, and therefore J' (0) = O. From the above assumptions concerning w (x, e) it follows that differentiation ander the sign is allowed and that cxe exists and is equal3 to ax. 1Provided always that the limits are fixed and that the ordinary rules for the differentiation of a definite integral with respect to a parameter are applicable. 2 Such variations were already considered by LAGRANGE, Oeuvres, Vol. X, p. 400. 3 Compare E. II A, p. 73. ~1] FIRST VARIATION 19 Hence we obtain1 also in the present case J' (0) = (F, + F,') ax, which leads immediately to (6). For variations of type (5a) the definition of the symbol a must be modified. In order to cover also the case of variable end-points, we suppose that x0 and xl are functions of e which reduce to x, and x1 respectively, for e 0. Putting then as before = () + w (x, E), Y = y'(x) + x (X, E), we define2 k X= 8"J F (x,y, )dx.e and similarly if 0 is a function of x, y, y',.. and Xo, xi, E=0 ak, (x Y, Y, *, Xo, ) k The definition of the symbol a given under b) is a special case of this general definition. The method of differentiation with respect to e, especially when combined with the consideration of variations of type (5a), seems to reduce the problem of the Calculus of Variations to a problem of the theory of ordinary maxima and minima; only apparently, however; for, as will be seen later, the method furnishes only necessary 1 For variations of the special type (5) equation (6) may also be written AJ= (FyTo +Fy, ow') dx+ e (e). (6a) 0 This formula remains true for variations of the more general type (5a). For from the properties of o (x, e) it follows that the quotients (w (x, e)-o(x, O))/ and ((x, E) - O(x, O))/e approach for L e =0 their respective limits E (x, 0) and wxe (x, 0) uniformly for all values of x in the interval (xoyl) (compare E. II A, pp. 18, 49, 52, 65; J. I, Nos. 62, 78 and P., Nos. 45, 100). Hence it follows that w(Fyc+ Fy,W') dx=e (F? r+Fy, ') dxc r ), 0O V XO which proves the above statement. 2 Always under the assumption that all the derivatives occurring in the process exist and are continuous. 20 CALCULUS OF VARIATIONS [Chap. I conditions, but is inadequate for the discussion of sufficient conditions, whereas the method based upon Taylor's formula, though less elegant, furnishes not only necessary but also sufficient conditions, at least for a so-called weak minimum (compare ~ 17, b). e) Transfomatior on of the first variation by integratio 1 by parts: For the further discussion of equation (7) it is customary to integrate the second term of 8J by parts: - i 1 + ( -- - ) F x. (9) -am0,"co Since v vanishes at x0 and xl, this leads to the result that for an extremum it is necessary that X - (F - F,) dx 0 (10) for all functions X of class D' which vanish at x0 and xl. The integration by parts presupposes, however, that not only y' but also y" exists and is continuous in (x0xZ), and for the present we shall make this further restricting assumption1 concerning the minimizing curve. ~5. THE FUNDAMENTAL LEMMA AND EULER'S EQUATION To derive further conclusions from the last equation we need the following theorem, which is known as the Fundamental Lemma of the Calculus of Variations: If M is a finction of x which is continuous in (x0x1), and if I M dx =0 (11) a0 1 The necessity of this assumption was first emphasized by Du BoIs-REYMOND in the paper referred to on p. 9). If y" does not exist, the existence of - Fy, becomes doubtful. The restriction will be dropped in ~6. Discontinuities of v' of the kind here admitted do not interfere with the above results (9) and (10), since v itself is continuous. For the principles involved in the integration by parts, compare E. II A, p. 99, and J. I, Nos. 81, 84. ~5] FIRST VARIATION 21 for all functions r which vanish at Xo and xl and which admit a continuous derivative in (xxl), then n (xox. M 0 (12) m12 (XQOX^ For suppose M(x') O, say >0, at a point x' of the interval (xoxl); then we can, on account' of the continuity of M, assign a subinterval ( oo0) of (x0xl) containing x' and such that M >0 throughout (Io:l). Now choose 7 0 outside of (&08o) and V = (x - 0o)(x - I1)2 in (fofi); this function admits a continuous derivative in (xoxl), vanishes at x0 and xl. and nevertheless makes X1. Mdx > 0, '0 contrary to the hypothesis (11); therefore M (x') 0 is impossible.2 The conditions of this lemma are fulfilled for equation (10); for, since we suppose y" to exist and to be continuous in (xoXl), the function M d is continuous3 in (xoxl). — F 1Compare P., No. 17. 2 This proof is due to Du BOIS-REYMOND (Mathematische Annalern, Vol. XV (1879), pp. 297, 300). In the same paper he proves that the conclusion M= 0 remains valid even if the equation (11) is known to hold only: 1. For all functions q having continuous derivatives up to the nth order, inclusive: proceed as above and choose, for (40,1), X = (X- -0)n (t1- )i 1 - 2. For all functions having all their derivatives continuous. H. A. SCHWARZ goes still farther and proves the conclusion valid if the 7?'s are supposed regular in (x0x1), i. e., developable into ordinary power series t (x - x') in the vicinity of every point x' of the interval (x0ox) Lectures on the Calculus of Variations, Berlin, 1898-99, unpublished.) On the other hand, the proof given in most text-books, in which 71 = (X - X) (x1- -)M is used, assumes that (11) holds for all continuous functions 7r vanishing at x0, x1, or else, if the assumptions of the lemma concerning -7 are not changed, that M' exists and is continuous. This last assumption would, in our case, imply that y"' exists and is continuous. Also HEINE'S proof (Mathenmatische Annalen, Vol. II (1870), p. 189) could be applied to our case only after further restricting assumptions concerning y. 3 Compare J. I, No. 60, and P., No. 99. 22 CALCULUS OF VARIATIONS [Chap. I Hence we obtain the first necessary condition for an extremum: FUNDAMENTAL THEOREM I:1 Every function y which mminimizes or mnaximizes the integral X $1 J = F(x, y, y')dx must satisfy the diffeiietial equation Y-d F,=0. (I) This differential equation was first discovered by EULER2 in 1744, and will be referred to as Euler's (dlfferential) equation.. X6. DU BOIS-REYMOND'S AND HILBERT'S PROOFS OF EULER S EQUATION The preceding method, which was based upon the integration by parts of ~ 4, furnishes only those solutions of our problem which admit a continuous second derivative. The question arises: Do there exist any other solutions and if so, how can we find them? In order to answer this question, we return to the equation 0J —0 in the original form (7) and, with Du BOIS-REYMOND and HILBERT, integrate the first, instead of the second, term by parts. Since r- vanishes at both end-points, we get: v (FI J,- Fdx)dx=. (13) iO 0XO 1 We have pro red this theorem only for functions y having a continuous second derivative. The extension to functions having only a continuous first derivative follows in ~6, to functions of class D in ~9. 2EULER, Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, chap. ii, art. 21; in STXCKEL'S translation in OSTWALD'S Klassiker der exakten Wissenschaften, No. 46, p. 54. 3 KNESE, HILBERT, and others call it "Lagrange's Equation." LAGRANGE himself attributes it to EULER. See Oeuvres de Lagrange, Vol. X, p. 397: " cette equation est celle qu'EULER a trouv6e le premier." ~6] FIRST VARIATION 23 This integration by parts is legitimate, even if y" should not exist, since it presupposes only the continuity1 of Fy and X'. We are thus led to the problem: If N(x) be continuous in (xoxi), and if r'Ndx = O (14) JX for all functions q of class C' which vanish at Xo and x1, what follows with respect to N? The answer is that N must be constant in (xoXi). a) Du BOIs-REYMOND2 reaches this result by the following device: Let v be any function which is continuous in (xoX1) and satisfies the condition tdx = 0 (15) Xo then the function Xe is of class C' in (XsXi) and vanishes for x=Xo and x=x,, and therefore, according to our hypothesis, satisfies (14), that is, f INdx O =. (16) o Thus it follows from our hypothesis that every continuous function which satisfies (15) necessarily satisfies (16) also. Now let Ai be any continuous function of x; and c the following constant: 1 rX1 c — I dx X1 - Xo. X then the function ~-~i-c is continuous and satisfies (15), hence it must satisfy also (16), therefore 1 The continuity of F5 follows from the continuity (compare the beginning of ~4) of y' and from our assumption (B) concerning F; and -/' may be supposed continuous, since,(9) must hold for all functions -q of class D' which vanish at x0 and xl, and therefore afortiori for all functions v of class C' which vanish at x0 and x1. 2Loc. cit., p. 313. 24 CALCULUS OF VARIATIONS [Chap. I f Ndx = f gl(N-X)d=~, (17) if we denote by x the constant X = Ndx/(x - Xo) 0X But from (17) it follows by the Fundamental Lemma that1 N-X, i. e., constant, Q. E. D. b) Another, more direct, proof has been given by HILBERT 2 in his lectures (summer 1899). He selects arbitrarily four values, a, P, a', 3' satisfying the inequalities Xo < a< / < a' < 1' < X1, and then builds up a function3 1 of class C' which is equal to zero in (xoa); increases from 0 to a positive value k as x increases from a o a 13 a ' a';, to3; remains constant, = k in (/a'); FIG~. 3 decreases from k to 0 as x increases from a' to 3', and finally is equal to zero in (P'xi): Substituting this function in (14), we obtain 'Ndx +- 'Ndx=O; I' being positive in the first, and negative in the second, integral. we can apply to both the first mean-value theorem4 which furnishes k N (a+ f (/3-a)) -N (at+ 9t(/t-A)) =O, where 0< < 1 and 0< 0'< 1. Finally, let P and 3' approach a and a' respectively; then it follows, since N is continuous, that 1 This result is a special case of the isoperimetric modification of the Fundamental Lemma, see below chap. vi. 2 See WHITTEMORE, Annals of Mathematics (2), Vol. II (1901), p. 132. 3 Nothing more than the existence of such a function- which is apriori clear -is needed for the proof: HILBERT gives a simple example, see WHITTEMORE'S presentation. 4 Compare E. II A, p. 97; J. I, No. 49; and P., No. 191, IV. ~6] FIRST VARIATION 25 N(a)=N(a'), i. e., N is constant in (xoXl). c) Applying this lemma to (13) we get Fy-. F, dx =, F, d a constant; or F,= + Fdx. (18) The right-hand side of this equation is differentiable and its derivative is F.; hence the same must be true of the left-hand side, i. e., the function Vy, (,x y (x), y' (x))- F,, [x] is differentiable in (xoxi) and dx F F Thus we find the important corollary to Theorem I that every solution of our problem with continuous first derivative -not only those admitting a second derivative —must satisfy Euler's equation. From the fact that F?', is differentiable follows the existence2 of the second derivative y" for all values of x for which FI,,' (x, (x),Y ' (x)) o. (19) For, if we put y (x + h)- y (x) =, y' (x + h) - y' (x) = I, then, since the theorem on total differentials3 is applicable under our assumptions, and since y' is continuous, we have 1 HILBERT'S proof can easily be extended to the case where N, while finite in (xoxl), has a finite number of discontinuities. For, if a and a' are points of continuity, we can always choose / and /' so near to a and a' respectively that N is continuous in (a/3) and (a'/3'); it follows then as above that N(a) =N (a'), i. e., under the present assumptions N has the SAME constant value in all points of continuity. Hence it follows further that in a point of discontinuity, c: N (c-0) = N(c+0) 2First pointed out and emphasized by HILBERT in his lectures; see WHITTEMORE, loc. cit. 3 Compare E. II A, pp. 71, 73; J. I, Nos. 86, 127; and P., No. 105. CALCULUS OF VARIATIONS [Chap. I F [ + h] F.,, [] = (FJ + a) + k (I F, + B) + a (,y, + h h h where a,,, a approach zero as lb approaches zero. Hence it follows that if (19) is satisfied, L, i.e., Lh=0 h exists, and that,, F, - F - y' F,, gtY -j- ^ WJ 1l; (20) moreover, (20) shows that y" is continuous in (xox,). ~7. MISCELLANEOUS REMARKS CONCERNING THE INTEGRATION OF EULER'S EQUATION a) Eutler's differential equation (I) is of the second order,l as can be seen from the developed form Fy - F1 ' -. y'Fy, - y"Fyy, 0; (21) its general solution contains, therefore, two arbitrary constants, y f(x, a, f). (22) The constants a, / have to be determined2 by the condition that the curve is to pass through the two points A and B: y0o =f(x, a,/3) (23) - f(X, a, ). (23 Every solution of Euler's equation (curve as well as 1Unless F?,J/Y (x, y, y') should be identically zero. In this case Euler's differential equation degenerates either into a finite equation or into the identity: 0 = 0. but never into a differential equation of thefirst order. For if F i,,=0, F must be of the form: L(x, y)- M(x, ) yy' and (21) reduces to: Ly - = 0. See also below, under d). If Euler's differential equation degenerates'into a finite equation, it is in general impossible to satisfy the initial conditions when the end-points are fixed. Also in the general case when F contains higher derivatives, Euler's differential equation can never degenerate into a differential equation of odd order; compare FROBENIUS, Journal fi r Mathemzatik, Vol. LXXXV (1878), p. 206, and HIRSGc, Mlathenmatische Annalen, Vol. XLIX (1897), p. 50. 2 This Determination may be impossible; in this case there exists no solution of the problem which is of class C' and lies in the interior of B. ~7] FIRST VARIATION 27 function) is called, according to KNESER, an extrewmal; there is then a double infinitude of extremals in the plane. In the special case when F does not contain x explicitly, a first integral of (I) can be found immediately.l For, if F does not contain x explicitly, we have d (F- ':F) y'(Fy-d FY) dx - Y and therefore every solution of (I) also satisfies F- y'F,, = const. (24) Vice versa, every solution of (24), except y — const., also satisfies (I). b) EXAMPLE I (see p. 1): F= y 1/1 -2. Hence Fy = I/i +Y2 F -- YY7 771 Y, 3 and Euler's equation becomes: 1 Y - 2 d Y. (I) dx i/ 1 _ y'2 or, after performing the differentiation, +dy 2 d2y dy_ By putting c- = p, the integration of this differential equation is reduced to two successive quadratures, and the general integral is easily found to be x-P y = a osh. a The extremals are therefore catenaries with the x-axisfor directrix. Since F does not contain x, a first integral could have been obtained directly by the corollary (24); F-y = y/ =Y a -' -y' —l+y 12 1 Noticed already by ETJLER, loc. cit., p. 56, in STACKEL'S translation. 28 CALCULUS OF VARIATIONS [Chap. I If a 4 0, this leads to the same result as above; for a=0 we obtain y=0, which, however, though a solution of (24), is not a solution of Euler's equation. The general solution of (I) being found, the next step would be_ so to determine the two constants of integration that the catenary passes through the two given points.' c) Throitgh a given point a, b in the interior of the region2 T one and but one extremal of class C' can' be (drtawn in a given direction of amplitude3 o (, provided that Fa, (a, b, b') 0 (25) where b'- tan 0o. For, if we solve (I) with respect to y", we obtain for y" a function of x, y, y' which, according to our assumptions (B), is continuous and has continuous partial derivatives with respect to y, y' at all points of the domain2 C which satisfy (25). Hence the statement follows from CAUCHY'S general existence theorem4 for differential equations. 1 For this interesting problem we refer to: LINDELOF-MOIGNO, loc. cit., No. 103; DIENGER, loc. cit., pp. 15-19; TODHHUNTER, Researches in the Calculus of Variations, pp. 55-58; CARLL, A Treatise on the Calculus of Variations, Nos. 60, 61. For SCHWARZ'S solution see HANCOCK, "On the Number of Catenaries through Two Fixed Points." Annals of Mathematics (1), Vol. X (1896), pp. 159-174. 2 See 3, c). See ~2,.i). 4"Suppose the functions fi (x, Y,, YI *', y,,) and their first partial derivatives with respect to yl, Y2*,,,to be continuous in the domain I x -a [ p, ] J-bl r, *, yn- b, [ r; let M be the maximum of the absolute values of the functions fi in this domain, and let I denote the smaller of the two quantities p and r/M. Then there exists one, and but one, system of functions yl (x), Y2 (),- * *, y,, (X) which in the interval I x - a i < 1 are continuous and differentiable, satisfy the differential equations dyi - - =fi(x Yl,Y2,. -Y); (i=1, 2, * * -,n) and the inequalities [ yi (x) - bi I r, and take for x = a the values Yl = bl, Y2 = b2 ' I ', Y- = — b." Compare E. II A, pp. 193 and 199, and J. I[I, Nos. 77-80; also PIC.ARD, Traie' d'Analyse, Vol. II, chap. xi. In order to apply the theorem in the present case, replace (21) by the equivalent system. dy dy'F — d — =', -- (Fy -- Fyx — yFy F, y, dx~y dx~ ' *y"yv ~7] FIRST VARIATION 29 If, therefore, Fy,, (a, b, p) # 0 for every finite value of p, one extremal can be drawn from (a, b) in every direction, except the direction of the y-axis. A problem for which F.,,(x, y,p) A 0 at every point (x, y) of the region T for every finite value of p, is called, according to HILBERT, a regular problem. d) We consider next the exceptional case in which Eiler's differential equation degenerates into an identity. Suppose the left-hand side of (21) vanishes for every system of values x, y, y', y". Then, since y' does not occur in the three first terms, it follows that the coefficient of y" must vanish identically, so that we must have separately FY,. 0, F1-FY- y'F,^ 0 for every x, y, y'. From the first identity it follows that F must be an integral linear function of y', say F (x, y,y')=M (, y)+ 1(x, y) y'. Substituting this value in the second identity, we get Miy = NN the well-known integrability condition for the differential expression Mdx + Ndy. Hence we infer: If M and N and their first partial derivatives are single-valued and continuous in a simply-connected region * of the x, y-plane, then there exists' a function V (x, y), single-valued and of class C' in * and such that = M, N, and therefore d F (x, y, y') = Vt+ Vy' = v (x, y) Hence if (S: y =f(x) be any curve of class C' drawn in 6 between the points A (xo, yo) and B (xl, yl) our integral Js has the value 1See PICARD, Traite d'Analyse, 2d ed., Vol. I, p. 93. 30 CALCULUS OF VARIATIONS [Chap. I Xl J(w-J - 'F (x, y, y' ) d V (x1, Y li) -T V(x0, yo), o and is therefore independent of the path of integration (S and depends only upon the position of the two end-points. On account of the continuity of V (x, y), the result remains true for curves ( with a finite number of corners, as is at once seen by decomposing the integral J in the usual manner.1 Vice versa. If the value of the integral Je is independent of the path of integration C as long as ( remains in the interior of a region F contained in iE, then the function F must be of the form M(x, y)+VN(x, y)y', where M~= N, for every point (x, y) in the interior of - for which Xo E x X x1. For let.(X2, y2) be any inner point of whose abscissa x2 lies between X0 and xi and y', y2' two arbitrarily prescribed values; then we can always draw in ~ a curve S: y = f(x), of class C" which passes through (x0, yo), (x1, y1), (x2, y2), and for which f'(x2)=y2, f" (X2) = 2' According to our hypothesis, AJ must vanish for every admissible variation of S, whence we infer by the method of ~~ 4, 5 that y-f(x) must satisfy Euler's differential equation. The lefthand side of the latter must therefore vanish for the arbitrary system of values x = 22, y = y, y = y " ', which proves the above statement. We thus reach the result: In order that the value of the integral J F(x, y, y')dx 0 may be independent of the path of integration it is necessary and sufficient3 that Euler's differential equation degenerate into an identity. It is clear that in this case there exists no proper1' extremum of the integral J. e) We conclude these remarks by considering briefly the inverse problem: Given a doubly infinite system of curves (functions) y -f(x, a, ), 1 Compare p. 12. 2 Compare J. III, Nos. 362, 363, and KNESER, Lehrbuch, ~51. 3 Sufficient only if the region X is simply-connected. 4 Compare ~3, b). ~ 71 FIRST VARIATION 81 to deterntine a fm tnction F (x, y, y') so that the given systemn of curves shall be the extremals for the integral J j5F (x,Yy') dx This problem has always an infinitude of solutions which can be obtained by quadratures.1 For if y" zG(x yy') (26) is the'differential equation of the second order' whose general solution is the given function y f (x, a, 3,) (with a, P as constants of integration), then we must so determine the function F (x, y, y') that (26) becomes identical with Euler's differential equation for F, i. e., according to (21) F~11 - Fy1x - F111,y y'- GFy15y. (27) If we differentiate (27) with respect to y', we get for M - F11y, a linear partial differential equation of the first order, vii., + y + G + Gly, M - 0. (28) DX ay ay' if a = 6(xi, zJt J ) (x, 2z') is the solution of the two equations y f (xi a, 8) y y' fX (Xi a, p with respect to a and P, and if further 0 Gfa (x, f (x, P), f ( a,!))dx 0 (xi a, e)= and X (Xi Y, Y') 0 (x, ( (Xi Y-, 0', 1P(xi Y, Y,)) I 1DARBOUX, I'horie des surfaces, Vol.III, Nos. 604,605. For the analogous problem in the more general case when F contains higher derivatives, compare HIRSCH, Matheiaatische Annalen, Vol. XLIX (1897), p. 49. 2Obtained by eliminating a, P between the three equations y=f(x,-, ) y'=fX (x, a, p), y=fXX(X, a, P) eompare, for inst., J. I, No. 166. 32 CALCULUS OF VARIATIONS [Chap. I the general integral of (28) is found to be, according to the general theory1 of linear partial differential equations of the first order, Mx = - ( (, y, y ),, (X, yJ )), where ~ is an arbitrary function of P and '. After the function M has been found, F is obtained by two successive quadratures from the differential equation 32F ay, =:(x, y,y') ayf2I Y Finally the two constants of integration x, t/ (which are functions of x and y), introduced by the latter process, must be so determined that F satisfies the original partial differential equation (27) from which (28) was derived by differentiation. EXAMPLE: To determine all functions F for which the extremals are straight lines y = ax +. The differential equation (26) becomes, in this case, y -=0 Accordingly, we obtain - q= y', - xy' X = const. Hence M= (y', y - xy/), and therefore (y'-t) (t, y- xt) dt + y'X (x, ) + (x, y) The condition for X and / becomes in this case ax ay The most general expression for and is therefore The most general expression for X and ~ is therefore av av A- = y, x = a where v is an arbitrary function of x and y. 1 Compare, for inst., J. III, No. 242. 2 Compare DARBOUX, loc. cit., No. 606. ~8] FIRST VARIATION 33 ~8. WEIERSTRASS'S LEMMA AND THE E-FUNCTION Before proceeding to the consideration of so-called discontinuous solutions, we must derive a lemma, due to WEIERSTRASS,1 which is of fundamental importance for many investigations in the Calculus of Variations. Suppose there are given, in the region E, an extremal ( of class' C": — f(x), and a curve d of class C' y-f(x), meeting C at a point3 2: (x2, Y2). Besides there is given a point 0: (x0, Yo) on E, before 2, that is, Xo< X2. Let 3 be that point of ( whose abscissa is xA- -h, h being a positive infinitesimal, and select arbitrarily a function r of class C' satisfying the conditions, o - 77 (XO) 0,- o2, - r (x2) = 0. Then we can so deter- 6 mine e that the curve 7; Y - + = FIG. 4 which necessarily passes through the point 0, also passes through the point 3. For this purpose we have to' solve the equation f (x2 + h) + r7 (x + h) = f(x2 + h) with respect to e. Since f (x2)=f (x2), we have f(xZ + h) -f(x2 + h) = (y2 -' y) h +h (h), where y -f' (X2), 2 =f' (x2) and (th) is an infinitesimal for L h - 0. Hence we obtain e - y[ + (h)] It is proposed to compute the difference A J = J3 - (Jo2 + J23), lThe lemma here given is a modification of the corresponding lemma given by WEIERSTRASS in his lectures (1879) for the case of parameter-representation; see ~28. 2This assumption must be made on account of the integration by parts which occurs below; compare ~ 4. 3For the notation compare ~2, e). 34 34 ~~CALCULU'S OF VARIATIONS [hp [Chap. I the integrals J, J, J being taken along the curves CS, -, 6 respectively, from the point represented by the first index to the point represented by the second. AJ may be written /~) x x2h JJ(F -F) dx+J (F -F)dx, whereF,F,ForF[x],F[x],FP[x] stand forFxyx,(x).F(x, -(x), -'(x)), F(x, ~(x)., ~'(x)) respectively. The first integral, treated by the method of ~4, becomes, since Gs is an extremal, f2(F - F) dx- 'E72 FY [X21 + 'E (E) -h [2- Y2' )Fy1 [X2]+ (h)] To the second integral we apply the first mean-valne theoreni and obtain, on account of the continuity of FI[X] and EP[x] I f2~ - F) cdx h [F[X21 F[X21+ (h)J Collecting the terms, we reach the result JO 3-(JO 2 + '3 ) - h (j2t- y2') x2] + F[X21 -FP[X2] +(h)~ Similarly let 4 be that point of 6 whose abscissa is It)-, and determine E' so that the curve 'Y -[- +IE -q passes through 4. Then we obtain by the same process J041+ J42 -J02 -- h (92 - Y2')F.,,[X21+F[X21 -FP X2] +(h) If we pnt f or brevity X, y, p, j5being considered as four independent variables, the, preceding results may be written: S,~ 8] FIRST VARIATION 35 -3 (J - J) -ht E ( Y2 y 2, S, P) + (h) ( J04- (J42 - J52) - +h E(X2, Y2 2~ 2) A-h We shall refer to these two formulae as Weiersirass' s Lemmnia. The function E(x, y; jp, _f) defined by (29) will play a most important part in the sequel; it is called Weierstrass's E-foieCtion.1 The same results (80) hold if the curves 03 and 04 are of the more general type (Sa):?J - f (x) + - (X, E) where the function w (x, c) vanishes identically for E 0, has the continuity properties ennmerated on p. 18, and satisfies besides the conditions: w (xs, E) -0 for every E, and WI(x2, 0) # 0 For the determination of E we have, in this case, the equation: f (X2+ h)+ w (X2+ h, E)-f(X2+h) — The resulting value of E is of the same form as above. This follows from the theorenm2 on implicit functions; for if Icompare ZERMELO, Dissertation, p. 66. 2 "If f(x, y) is of class C' in the vicinity of (x5, yo) and then a positive quantity k being chosen arbitrarily but sufficiently small, another positive quantity hk can be determined such that for every x in the interval (xs- hk, xs+ hk) the equation f (x, y) = 0 has one and but one solution y between yo- kC and y(o+ k. The single-valued function y =4 (x) thus implicitly defined by the equation: f (X, Y) = 0, is of class C' in the interval (x5- h7k I XO+ h) and dy fx dx Hnece r ro=("-~o)fX (X0, Yo) where L a=O." X=XO (Compare E. II A, p. 72; J. I, No. 91; P., No. 110). If f (x, y) is regular in the vicinity of(xo, yY), also the function yq=4 (') is regii - lar in the vicinity of x0. (Compare E. II B, p. 103, and HARKNESS AND MORLEY, Jut roduction to the Theory of Analytic Functions. No. 156.) For the extension of the theorem to a system of m equations between m + n unknown quantities, see the references just given. 36 CALCULUS OF VARIATIONS [Chap. I we denote the left-hand side of the preceding equation by F (h, e), this function is of class C' in the vicinity of it 0, e 0; further: F(0, 0) 0 and finally F (0, 0) 0. Incidentally we notice here the formula _ Z2 _ rj2+h _ Jo3- J1,2= Y (F-F)dx+ F d = h [(y - y2) F [x2] + F [2] + (h)] which holds for negative as well as for positive values of h. Hence it follows that if the arc 02 of the extremal ( minimizes the integral J, the end-point 0 being fixed while the end-point 2 is movable on the curve (, then the co-ordinates of the point 2 must satisfy the condition F+( '-y')' 0. ("Condition of trcnsversality," compare the detailed treatment of the problem with variable end-points in ~23.) ~9. DISCONTINUOUS SOLUTIONS We must now free ourselves from the restriction1 imposed upon the minimizing curve at the beginning of ~4, viz., that y' should be continuous in (xoxl), and we propose to determine in this section all those solutions of our problem which present corners-so-called "discontilnuous solutions." a) In the first place, the theorem holds that also discontinulbs solutions 'must satisfy Euler's differential equation. Suppose for simplicity2 that the minimizing curve (E has only one corner C(x2, Y2) between A and B. According to ~3, c) the integral J, is then defined by rx2-0 rX1 J= fJ F(x, y,)dx+ + F(, y, y')dx, (31) 1 The assumption that the curve shall lie entirely in the in t e r i o r of the resion *1 will still be retained in this section. 2 The results can be extended at once to the case of several corners. ~93 FIRST VARIATION 37 the notation indicating that y'(x2) is defined in the first integral by y' (x2 - 0), in the second by y' (x2 + 0). The theorem in question is most easily proved by the method of partial variation, which is very useful in many investigations of the Cal- A -- --- --- culus of Variations: - We consider first such spe- c ia 11 variations A D C of type (5) as leave the ar C B unchanged and vary only A C. a x FIG. 5 To such variations all the conclusions of ~~4-6 can be applied, and it follows as before that for the interval (xo, x —0) Euler's equation must hold. The samer result follows for (x2 0, xi) from the consideration of variations which leave A C unchanged; hence it is true for the whole interval (oXl).2 b) A discontinuous solution with one corner is therefore composed of two extremals involving in general different constants of integration: y = f (x, al, /) in (Xo, x - 0) y =f(x, 2a, /2) in (X2 +0, X1) For the determination of x2 and of the constants of integration we have in the first place the initial conditions Yo =.f (Xo. al, il), Y1 = f (X a2, 2); further the condition that y is continuous at x2: f (X2, al 1A) = f(x2, a2, 8); and finally two further conditions which are furnished by the following theorem due to WEIERSTRASS and ERDMANN:3 1 Compare the remark on p. 15, footnote 2). 2 With the same understanding as in (31) concerning the meaning of y' at the corner. 3WEIERSTRASS,Lectur-es at least as early as 1877; ERDMANN, Journal fir Mathematik, Vol. LXXXII (1877), p. 21. Another demonstration has been deduced by 38 CALCULUS OF VARIATIONS [Chap. I THEOREM: At every corner of a minimizing curve the two limiting values of Fy. are equal: a2 -0 aX2+~ Ft., =- F,(; (32) and likewise x12- 312+~ F- y' F = F- y.' | * (33) To prove (32) consider a variation A G B of type (5) for which the function X is of class C' in (XoX1) and qV (x2)= 0. The integral AJ breaks up into two integrals taken between the limits (xo, x2 - ) and (x2 +0, i1) respectively. Applying to each of these the methods of ~4 we find that also in this case 7J= 0, -and further we obtain2 from (9), since (I) is satisfied: 8 J =,,) (?2) (Fl, [x2 - 0] - Fy, [xz + 01]) where Fy [x] stands again for Fy (x, f(x), f'(x)). Since SJ=0, (32) is proved. The proof of (33) follows from Weierstrass's Lemma (30) if we identify the arcs AC and CB of Fig. 5 with the arcs 02 and 21 of Fig. 4, respectively, and consider successively the variations 031 and 04231 of the arc 021. The corresponding values of the total variations AJ are given by the two equations (30), the values of y, Y2 being in the present case 2 ' (x2 - 0) = Y2; 2 = y' (X2 + 0) = Y ' Hence it follows that for an extremum it is necessary that WHITTEMORE, loc. cit., from Hilbert's proof of Euler's equation: By means of the extension of the lemma of ~6 to discontinuous functions (see p. 25, footnote 1), it can be shown that equation (18) holds with the same value of the constant X for both segments (x, x2 -0) and (x2+0, x1). Hence follows Euler's equation as well as equation (32). This method can be applied to discontinuities of a much more complex character and even to the case of an infinitude of points of discontinuity; see WHITTEMORE, loc. cit. 1For the notation compare ~2, b). 2 The integration by parts is legitimate since by the method of ~6 the existence of d-Fy, is established for each of the two segments (xo, x2 -0) and (x2+O, x1). __ ~ 9] FIRST VARIATION EB(x2,,y2; 112 YY2-0; and on account of (32) this is equivalent to (33). c) EXAMPLE' III: To minimize the integral VXI1 Here -O, F,5 = 4y" + 6y'2 + 2y', F - y'F5 -3y' 4-y'3-y'2 Hence a first integral of Euler's differential equation is 4u"3 + 6y 12 + 2y' const. therefore y -- ax +, i. e., the extremals are straight lines, and the line AB joining the two given points is a possible continuous solution. In order to obtain all discontinuous solutions with one corner, we have to find all solutions Pi, P2 of the tvo equations 4p3 + 6p2 + 2pl, -4p + G6p2 + 2pC, 1 p2 2 2LJ r i/ 4 _ 4p33 2 3p-4 - 4P3 - 3PL1 IL1 - Pi -' 2 2 P2 where p1i' (c- 0) and p,2y'(c+0) and pIp, Dividing out by PI - P2 and putting P1+P2, P+1 M P2+P2 -we get 2w ~ 3it + 1 — 0 - 3u+uw + 4w +u 0 These equations have one real solution, it -1, I= + 1, from which we obtain pi:::0 P2 or PI= P~, - 0 IA special case of the example given by EEDMANN, loc. cit., p. 24. _ 40 CALCULUS OF VARIATIONS [Chap. I Every discontinuous solution must therefore be composed of straight lines making the angles O or 37r/4 with the positive x-axis. If the slope m? = (yl- yo)/(x1- xo) of the line AB lies between 0 and - 1, there are indeed two such solutions, A C, B and A C2B with one. A s- - - - -- corner and an infinity with n 2 corners. Since Fy=2 (y'+1)2, these \. ^-\ discontinuous solutions furnish c,.- - _ B for J the value zero and therefore the absolute minimum.1 -^ — 0 -- -, dcl) In many cases the imposFIG. 6 sibility of discontinuous solutions can be inferred from the following Corollary:2 If (x9, y2) is a corner of a minimizing curve, then the functioZ. L Fy.:' (X2 y, 2p),must vanish for sonme finite value of 2p. For the function K(p) =F5-( 2, 2, P) is a continuous function of p admitting a finite derivative for all finite values of p; further, if we put y (x2- 0) p1, ' (X2 + 0) = 2, we have P] P)2, and, according to (32), + (p1) = + (p2) ~ Hence by Rolle's Theorem the derivative <'(P) = Ft,(x2, y2, p) must vanish for some value of p between P1 and p2. If therefore the problem is a "regular problem," i. e., if FJ (xy, p) * 0 for every point in the interior of I and for all finite values 1The minimum is, however, "improper" (compare ~3, b)), because in every neighborhood of A C1 B (or A C2 B) broken lines can be drawn, joining A and B, whose segments have alternately the slopes 0 and -1. For such a curve A J= 0. 2 Compare also WHITTEMORE, loc. cit., p. 136. ~10] FIRST VARIATION 41 of j), we infer that no discontinuous solutions are possible in the interior of I. EXAMPLE I (see p. 1): F =y i 1 y, 2 is the upper half-plane (y 0).1 Here i '0 and cy no d2)i is i= 0 in the i n terior of i8, and consequently no discontinuous solutions are possible in the interior2 of i. ~10. BOUNDARY CONDITIONS In all the preceding developments it was assumed3 that the minimizing curve should lie entirely in the interior of the region i. But there may also exist solutions of the problem as formulated in ~3 which have points in common with the boundary of i. To determine these solutions is the object of the present section. For this investigation it is convenient to make use of the idea of a point by point variation of a curve which played an important part in the earlier history of the Calculus of Variations. Between the points of the two curves 6~: y =f(x), and; y ad y + A y we may establish a one-to-one correspondence by letting two points correspond which have the same abscissa x. And we may think of the second curve as being derived from the first by a continuous deformation in which each individual point moves along its ordinate according to some law, for instance, if in y + a A y. we let a increase from 0 to 1. A point of ( whose abscissa is x', is called a point of free variation if Ay(x') may take any sufficiently small value; otherwise, a point of unfree variation. For a curve G which lies entirely in the interior of i all points except the end-points are points of free variation,4 and this freedom was essential in the conclusions of ~~4 and 5. 1 Compare ~1, c). 2 Compare the next section. 3 See the beginning of ~ 4. sIn our formulation of the problem, ~3. 42 CALCULUS OF VARIATIONS [Chap. I This is not true for a curve which has points in common with the boundary. For simplicity let us suppose that the boundary of A contains an arc ( representable in the form = f(x), f(x) being of class C". In order to fix the ideas suppose that I lies above (. Then if d has a point P in common with (, the variation of P is unfree and restricted by the condition A yO. (34) Suppose the minimizing curve 0231 has the segment 23 in corn1 mon with the boundary. Then the method of partial varia-;i,tion applied to 02 and to 31 shows that these two arcs muzst be extremals.:f!.....7X/ Consider next a variation of type (5):-...'* which leaves 02 and 31 unchanged and varies only 23. Since Ay- ev must be 0, X cannot change sign and if we choose n 0 then e must be taken positive; hence we can no longer infer from that J =0, but only that // (6) 3SJO. After the integration by parts of ~4 we obtain therefore F-X3xFd)dO (35) for all functions n of class D' which vanish at x2 and x3 and satisfy besides the condition V,0 The lemma of ~ 5, slightly modified, leads in the present case to the 1 Moreover at the end-points 2 and 3 the following condition must be satisfied: E(x2,Y2; Y2', 2')=0; E(x3,y3; y3', u3')=0 The proof follows easily from We i e r s t r a s s's Lemma (see Fig. 7). Compare also the treatment of the problem in parameter-representation, ~29. The question of sufficient conditionsfor one-sided variations has recently been considered by BLISS in a paper read before the Chicago section of the American Mathematical Society. He finds that for a so-called regular problem (~7, c) the arc 23 of the curve Z furnishes a ~10] FIRST VARIATION 43 Theoremn:' If the minimizing curve has a segment1 23 in common with the boundary of Ag, then along this segment the following condition must be satisfied., dx F- d - -,, F O, if i lies above 23, (36a) Fy - d F O,,. 0, if 1 lies below 23. (3b) smaller value for the integral J than any other curve of class D' joining the two points 2 and 3, lying in a certain neighborhood of the arc 23 and satisfying the condition A y - 0, provided that the condition F -d FY, >0 dx is fulfilled along the arc 23. The proof is based upon the construction of a "field" (see ~~ 19,20, 21) of extremals each one of which is tangent to the curve Z and lies entirely on one side of i. 1 Of the properties specified above. CHAPTER II THE SECOND VARIATION ~11. LEGENDRE'S CONDITION THE integration of Euler's differential equation and the subsequent determination of the constants of integration' yield in general a certain number2 of curves 6 as the only possible solutions of our problem; that is, if there exist at all curves which minimize the integral J, they must be contained among these curves. We have now to examine each one of these curves separately and to decide whether it actually furnishes a minimum or not. We confine ourselves in this investigation to curves which lie entirely in the interior of the region I and have no corners. a) Generalities concerning the second variation. We suppose then we have found an extremal C0o = fo() f(, x0 x x (1) of class C' which passes through the two points A and B, and which lies entirely in the interior of the region TR. Then we replace, as in ~4, the curve 0o by a neighboring curve and apply to the increment AJ Taylor's formula,3 stopping, 1By the initial conditions (23), the corner conditions (32) and (33), and. the boundary conditions. 2 The number may be infinite (see Example III, p. 40); but it may also be impossible so to determine the constants as to satisfy the conditions imposed upon them; this happens, for instance, in Example I for certain positions of the two given points; see the references given on p. 28. 3 If F is an analytic function, regular in the domain T, expansion into an infinite series may be used instead. 44 ~11] SECOND VARIATION 45 however, at the terms of the third order. If we put for brevity F, (x, o,(), (x)) -= P Fyy, (x, fo(x), f(x))=Q (2) Fy,V, (x, f(x), fo(x))= R and remember that 3J= 0, since (o is an extremal, we obtain A J = (Po2 + 2Qoo'-+ Ro'2) dx + o- (wl), x, (3) (to, o')3 being a homogeneous function of dimension three of W, o'. Considering again special variations of the type wo = E and reasoning as in ~4, we obtain aJ= C2 [1 ' (P + 2Q-,'+ Rv'2) dx + (E)] (4) where (e) is again an infinitesimal. Hence we infer the theorem: For a minimum (maximum) it is necessary that the second variation be positive (negative) or zero: 82J 0 ( 0) (5) for all functions q of class D' which vanish at.x anc d x1. For according to the definition given in ~4, c), 82J = -2 4 (Pr' + 2Qrq'' + Rq ") dx. (5a) The same result can also be obtained by the method of differentiation with respect to e, explained in ~4, b); see p. 16, footnote 2. From our assumptions concerning the functions F(x, y, p) and fo(x) it follows1 that the three functions P, Q, R are continuous in the interval (Xoxl). We suppose in the sequel that they are not all three identically zero in (xo0x). 1 Compare J. I, No. 60, and P., No. 99. CALCULUS OF VARIATIONS [Chap. I] b)) Legendcre's condition. For the discussion of the sign of the second variation, LEGENDRE1 uses the following artifice: He addsto the second variation the integral E2 4 (2rvJ'w + r2w) dx, where w is an arbitrary function of x of class C' in (xo'i). This integral is equal to zero;2 for it is equal to E2 - e(V)dx=2 2 t dx * L o and y vanishes at x0 and x1. He thus obtains 82J in the form VJ - 2 e [(P + t') rq + 2 (Q + w) A.' + KR ~] dx. And now he determines the arbitrary function w by the condition that the discriminant of the quadratic form in 7, a' under the integral shall vanish, i. e., (Q + )-R(P+w') =O. (6) This reduces 82J to the form 82J=-E2 R + ( + )dx (7) from which he infers that R must not change sign in (x0x) and that 82J has then always the same sign as R. These conclusions are, however, open to objections. For, as LAGRANGE' had already remarked, Legendre's transformation tacitly presupposes that the differential equation ILEGENDRE: "M6moire sur la manibre de distinguer les maxima des minima dans le calcul des variations," Memoires de l'Academie des Sciences, 1786; in STXCKEL'S translation in OSTWALD'S Klassiker der exacten Wissenschaften, No. 47, p. 59. 2 This holds true also when -q has discontinuities of the kind which we have admitted (~3, c)); compare p. 12, footnote 5), and remember that T and wv are continuous in (x01X). 3 In 1797; see Oeuvres, Vol. IX, p. 303. ~11] SECOND VARIATION 47 (6) has an integral which is finite and continuous in the interval (xox0), and that R does not vanish in (xoxl). Nevertheless, by a slight modification' of the reasoning, the first part of Legendre's conclusion can be rigorously proved, i. e., the FUNDAMENTAL THEOREM II: For a minimum (maximum) it is necessary that R (x) F,Y, (x, fo (x), /o (x)) 0 ( 0) in (ox ). (II) For, suppose R(c) < 0 for some value c in (xox0); then we can assign a subinterval (0 o) of (xox0) for which the following two conditions are simultaneously fulfilled: 1. R(x) <0 throughout (o0l); 2. There exists a particular integral w of (6) which is of class C' in (fo) For, since R(x) is continuous in (xx1) and R(c)< 0, we can determine a vicinity (c 8, c + 8) of c in which R (x) < 0. Hence it follows that if we write the differential equation (6) in the form div (_ + _ _2 da = P + + (6a) dx RI the right-hand side, considered as a function of x and w, is continuous and has a continuous partial derivative with respect to w in the vicinity of the point x=c, w -w0o, zw being an arbitrary initial value for w. Hence there exists, according to CAUCHY'S existence theorem,2 an integral of (6) which takes for x= c the value w =- w, and which is of class C' in a certain vicinity (c - ', c -+8') of c. The interval (o:0) in question is the smaller of the two intervals (c- 8, c + 8) and (c — 8', c + '). This point being established, we choose for V a function which is identically zero outside of (&01), and equal to iThe proof in the text follows WEIERSTRASS'S exposition, Lectures, 1879. 2 Compare p. 28, footnote 4. 48 CALCULUS OF VARIATIONS [Chap. II (x-to:) (x- I) in (~o0i). The function D thus defined furnishes an admissible variation of the curve C0, since it is of 1B9~~ ~~class D' in (x0Xl), and vanishes _,_ ^-^ __, at x0 and x1. a. ' f. I' For this particular function FIG. 8 FIG.:, 6 J becomes 82J = 2 - (Pv2 + 2Q' q- + R'q2) dx To this integral Legendre's transformation is applicable. Accordingly 82J -E2f R ' +- V 2. The function r'+ Q v is certainly not identically zero throughout (0o1)); for it is different from zero for x - 0 and Hence if R (c) were negative, a variation of (0 could be found for which 2J < 0, which is impossible if 0o minimizes the integral J. Therefore R (x) 0 in (x0X1), Q. E. D. Leaving aside the exceptional case' in which R(x) has zeros in the interval (xox0), we assume in the sequel that for the extremal (o the condition R> 0 in (x,x,) (IIr) is fulfilled. A consequence of this assumption is that not only fo (x) but also fo'() is continuous in (x0x}), as follows immediately from equation (20) at the end of ~6. Hence we infer that not only the functions P, Q, R themselves but also their first derivatives are continuous in (xo04). EXAMPLE2 I (see p. 27): F y V 1-y'2"; hence 1 An example of this exceptional case is considered by ERDMANN, Zeitschriftfilr Mathematik uncd Physik, Vol. XXIII (1878), p. 369, viz., F= y' cos2x and x< < <x 2 All the square roots are to be taken positive, see p. 2, footnote 1. ~11] SECOND VARIATION 49 Y y F =0, = FF:/' F,,y = ~X 1/I+ Yy,2. (1/1 _ y,2): Further 0'o: Y = ao cosh x, ao hence P = 0, Q =tanh, R a,/cosh2 a0 ao Since we suppose y >0, it follows that ao >0 and therefore R > 0 for every x. c) Jacobi's form of Legencre's dZffereltial equation. We have now to examine the second part of Legendre's conclusion, viz., that, if R > 0 throughout (x0x1), then 862Js for all admissible functions 1. The conclusion is correct, as follows immediately from the preceding developments, whenever there exists an integral of the differential equation (6) which is finite and continuous1 throughout (xoxl); it is wrong, as will be seen in ~16, if no such integral exists. It is therefore necessary to enter into a discussion of the differential equation (6). For this purpose JACOBI2 reduces the differential equation (6) to a homogeneous linear differential equation of the second order by the substitution2 U' w=-Q-R-, (8) which transforms (6) into (P - ')u - (Ru') =. (9) We shall refer to this differential equation as Jacobi'S differential equation and shall denote its left-hand side by T(z): 1 Since R =i0, the continuity of tw implies the continuity of wt', compare (6a). 2 " Zur Theorie der Variations-Rechnung und der Differentialgleichungen," Journal fur Mathematik, Vol. XVII (1837), p. 68; also Ostwald's Klassiker, etc., No. 47, p. 87. 3 Notice that also the derivatives of Q, R exist and are continuous, as shown above. 50 CALCULUS OF VARIATIONS [Chap. II q (') - P - x ] c (- R dw.) I cQ\ d I, dci (10) \ dx dx \ dx; If we write (9) in the form Cd2u R' du Q'- P 2 + R dc + /d tt = ~(, (9a) dx' R dx R the coefficients are continuous in (xox0). Hence it follows, according to the general existence theorem1 on linear differential equations, that every integral of (10) is continuous and admits continuous first and second derivatives in (XOX1). Hence we can infer that if the condition: R > 0 in (Xoxl) is satisfied and if the differential equation (9) has an integral tu which 'is different from zero throughout (xox0), thein 82J> 0 for every admissible function vj not identically zer0o. For if u is such an integral, then (8) furnishes an integral 10 of (6) of class C' in (xox1), and therefore 82J O0. In order to show that the equality sign must be excluded, we introduce u instead of 1o in (7), and obtain 2J E2 f R (x. (D1) 2 This shows that c2J can be equal to zero only when I'u — if 0' O throughout (xoxl), i. e., when - Const. it, which is impossible since V? vanishes at x0 and xz, and u does not. If, on the contrary, every integral of (9) vanishes at least at one point of (xoxl), Legendre's tranformation is not applicable to the whole interval. We shall see (in ~16) that in this case 82J can, in general, be made negative. 1 Compare E. II A, p. 194, and PICARD, Traite d'Analyse, Vol. III, pp. 91, 92. If F and consequently also P, Q, R are analytic functions, the existence theorems for analytic differential equations may be used instead. For linear differential equations in particular, see SCHLESINGER, Handbuch der Theorie der linearen Diffe?entialgleichungen, Vol. I, p. 21. ~12] SECOND VARIATION 51 ~12. JACOBI'S TRANSFORMATION OF THE SECOND VARIATION The proof of the statement made at the end of the preceding section is based upon a second transformation of 82J due to JACOBI.' a) Let (Io01) be either the interval (xoxl) itself or a subinterval of (xoxl), and let D be identically zero outside of (i001), and in (0&o) equal to some function of class C" which vanishes at to and:1. Then if we denote by 21f the quadratic form of 77,? ': 20 = Pq- + 2Q vv + R,12 and apply Euler's theorem on homogeneous functions, we may write 82J in the form 82J =' 2 f (~aaQ + dtxa ) ~ The second term can be integrated by parts since A" is continuous, and we obtain 827j 2^ ^t^ 1 C^ /an 2 a 2J = e2 V [I 4 ( rj ( - - d ) dd } 1 Journal filr Mathematik, Vol. XVII (1837), p. 68. JACOBI derives (8) as well as the integration of (10) from the remark that 62J= 8 (8J), hence {2J- - e ( I ' l X 82J=E 8, ]XSiF~ + I i S3irdx, where M=Fy-d F. But AM= ( y )= e(). Jacobi s paper, which is not confined to the simple case which we are here considering, but which also treats the case in which the function F contains higher derivatives of y of any order, marks a turning point in the history of the Calculus of Variations. It gives, however, only very short indications concerning the proofs; the details of the proofs have been supplied in a series of articles by DELAUNAY, SPITZER, HESSE and others (see the list given by PASCAL, loc. cit., p. 63). Among these commentaries on Jacobi's paper, the most complete is that by HESSE (Journ al fi r Mathematik, Vol. LIV (1857), p. 255), whose presentation we follow in this section. Jacobi's results have been extended to the most general problem involving simple definite integrals by CLEBSCH and A. MAYER (see the references given in PASCAL, loc. cit., pp. 64, 65, and C. JORDAN, Cours d'Analyse, Vol. III, Nos. 373-94). 52 CALCULUS OF VARIATIONS [Chap. II But V vanishes at o0 and i1, and a &2 _ ( _ ') = ( R_') d= - (cP) — dx a' -q Ix' Hence we obtain JACOBI'S expression for the second variation: 82J= - 2 r( r)dx, (12) which leads at once to the following result: If there exists an integral i of the differential equation (9) zohich vanishes at twzo points to and zcl of (xo01), we cant mtake' 82J 0, viz., by choosing - 5' in (v,), 77- } 0 outside of (e0). b) In the sequel we shall need an extension of formutla (12) to the case when r is of class D". Let cl, c,, *, cn be the points of discontinuity of n' or 7". Then the integral for 82J must be broken up into a sum of integrals from o0 to cl, from cl to c2, etc., before the integration by parts is applied. Hence we obtain in this case 82J = Ee V a 02 A Jl (C/ or, if we substitute for a its value and remember that, Q, R are continuous at Cl, c2, *., c.,: 7 n 82j = 2 (c~ ) R (c.) ( '(cv- 0) - '(c+ 0)] V=1 -- + r' ( (I). (12a) c) From (12) a second proof'2 of (11) can be derived; this proof is based upon the following property of the differen1 It will be seen later on that it follows from this result that, in general, there can be no extremum in this case, see ~~14 and 16. 2 Due to JACOBI, see the references on p. 51, footnote 1, in particular to HESSE. ~12] SECOND VARIATION 53 tial operator 1: If u and v are any two functions of class C"l, then d t ' (V) - v', (u) - -C- R (v' -u v). (13) Hence if u satisfies the differential equation (u) = O, we get (V) -=- -- R (Iv'-u' (v), dx and if we put V =- pt, p being any function of class C", and multiply by p, we obtain (pu) 9 (P,)= - -p (-R- t 2) dxR *- -(Rpp' 2) + R (p' t.)2 (14) But since dx p~,2 + 2Qv' + Rv'2 =- v (v) + d v (Qv + Rv') we obtain from (14): P (p,)2 + 2Q (p) d (pu) d ( dx + dR\(d = R (p'u)2 + d (Pt (Qt + R')). (5) Now suppose Cmoreover that u is different from zero throughout (lo)0. Then we may substitute in (15) for the arbitrary function p the quotient p=. and since Vq vanishes at 0o and l, also p will vanish at 54 CALCULUS OF VARIATIONS [Chap. II 0o and:1, Hence, on integrating (15) between the limits:o and:l, and substituting for p its value, we obtain' 82J = f 1 ('x. (a) ~13. JACOBI'S THEOREM By the developments of the last two sections, the decision regarding the sign of the second variation is reduced to the discussion of Jacobi's differential equation (9). It is therefore a theorem of fundamental importance, discovered by JACOBI2 in 1837, that the general solution of the differential equation (in) = 0 can be obtained by mere processes of differentiation, as soon as the general solution of Euler's differential equation is known. a) Assumptions3 concerning the general solution f(x, a, 3) of Euler's differential equation: We suppose for this investigation that the extremal C0 is derived from the general solution by giving the constants a, 3 the special values a0,,o0, so that fo(x) =f(X, a, Po) Further, we suppose that the function f(x, a, 3), its first 1 Notice that in the present proof we have to suppose -q to be of class C" in (0il). It can, however, be easily proved that the result is true also for functions 7 of class C' and even D', in accordance with the results of ~11, c). This follows from the fact that p" does not occur in the identity (15) and that p2u (Qu-+-Ru') is continuous even at the points of discontinuity of -' or '". 2 See the reference on p. 51, footnote. 3If the interval (xzxl) is sufficiently small, these assumptions are a consequence of our previous assumptions concerning the function F (p. 12), the extremal c0 (p. 44) and the function R (p. 48). This follows from the theorems concerning the dependence of the general solution of a system of differential equations upon the constants of integration; compare PAINLEVE in E. II A, pp. 195 and 200, and the references there given to PICARD, BENDIXSON, PEANO, NICOLETTI, and V. EscHERICHr; also NICOLETTI, Atti della R. Acc. dei Lincei Rendiconti, 1895, p. 816. For the case when F is an analytic function, compare E. II A, p. 202, and KNESER, Lehrbuch, ~27. For certain special investigations concerning the "conjugate points," the additional assumption is necessary that also faa,,fap, ff exist and are continuous in A; compare p. 59, footnote 1, and p. 62, footnote 4. ~13] SECOND VARIATION partial derivatives and the cross-derivatives fx,, fxa are continuous, and that fx exists in a certain domain At:' Xo x- X1, I a-a- l I d, 13 -o3 c d where Xo < x, X1 > xr and d is a positive quantity. From these assumptions, together with our previous assumptions concerning the function F, the assumption that ~o lies in the interior of the region X and the assumption that R(x)> 0 in (0xxl) it follows: 1. That1 also the partial derivatives fax, fx exist, are continuous and equal to f,,, fx respectively, throughout A; 2. That if we replace in the first and second partial derivatives of F the arguments y, y' by f(x, a, 3), fx(x, a,/3), these partial derivatives are changed into functions of x, a, i3 which are continuous and have continuous first partial derivatives with respect to a and /; 3. That2 Fy (x, f(xS, a, p), f (X, a, 13))>0, (16) the last two statements being true throughout the domain A provided that the quantity d and the differences x0-X0, Xi-xl be taken sufficiently small; 4. The quantities d, 0- X0, X1 — x being so selected, it follows further from equation (20) in ~6 that also the partial derivatives fx, fxa, fxxP exist and are continuous in A. b) The general integral of Jccobi's differential eqaa.. tion (9) can now be obtained according to JACOBI (loc. cit.) as follows: If we substitute in Euler's differential equation for y the general integral f(x, a,,/) we obtain 1 Compare E. II A, p. 73, and STOLZ, Grundzilge der Differential- und Integral. rhechnung, Vol. I, p. 150. 2 Since R(x) has a positive minimum value in (.ox1) and Fy11 (.x, f(x, a, /), f,(x, a,3)) is uniformly continuous in A. 56 CJALCULUS OF VARIATIONS [Chap. II F,(x f (x, a, /3), (X, a,,)) F- F.(x Sf(sx, a, /?),.f, (, a, 3))= 0 7 an identity which is satisfied for all values of x, a, / in the domain A and which may therefore be differentiated with respect to a or /. On account of the preceding assumptions, the order of differentiation with respect to x and a (or /) may be reversed1 and we obtain (Fyy - d F:,,:. fa-d x i:'f fa)= O (17) (F11i-d - F ) fa - (F5 Afi) (1)= O where the accents denote again differentiation with respect to x. If we give in (17) to a, /3 the particular values a-ao,,/3/30 and remember the definition of P, Q, R in ~11 equation (2), we obtain JACOBI'S Theorem: If y =f (x,, a 3) is the general solution of Euler's differential equation, thefn the differential equation T (t) -(P - Q')u - (Ru') = 0 admits the two particular integrals 2)1 = fa (Xa, ao, /o)). (18),2 = f3 (X, ao, a).. Corollary:2 The two particular integrals ri and r2 are, in general, linearly independent. For, in order that rl and r2 may be linearly independent, iFrom the existence and continuity of d (Fy,,fax) and Fy, follows the existence and continuity of f,,xx on account of (16). 2 See PASCAL, loc. cit., p. 75. ~14] SECOND VARIATION it is necessary and sufficient that their "Wronskian determinant 1 __ r, (x) r2 (x) D ( (x ) r(x) be not identically zero. On the other hand, since f(x, a, /) is supposed to be the general solution of Euler's differential equation, it must be possible so to determine a and 8/ that y and y' take arbitrarily prescribed values Y2 and y' for a given nonsingular value of x, say xa2. The two functions f(x2, a, /3) and f (x2, a, /) of a, 8 must therefore be independent, and consequently2 their Jacobian a (, f/) fXa fa cannot be identically zero for all values of a, /3. But for a =a0, /3= -/o, this Jacobian is identical with the determinant D(x), since fax -fx,, fox -fx, and therefore ir and r2 are linearly independent, except, possibly, for singular systems of values a0, io, i. c., for singular positions of the two given points A and B. We exclude in the sequel such exceptional cases and assume that i1 and r2 are linearly independent. Then the geneiral integral of Jacobi's differential equation is u = Crl + C2r2, (19) C1, C2 being two arbitrary constants. ~14. JACOBI'S CRITERION By Jacobi's theorem the further discussion of the sign of 8'J is reduced to the question: Under what conditions is it possible so to determine the two constants (C, C2 that the function u - C17r1 + 02r2 shall not vanish in (x0xl)? 1 Compare E. II A, p. 261, and J. III, No. 122. 2 Compare P., No. 122, IV and J. I, No. 94. CALCULUS OF VARIATIONS [Chap. II In order to answer this question, we construct the expression (x, x,) - r1 (x) 12 (x0) - 12 (x) 1r (x0); (20) it is a particular integral of (9) and vanishes for x =X; if it vanishes at all for values of x > x0, let x0 be the zero next5 greater than x0, so that A (xo, x) = 0, A (x, X) 4 0 for x, < x < x', (21) A (, x,)-= Thei it follows from a well-known theorem on homogeneous linear differential equations of the second order due to STURM2 that every integral of (9) independent of A (x, x0) vanishes at one and but one point between x0 and xQ. We have now to distinguish two cases: Case I: x'o x. Then every integral of (9) vanishes at some point of (x1oZ ) and we obtain according to ~12, a) the Theorem: If x-o xl, it is possible to make 82J —0 by a proper choice of the function 7. Compare HESSE, loc. cit., p. 258, and A. MAYER, Journal fiur Mathematik, Vol, LXIX (1868), p. 250. 2 If ul, u2 are two linearly independent integrals of d2_I du -UP -+ q t = 0 dx2 dx where p and q are functions of x, then between two consecutive zeros of ul there is contained one and but one zero of u2, provided that these zeros are comprised in an interval in which p and q are continuous." See STURM, " M6moire sur les 6quations diff6rentielles du second ordre" (Journal de Liouville, Vol. I (1836), p. 131); also STURM, Cours d'Analyse, 12th ed., Vol. II, No. 609. The theorem follows easily from the well-known formula du2 dul -fpclx 29 uc d - 2 -l- = Ce f x V ) where Cis a constant * 0. From the same formula it follows that ua and uz cannot du1 vanish at the same point, and that ul and d- cannot vanish at the same point. Compare also DARBOUX, Theorie des Surfaces, Vol. III, No. 628, and BOCHER, Transactions of the American Mathematical Society, Vol. II (1901), pp. 150, 428. It seems that WEIERSTRASS was the first who used Sturm's theorem in this connection. HESSE (loc. cit., p. 257) reaches the same results in a less elegant way by making use of the relation (22). 3 Compare Addenda at end of book. _ ~_ _~ __ ~14] SECOND VARIATION 59 For instance, by taking r-A (x, x0) in (xox') and identically zero in (xOxl). Hence JACOBI inferred that an extremum is impossible if x- axl; for, 8J and 82J being zero, the sign of AJ depends upon the sign of 83J which can be made negative as well as positive by choosing the sign of e properly. This conclusion is, however, legitimate only after it has been ascertained' that the particular variation which causes 82J to vanish does not at the same time make 83J- 0. Case II: xQ > xi or else X0 non-existent. In this case the particular integral a (x, x,) =,1 (x) r2 (x,) - 2 (x),1 (x,) of (9) is linearly independent of A (x, x0) since A (xo, x0) 0, whereas A (0, x,) = - a (xi, X0) i= 0 Hence it follows from Sturm's theorem that A(x, xl) p0 for xo0x < x, and therefore also (on account of the continuity of A (x, xl) ) for x0- x < xa, 8 being a sufficiently small positive, quantity. Now choose x~ between x — and x0 and so near to Xo that2 X0< xO< x0. Then we can apply Stu-rm's theorem to the two particular integrals A(x, xj) and A (x, x~) = r1 (x) r (x~) -- r2 (x) r (x) and obtain;f... the result that,o \, a (x,x,) A(a,a,) A(a,,a,) A (X, X0) == 0 ill (X1X). FIG. 9 iThe value of 83J for this particular function '1 has been computed by ERDMANN (Zeitschrift fur Mathenatik und Physik, Vol. XXII (1877), p. 327). He finds, in-the notation of ~ 15 63J=-e3R(x0')C (X0), 0O) yyo) O (X Y'); (23) R (X0') and PB (xo', y7) are always different from zero; and fyy (x', y)) is also different from zero except when the envelope of the set (28) has a cusp at A' or degenerates into a point. With the exception of these two cases then, JACOBI'S result is correct. Compare also ~ 16. 2See ~13, a). On account of (16), R(x)>0 and, therefore, rl(x) and rs(x) are continuous not only in (xoxl) but also in the larger interval (XoX1). 60 CALCULUS OF VARIATIONS [Chap. II We obtain, therefore, according to ~11 c), the Theorem: If R > 0 throughout (xox1), and either xx< x0 or xS non-existent, then 82J is positive for all admissible funzctions v. Hence JACOBI inferred that in this case a minimum actually exists, and this was generally believed until WEIERSTRASS showed the fallacy of the conclusion (1879) (see ~17). The above two theorems constitute "JACOBI'S CRITERION." The value xQ is called the conjugate of the value X0; and the point A' of the extremal @0 whose abscissa is X0, the conjugate of the point A whose abscissa is x0. ~1D5. GEOMETRICAL INTERPRETATION OF THE CONJUGATE POINTS JACOBI1 has given a very elegant geometrical interpretation of the conjugate points, which is based upon the consideration of the set of extremals through the point A. a) This set is defined by the two equations y =f(x, a, ), (24) y0J=f(xO, a, ). (25) The second equation is satisfied by a-ao, /= /,8; and at least one of the two partial derivatives fa (xo, oa0, /o) = ri (x0) and fs (x0, a0, /) = r2 (x0) is + 0 since r1 (x) and r2(x) are two independent integrals of (9) and R(xo) =0 (see p. 58, footnote 2). According to the theorem2 on implicit functions we can therefore solve (25) either with respect to a or with respect to /3. But we obtain a more symmetrical result if we express a and/3 in terms of a third parameter y. If we choose, for instance, 1 Loc. cit., and Vorlesungen Aiber Dynamik, p. 46; also HESSE, loc. cit., p. 258. 2 Compare p. 35, footnote 2. ~15] SECOND VARIATION 6I Y = fx(Xo, a, P) (26) and denote by yo the value o = fx(xo, at Po) we can solve' the two equations (25) and (26) with respect to a and /3, and obtain a unique solution a= a(y), / =/ (y), which is continuous in the vicinity of the point y7-y7 and satisfies the condition ao = a (yo), o = -/ (Y0o) Moreover the functions a (), 13(y) admit, in the vicinity of of 7o, continuous first derivatives. Hence it follows that if we put f/(x, a (y),/ (y)) =, (x y), the function O(x, 7), its first partial derivatives and the derivatives2,Cx, cfy will be continuous in the domain Xo0 X1, [ y - y- o dl, d1 being a sufficiently small positive quantity. Furthermore, the equation yo -= (~, y) (27) is satisfied for all sufficiently small values of i - 7o The equation y (x, y) (28) represents, therefore, the set of extremals through A in a certain vicinity of the extremal o0, the latter itself being represented by (0: y= - (x, yo). (29) By differentiation with respect to y we get 1 All the conditions of the theorem on implicit functions are fulfilled at the point a=ao, P3=Po, y=y0. In particular, the Jacobian of the two functions f(Xo, a, 3)- yo and fx(xo, a,P )-y with respect to a and p is: O for a =.a0, = P,7 -- O, its value being D (x0) =rl (x0) r2' (x0) - r2 (x) r1' (x0), which is different from zero. since r1, r2 are linearly independent and x0 is a non-singular point of the differential equation (9). 2 Also Byy will be continuous if faa, fa, f3a are continuous in A. 62 62 ~CALCULUS OF VARIATIONS [hp [Chap. 11 cdy d/3 a-nd therefore, on pntting 7Y-'O r, ~ (x5) r, (x) -+ r, (x5,) r2 (x) The fulletions ~, (x, yo) and A (x, x0) differ, therefore, only by a constant f actor:' (Ay(x yo - A XIC0)0 (30) and consequently the conjugate value x0' may also be (lefipied2 a s the root next greater than x0 of the equation From (30) and the properties' of A (x, x0) it follows further that 0,/ (X0, Y/o) = 0 ckYX (Xi, Iy-/) ~ 0 (31) b) According to the preceding results, the co-ordinates X, y' of the conjugate point A' satisfy the two equations - ~ (X0/I6y, Iyo) ~(X6' y/) 0I and the determinant 4-/x 41yy is different from zero for x-x, ' y r ' yo, its value being 0, (xo, Yo7). Hence we obtain, ac~cordin g to the, theory of envelopes,' the following geometrical interpretation. II The same results concerning P (xa, y) hold if, instead of the particular parameter -y chosen ahove, we introduce auother parameter -y' connected with y hy a relation of the form where x, (y) and its first derivative are continuous in the vicinity of yo, and x '(vO) tO0. 2 ConaeERDMANN, Zeitschrift flir Mathematik und Physlik, Vol. XXII (1877), p, 325.1 3 Compare p. 18, footnote 2. 4Copr E. III ID, p. 47. The proof presupposes the continuity of, ~D 4 %D ~Dyx 4 hD in the vicinity of the ppint x= x'5, y= yo, y=,y5. These ~15] SECOND VARIATION 63 Consider the extremal C": g~y = 0(X,yo) and a neighboring extremal of the set (28): y = (x, y + k). Then if 1 k be chosen sufficiently small, the curve C will meet (i0 at one and but one point P in the vicinity' of A'. And as k approaches zero, the s point P approaches A' as lim- iting position. Hence we have the Theorem: The conjugate A' of the point A is the point where ao 5-8 aCo 04 the extremal (o meets for the FIG. 10 first time the envelope of the set of extrenmals through A. d) EXAMPLE IV: F=g(y'), a function of y' alone. The extremals are straight lines; the set of extremals (28) is the pencil of straight lines through A; hence there exists no conjugate point. The same result follows analytically: The general solution of Eul e r's equation is y ax + /, hence r, x, r2 = 1, conditions are satisfied in our case provided that xo' lies in the interval (XOX1), and provided that we suppose that not only the derivatives mentioned on p. 55, but also fea, fa, fpB are continuous in A (compare p. 54, footnote 3). 1This means: If we choose a positive quantity 8 arbitrarily but sufficiently small, and denote by M1 and M2 the points of (% whose abscissae are x0 - and x0 S then another positive quantity a can be determined such that every extremal ~( for which I k < a meets C0 at one and but one point P between M1 and M2. Compare p. 35, footnote 2. If, on the contrary, x2 be any value in the interval (XoX,) for which P, (a2, 70) 0, then two positive quantities 8' and a' can be determined such that no extremal S for which 1 k I < a' meets C0 between the points whose abscissae are X2 - ' and x2 + 8'. For in this case the difference ( (x2+h, yo+k)- + (X2+h, yo) =-k(, (2+h, yo+Ok), where 0 < 0 < 1 is different from zero for all sufficiently small values of [ h I and I k I 64 CALCULUS OF VARIATIONS [Chap. II and A(x, o) = x- - EXAMPLE I (see p. 27): From the general solution of Euler's equation y = a cosh - a we get A (x, Xo) = sinh v cosh vo - sinh Vo cosh v + (v - v,) sinh v sinh vo, where x -, y, - go x ---, o = -- ao a Hence we obtain (if vo # 0) for the determination of xo the transcendental equation coth v - v = coth 0 - vo. (32) Since the function coth v- v decreases from + co to -co as v increases from -cc to 0, and from + co to - co as v increases from 0 to + o, the equation (32) has, besides the trivial solution v = vo, one other solution va, and vo and vo have opposite signs. Hence if v, > 0, i. e., if A lies on the ascending branch of the catenary, there exists no conjugate point: A (x, Xo) * 0 for every x > Xo. The same result follows for t0 = 0. If, on the contrary, vo < 0, i. e., if A lies on the descending branch of the catenary, there always exists a conjugate point A' situated on the ascending branch. It can be determined geometrically by the following property, discovered by Lindelof:' The tangents to the catenary at A and at A' meet on the x-axis. For the abscissae of the points of intersection of these two tangents with the x-axis are X\~~\? X, - x-a coth 0-/ ao A\ // and, O- Po / X'=0- a0 coth —, ao X=o - and they are equal on account FIG. 11 of (32). ILINDELOF-MOIGNO, loc. cit., p. 209, and LINDELOF, Mathematische Annalen, Vol. II (1870), p. 160. Compare also the references given on p. 28, footnote 1. ~16] SECOND VARIATION ~16. NECESSITY OF JACOBI'S CONDITION It has already been pointed out that the two theorems of ~14 which constitute Jacobi's Criterion, though giving important information concerning the sign of the second variation, contain neither a necessary nor a sufficient condition for a minimum or maximum. But at least a necessary condition can be derived from the first of the two theorems by a slight modification of the reasoning: If xQ < xl, then 82J can be made not only zero but even negative. This was first proved by WEIERSTRASS in his lectures; the first published proof is due to ERDMANN.1 The following is essentially Erdma n n's proof: 1Zeitschriftfii r Mathenmatik und Physik, Vol. XXIII (1878), p. 367. SCHEEFFER'S proof (Mathematische Annalen, Vol. XXV (1885), p. 548), is not essentially different from ERDMANN'S. WEIERSTRASS writes the second variation in the form ( x1 XI l a2J=e2= | [(P+- k) 72+42 Q7q-'+ Rq'2]dx —k X 72dx 0o 0o k being a small positive constant, and applies to the first integral J a cob i's transformation: ^2J=e27 () q dx.- k 2 djx, where d ()= ((P + k) - ') - (R77') ~ Then he shows that there exist admissible functions 7 which satisfy the differential equation ' (7i) = 0. For such a function 7, 62J is evidently negative. H. A. SCHWARZ (Lectures, 1898-99) uses the following function q7: ( A(aX, o) +kwo in (XoXo'), r)~= } k co in (xo'xl), where k is a small constant and o is a function of class C' which vanishes at x0 and xl but not at Xo'. The corresponding value of 2J is of the form: 82J = 2 f 2cR (X') '(XAo', x) O (xo') + 2V, which can be made negative by a proper choice of c. (Compare SOMMERFELD, Jahresbericht der Deutschen Mathematiker- Vereinigung, Vol. VIII (1900), p. 189.) All these proofs presuppose xo'<xl; for the case Xo'=x-, so far as it is not covered by E r d m a n n's formula (23) for 83J, compare KNESER, Mathematische Annalen, Vol. L (1897), p. 50, and OSGOOD, Transactions of the American Mathematical Society, Vol. II (1901), p. 166. This case will be treated in parameter-representation in chap. v, ~ 38. 66 CALCULUS OF VARIATIONS [Chap. II Take x2 so that xo < xt < x and A (x2, x0) =, and put u = A (x, x), v = p (x, x'), where p= - 1 or — 1; u and v are particular integrals of (9) and linearly independent; hence the relation (22) holds and takes the following form for the differential equation (9): R (.uv'- u'v) K, (33) K being a constant different from zero. We choose p so that K> 0; this is always possible, for, if v is replaced by -v, KC is changed into -K. Further, since also ui and u- v are linearly independent, it follows from Sturm's theorem (see p. 58, footnote 2) that u-v vanishes for one value of x, say x c, between x0 and x 0; hence t (c) v (c) Now define X7 as follows: ^ ^ ><:f"^ {_ u in (x c ) / u;i,. e = A- v i nC 2 7), FIG. 12 0 in (x2x) ) This function D fulfils the conditions under which the formula (12a) for 82J holds, and since () — 0 for each of the three segments, formula (12a) becomes: 82J = E2R (uu'- v'), which may be written, since (c) v (c): 82J - 2R (Ruv - u'v) i = - K, and this is negative according to our agreements concerning the sign of v. Thus we have proved the FUNDAMENTAL THEOREM III: The third necessary condition for a minimtum (maximum) is that ~16] SECOND VARIATION 67 A(x, Xo) O (IrI) fo'r all values of x in the open interval x0< x <l. Corollary: The same condition may also be written Xl x-o, or else a'o non-existent, (III) i. e., if the end-point B lies beyond the conjugate point A', tlhere is no minimum or maxitmum. We shall refer to this condition as JACOBI'S co(dlition. CHAPTER III SUFFICIENT CONDITIONS ~17. SUFFICIENT CONDITIONS FOR A "WEAK MINIMUM"' WE suppose henceforth that for our extremal (0 the conditions R> (II') A (x, X0) 4 0 for 0 < x 2 (III') are fulfilled, and we ask: Are these conditions SUFFICIENT for a minimum? a) It seems so, and until rather recently it was generally believed to be so: For the reasoning of ~11 shows that after an admissible function y has been chosen, AJ will be positive for all sufficiently small values of e I; hence within the set of curves with parameter e: y = y + er (1) the curve @o does furnish a minimum. On the other hand, every curve ( may be considered as an individual of such a set, and therefore it seems as if we must actually have a minimum. But a closer analysis shows that the conclusion is wrong. For all we have proved so far is this: After a function X has been selected we can assign a positive quantity3 p, such that AJ>0 for every I e < p. And if 1 Compare for this section SCHEEFFER, "LUeber die Bedeutung der Begriffe Maximum und Minimum in der Variationsrechnung," Mathematische Annalen, Vol. XXVI (1886), p. 197. This paper has been of the greatest importance in clearing up the fundamental 'conceptions in the Calculus of Variations. 2Notice the equality sign which distinguishes (III') from (III); for the case xt = X', which we omit here, compare the references on p. 65, footnote. 3The notation p~ indicates that p depends on the function vq; compare E. H. MOORE, Transactions of the American Mathematical Society, Vol. I (1900), p. 500. 68 ~17] SUFFICIENT CONDITIONS we denote by mn, the maximum of I71 in (xor) and put k, =-, p,, we have Ay I <, for all curves of the set (1) for which | e < p,; and vice versa, if we draw in the neighborhood (k,) of 0o any curve of this particular set, the corresponding e satisfies the inequality ei < p, and therefore AJ> O. Now consider the totality of all admissible functions a; the corresponding set of values k, has a lower limit ko0 0. If it could be proved that 0o > 0, then we could infer that AJ > 0 for every admissible variation y for which I Ay I < ko, and we would actually have a minimum. But it cannot be proved that ko > 0 and therefore we cannot infer that eo minimizes J. It is even a lpriori clear that the method which we have followed so far can never lead to a proof of the sufficiency of this or any other set of conditions.' For, if we apply Taylor's expansion (either infinite or with the remainder term) to the difference AF= F (xy Z, t + Ay'+ -F (x, y, y') and integrate, we can only draw conclusions concernig the sign of AJ from the sign of the first terms, if not only I Ay I but also 1 Ay' 1 remains sufficiently small, or geometrically: if for corresponding points of ~0 and - not only the distance but also the difference of the directions of the tangents is sufficiently small. b) If there exists a positive quantity k such that AJ 0 for all admissible variations for which AylI <k and Ay'l < k, KNESER (Lehrbuch, ~17) says that the curve e0 furnishes a " Teak Minimum," from which he distinguishes the mini1 First emphasized by WEIERSTRASS. MO I 70 ~CALCULUS OF VARIATIONS [hp I [Chap. III nmun as we have defined1 it according to WEIERSTRASS, as ~'StrongM]Iiinimnuw)." If acurve fnrnishes astrong minimum, it always f urnishes at for-tiori also a weak minimum, but not Vice versaf. If we adopt temporarily this terminology, we can enunciate the following Theor-em: An ertrem)cti @0 for- wlhich, the, conditionts z~(x, x5) =~ 0 fr X0 < x ~ (I' aire fulilled, fur-nishes att least at " wveak mininiiu-a " for the integrtal J. The first proof of this theorem was given by WEIERSTRASS (Lectnr-es, 1879), the first published proof by SCHEEFFER (loc. c it., 1880). The following proof is due to KNESER V We return to equation (3) of ~ 11 which we write in the form: AJ ~(Pw2' + 2 QeW I + Rw' 2 )dx + - (LW2 + NW'2 ) dx where wj=- AY, and L, N are infinitesimals in the following sense: corresponding to every positive quantity o- another positive quantity p, can be assigned such that: ILK< a N <fa- ill (x0x1), provided that OjI < p, and IW'I< p, ill (X0x,) By Legendre' s transformation,2 the first integral may be thrown into the form: _L1R WI+ Q+W w) +(_P+ w,(Q +wq)2) W2]cd 1 Compare ~3, b). 2Jahresbericht der Deutschen Mathernatiker- Ver-einiguis~t, Vol. VI (1899), p. 95. The theorem can also be proved by means of We ie rs t rass '5 Theorem (~ 20); compare KNESER, Lehrlnuch, ~~20-22. 3Compare ~11, b). ~ 17] SUFFICIENT CONDITIONS Since the conditions (II') and (III') are fulfilled, there exist' solutions of the differential equation +dw _ ( Q+ w) _ dx R which are of class C' in (xoxl); hence it follows2 that, provided the constant c be taken sufficiently small, there also exist integrals of the differential equation + dw (Q + w)2 2 (2) +'dx R which are of class C' in (xoxl); let w be such an integral, and introduce, g+?c '= 0~ + ----- instead of a'. Then AJ takes the form AJ= i r [(ca + X) 02 + 2V)b + (R + v) 21 d, where X,,/, v are infinitesimals in the same sense as L and N. But this may be written XI 2, + C2 + PI22 AtJ= -^12[(+ )+R+ )+(v2+W-R)+2] and since X, /, v are infinitesimals, we can choose a positive quantity k so that R A- v > O and c2 - X - 2/(R + v) > 0 in (x0x1), and consequently AJ> O, provided that lo < k and |w'1 <7k, Q.E.D. Remark. We have given this theorem chiefly for its historical interest: It marks the farthest point which the Calculus of Variations had reached before WEIERSTRASS'S I This follows from the connection between Legendre's and Jacobi's differential equations; see equation (8) in ~11, b). 2 According to a theorem due to POINCARI (M6canique celeste, Vol. I, p. 58; compare also E. II A, p. 205, and PICARD, Traite, etc., Vol. III, p. 157). A similar theorem was given by WEIERSTRASS in his lectures in connection with his proof of the necessity of J a c o b i's condition, see p. 65, footnote, 72 CALCULUS OF VARIATIONS [Chap. III epoch-making discoveries concerning the sufficient conditions for a "strong minimum." 'After these discoveries, only a secondary importance attaches itself to the "weak minimum;" for the restrictionimposed upon the derivative in the "weak minimum" is indeed a very artificial' one, only suggested and justified by the former inability of the Calculus of Variations to dispense with it. c) The terms "weak" and "strong" are sometimes also applied to the variations. A variation containing a paramete'l e A - o(x, E) is called weak if not only Lw(x, ) =O but also L o(x, ) =O eO= e=0 uniformly in (xoxT), strong if this condition is not satisfied. The variations of the form A y - er, as well as the more general variations which we have mentioned in ~4, d), are weak variations. WEIERSTRASS gives the following example2 of a strong variation: isy8 = E sin (X -- X0) 7\ n a positive integer; here the condition L Ay — 0 e=-. 1 Especially if we think of geometrical problems, for instance, the problem of the shortest curve on a given surface between two points. For the more general problem, however, where higher derivatives occur under the integral sign, such restrictions are of greater importance; compare ZERMELO, Dissertation, pp. 26-31. 2The following modification of WEIERSTRASS'S example has the advantage of vanishing at both end-points: Ay= - sin ( — x,,)mar' n m Xl- o X / 'rs mn and n being positive integers. ~18] SUFFICIENT CONDITIONS 73 is satisfied, but not the condition LAy' O. e=O Other examples of strong variations will occur in ~~18 and 22. ~18. INSUFFICIENCY OF THE PRECEDING CONDITIONS FOR A STRONG MINIMUM, AND FOURTH -NECESSARY CONDITION From the introductory remarks of the previous section, it follows that we have no reason to expect that the conditions (I), (II'), (III') are sufficient for a minimum in the sense in which we have defined it according to WEIERSTRASS (a "strong minimum" in KNESER'S terminology). a) As a matter of fact the three conditions (I), (II') and (III') are NOT sufficient for a strong minimum, and it is easy to construct examples' which prove this statement: EXAMPLE III2 (see p. 39): F = y12 (+ 1). Here (0 is the straight line joining the two given points A and B, say o y -= inx - n. Further: R = 2 (6m2 + 6,2 + 1) A (x, Xo) = - X; hence x' non-existent. Let inl, m2 be the two roots of the equation 6n2 + 6m + 1 = 0, viz., in = 2(- + ) = -0 2113... 1The first example of this kind was the problem of the solid of revolution of least resistance; already LEGENDRE had shown that the resistance can be made as small as we please by a properly chosen zigzag line; see LEGENDRE, loc. cit., p. 73, in STiACKEL'S translation, and PASCAL, loc. cit., p. 113. 2 Compare BOLZA, " Some Instructive Examples in the Calculus of Variations," Bulletin of the American Mathematical Society (2), Vol. IX (1902), p. 3. 74 CALCULUS OF VARIATIONS [Chap. II1 (l - ) O-07887... then R > 0 if mn > mi1 or mrt < m2, R <0 if /m2 < m <.n, In the former case, the first three necessary conditions for a m i i - m unm, in the latter for a maximum, are satisfied. Nevertheless, if - 1 < m < 0, neither a maximum nor a minimum takes place. For, in this case, if any neighborhood (p) of Co be given, however small, we can always join A and B by a broken line 6 made up of segments of = + straight lines of slope 0 and Cs ~~ — -" -\ -1, and contained in (p). But B Y for such a broken line J=0, whereas for (o the integral J is positive. This proves that ~; --- —--— v-o (0 cannot furnish a minimum. FIG. 13 That it cannot furnish a maximum will be seen later, in ~ 18, e). EXAMPLE V: To minimize -rX J= (y'2-+-y')x, 0 the given end-points having the co-ordinates (x,, yo)=(0, 0), (x1, yi) =(1, 0). The extremals are straight lines, and (E, is the segment (0 1) of the x-axis. Further, R = 2 —,.;.(X, X0) - X -,0o Hence the conditions (I), (II'), III') for a minimum are satisfied. Nevertheless AJ can be made - p negative. For, if we choose for - q S the broken line APB, the co- A b ] B ordinates of P being (1-p, q), where 0 <p <1, and q > 0, we FI obtain AJ q2 (11+ q q) P(1-P) 1-p 'p ~18] SUFFICIENT CON)ITIONS 75 Any neighborhood (p) of 0o being given, choose q < p; then 1) can always be taken so small that A J < 0. b) The insufficiency of the preceding three conditions being thus established, further conditions must be added before we can be certain that the curve o0 minimizes the integral J. A fourth necessary condition was discovered by WEIERSTRASS in 1879 and derived by him in the following manner: Through an arbitrary point 2: (x2, Y2) of ~0 we draw arbitrarily a curve (5: y —f(x), of class C'. Denoting by 4 that point of (- whose abscissa is x2 —, h being a small positive quantity, we draw, as in ~8, a curve -FIG 15: = y4-yer of class C' from o 0 to 4 and replace the are 02 of 0o by the curve 042. By taking h sufficiently small we can make the curve 042 lie in the neighborhood (p) of o0. For this variation of 0o we obtain in the notation, of ~8: AJ -Jo4 + Ja-2 J- (3) But according to ~8, equation (30), this is equal to J= - I E (x2, y2; y2/, 2) )+ (h), (4) where (h) denotes as usual an infinitesimal, and the Efunction is defined by E (x, y; p, p )=F(x, y, ) -F(x, y, p) - (1) -p))Fy.(x, y, p). Hence follows the FUNDAMENTAL THEOREM IV: The fourth necessary contdition for a minimum (maximum) is that 7 6 76 ~CALCULUS OF VARIATIONS rhp l [Chap. III along' the curve (~o for every finite value of 3). We shall refer to this condition as WEIERSTRASS'S5 condition. c) Applying Taylor's formula to the difference F (x, y, )F (x, y,p) we obtain the following important relation' between the E.Junction and F,,, E (X, y; p, ) -,"(,y p*E) (5) where P*=+0(~-J)) 0<0<1I This proves Corollary I: Condition. (IV) is always satisfied 'if for every point (x, y) on (Yo and for every finite value of3 y,,,(, 31) - 0 (Iha) Fnrthermore, if we define the function' E, (X, y; p, 3) by the equation El (x y; p,3) E Pxy~p (6) when j5#p, and by El(x, Y; p, p) =LE, (x, y; p,j) yp (Ga) when _3 -p, we obtain C1orollary II: Condition (IV) is equivalent to the condition along (Yo for every finite 3-. d) ZERIMELO' has given the following geometrical 0 1. e., if (a,, y) is any point of go and y' the slope of 5o at (x, y). 2 Due to ZERMELO, loc. cit., p. 67. 3 Compare ZERMELO, loc. cit., p. 60. 4 Loc. cit., P. 67. ~18] SUFFICIENT CONDITIONS 77 interpretation of the relation between the E-function and Fyy, Let F(p) denote the function F(x, y, p) considered as a function of p alone, x, y being regarded as constant, and consider the curve - F(p). (7)T Draw the tangent PoT at the point Po whose abscissa is p y'; and let P and Q be the R points of intersection with the line p -p of the curve and of the tangent PoT respectively. - r - ' Then FIG. 16 E (x, y; y', )) - F()) - F (y)- (-y') F' (y') is represented by the vector QP, and the condition E (x, y; y',P ) 0O (IV) means therefore geometrically that the curve (7) lies entirely above-or at least not below-the tangent PoT. In order that (IV) may hold it is therefore: a) Necessary that the curve shall turn its convex side downward at p- y', i. e., that " ( I') o. This is our old condition (II), which is consequently contained in the new condition (IV).,8) Sufficient that the curve shall everywhere turn its convex side downward, i. e., that F"(p) 0 for every p, which is the above condition (IIa). But neither is the first condition sufficient, nor the second necessary. e) EXAMPLE I (see p. 49): F= yV I y''; __ 78 CALCULUS OF VARIATIONS [Chap. Ill hence y By1 S' (1'1 + pY2)3 Since y > 0 along the catenary, condition (IIa), and therefore also (IV), is satisfied. EXAMPLE III (see pp. 39, 73): F = y2 (y' + )2; hence E(,; y, ) (P _ )2 [2+ 2 (y,'+ 1) + 3y2 + 4y_+ ]. (o is the straight line joining the two points 0 and 1, say: y = m x + n; hence along Cy0, y' = m. The quadratic in p ) + 2p3 (t + 1) + 3j 2 + 4 m + 1 is always positive if mn (m +1) > 0; it can change sign if m (m + 1) < 0; and it reduces to a complete square if m (mn+1) 0. Hence we obtain the result: If m - 0 or m, ' — 1, condition (IV) is satisfied; if -1 < m < 0, condition (IV) is not satisfied, and the line 01 furnishes no extremum, in accordance with the results of ~18, a). EXAMPLE V (see p. 74): F= y't2+ y'3 hence along the curve (: y = 0 we have E (x, y; ', 1) =2 ( +), which can change sign at every point of (@0. Condition (IV) is therefore not satisfied. ~19. EXISTENCE OF A "FIELD OF EXTREMALS" Before we can take up the question of sufficieniy?conditions, we must introduce the important concept of a "field of extremals." a) Definition of a "field." Consider any one-parameter set of extremals' y = (x, y), (8) 1 Here the symbol < (x, y) is used in a more general sense than in ~15. ~19] SUFFICIENT CONDITIONS 79 in which our extremal Co is contained, say for y 7 Yo. Suppose b (x, 7), its first partial derivatives and the derivatives Oxx, nxy to be continuous functions of x and y in the domain. X X1, y Y -o l c do, d0 being a positive quantity and X0, X1 having the same signification as in ~11. Let k denote a positive quantity less than d0, and *k the set of points (x, y) furnished by (8) as x and y take all the values in the domain,': ocx0x xc, y - yo I ck. O'k may also be defined as the strip of the x, y-plane swept out by the extremals (8) as y increases from 70 -k to 7Yo -k, x being restricted to the interval (xox1). Then 8k is called' a "field of extremals about the arc 'o0" 'if through every point (x, y) of Jk there passes BUT ONE EXTREMAL of the set (8) for 'which Iy - 7o 1 7ck. This means analytically that there exists a single-valued function - y= (x,) y) such that = X (x y)) (9) and 1( )for every (x, y) in 'k In addition to this principal property we shall include in the definition of a field the further conditions that the inverse function f (x, y) shall be of class C' in ~k, and that it shall be possible to choose a positive quantity p so small that the domain Ak contains the neighborhood (p) of the extremal (o. b) With respect to the existence of a field the following theorem holds: Whenever,Y (x, yo) i 0 throughout (XoX1), (10) According to KNESER, Lehrbuch, ~14; the notion of a field is due, in a more special sense, to WEIERSTRASS; in its most general sense to H. A. SCHWARZ, Werke, Vol. I, p. 225. Compare also OSGOOD, loc. cit., p. 112. so CALCULUS OF VARIATIONS [Chap. HII k can be taken so small that the extrenmals (8) furnish a field Fk about 0o. Proof:l From (10) it follows that y (x, 7) —being continuous in (x0xo)-cannot change sign in (xo0x). In order to fix the ideas suppose that Y (x,,yo)> O in (x0x1) Then it follows, according to well-known theorems2 on continuous functions, that k] can be taken so small that y(x, )> 0 in,. ( 1) Hence if we give x any fixed value x2 contained in (xiXl) and let 7 increase from 7 - k to yo -i k, ( (x2, y) increases continually from (x2, 7o - k) to < (x2, Yo + 7) and therefore passes once and but once through every intermediate value. Hence if 72 be any value of y in (7o- 7, 70-+ k) and we put (x2,, 72) -Y2, then the equation Y2 —(Z2, 7) has in (7yo-, 7yo+- ) no other solution but 7-Y72, which means geometrically that through the point (x2, Y2)-which is any point of k,-there passes but one extremal of the set (8) for which 7 -7yo l 7k. The existence of the single-valued function y7=- (x, y) being thus established, the existence and continuity of its first partial derivatives follows from the theorem3 on implicit functions, since ~t(x, Y) I 0 in, 1 Another proof is given by OSGOOD, loc. cit., p. 113. 2Viz., the theorems on "uniform continuity" and on the existence of a minimum. Compare E. II A, pp. 18,19, 49; J. I, Nos. 62, 63, 64, and P., Nos. 19 VI, VII, and 100 VI, VII. 3 See p. 35, footnote 2. The values of these partial derivatives are obtained from (9) by the ordinary rules for the differentiation of implicit functions: 'Y = -1, 7,,==^-.(12) In case the function k (x, y) is r e gul a r in Vk, also the function i (x, y) will be regular in dic; compare E. II B, p. 103, and HARKNESS AND MORLEY, Introduction to the Theory of Analytic Functions, No. 156. ~19] SUFFICIENT CONDITIONS 81 At the same time we see that the set of points.k is identical with the strip of the x, y-plane bounded by the two non-intersecting curves y = (x, -0- k) and y = (x, 0 + k) on the one hand, and the two lines x -x0 and x -=x on the other hand. Finally, a neighborhood (p) of the are (0 can be _ y=y assigned which is wholly contained in ~k. - Y=-k For each of the two continuous functions (x, v0 + k) - 0 (x, yo) and -...... (x,7o+At) -(t, 7o0) and x FIG. 17 (X, ry7o)- (, 7yo- A) has a positive minimum value in (x ol); hence if p be the smaller of these two minimum values, the neighborhood (p) of 0o is entirely contained in Sk. The region Ok has therefore the three characteristic properties of a "field,' and the above theorem is proved. * Corollary I: The slope at a point (x, y) of the unique extremal of the field passing through (x, y) is likewise a single-valued function of x, y, which we denote by (x, y). It is defined analytically by the two equations p (x, y) =, (x, Yy), = (x, y), (13) which show at the same time that p (x, y) has continuous first partial derivatives in *k. In case S (x, /) is regular in 2k, also p (x, y) is regular in A/.' Corollary II. The slope p(x, y) satisfies the following partial differential equation of the first order: ' (Px + PP1) Fy,, + + Fy + Fx - F = 0, (14) the argznents of the partial derivatives of F being x, y, p(x, y) 1This corollary forms part of HILBERT'S proof of W e i e r s t r a s s' s theorem; see below, ~21, and the references there given. 82 CALCULUS OF VARIATIONS [Chap. T11 Piroof: From (13) we obtain by differentiation Pr - oxr + PX'yYTX PY - X11 hence if we make use of (12) we get But since O(x, -) satisfies Euler's equation for every value of v, we have, for every value of x and Y, 7 + ~ Fy. + E - - 0 the arguments of the partial derivatives of F being x, (x, y), -o (x, y). Hence, -if we express y in terms of x, y by means of (9), we obtain (14). c) Application to the set of extremncls through the p)oint' A. We can now establish the following Tlheoremniz: Iffor the extrenmal Go the conditions R>0, (II') Ax,(x )*0) for X5<X-X1 (III') are fulfilled, and if a point A be chosen on the continuation2 of C% beyond A, but sufficiently near to A, then the set of extremals through A furnishes af ield 'about Go. It is only necessary to choose the point A (x5, y5) so near to A that 1. XO< x5 <xO 2. n(,x,)t* O in (x,x,) The possibility of such a choice of x5 has been established in 14. Under these circumstances, it follows by the method employed in 15 that there exists a set of extremals through A: Y = (A (X, Y) J (15) 1The introduction of the set of extremals through A instead of the set through A, which considerably simplifies the proofs, is due to ZERMELO, Dissertation, pp. 87, 88; compare also KNEsER, Lehr-buch, ~~14, 17 and OSGOOD, 10c. cit., p. 115. 2 Compare the assumptions in ~13 a). ~19] SUFFICIENT CONDITIONS 83 where1 ~ (x, y), its first partial derivatives and the derivatives x.P, koxy are continuous in the domain Xo< x X, y-yo ado, dto being a sufficiently small positive quantity. Moreover y (X, yo) O in (xox), since, corresponding to equation (30) of ~15, we have in the present case y(x, Yo) = C. (x, x5) where C is a constant different from zero. Hence the set of extremals through2 A satisfies the conditions of the lemma given under b) and furnishes therefore indeed a field about 0o. 1 Notice that in ~15 the symbol 0 (x, y) was used with a slightly different meaning, viz., for the set of extremals through A. 2 To the set of extremals through the point A itself the lemma cannot be applied, since for this set ky (xO, yO)= 0. Nevertheless it can be proved that in this case through every point of k, except the point A itself, a unique extremal of the set can be drawn. For in the present case we have: 4 (x0, y)= y for every y and therefore (y (xo, y)-O. Hence it follows that if we define X, )fy(x, 7)/(x-Xo), when x* o, x(x, )= 'yx(, y), 0Y (xO < Y) x when x=x, the function X(x,y) is continuous in the domain: Xo - x X1, I y —yo do, and x(x, yo) 4 0 in (XOX1), also for x=xo, since yx (xo, 'yo):0 according to equation (31) of ~15. We can therefore take k so small that X(x, y) t0 in the domain: X-Y x. x X1, y -- I c k.. Hence it follows that 4by (x, y) has the same sign throughout the domain: x0 < x = X1, I y-yo I - k. The further reasoning proceeds then as under b). It should also be noticed that in the present case it is impossible to inscribe in k a neighborhood (p) of %0, since the width of Ok approaches zero as x approaches xo. We shall say that the set of extremals through A forms an improper field about %. The inverse function q (x, y) and the slope p (x, y) are in this case single-valued and of class C' in 'k except at the point (x0, yo) where they are indeterminate. But if the point (x, y) approaches the point (x(, yo) along a curve e of class C' lying entirely in ik, then both functions approach determinate finite limiting values. The limit of 4r (.x, y) is the parameter y of that extremal of the set which is tangent to d at (x0, yo); the limit of p (x, y) is the slope of the curve ( at (x0, yo). 84 CALCULUS OF VARIATIONS [Chap. III ~20. WEIERSTRASS'S THEOREM We are now prepared to prove a fundamental theorem whose discovery by WEIERSTRASS in 1879 marks a turningpoint in the history of the Calculus of Variations. It gives an expression for the total variation of the integral J in terms of the E-function, from which sufficient conditions for an extremum can be derived. a) The gist of Weierstrass's method can be best understood from a simple example, in which the difficulties concerning the existence of a field, which complicate the proof of Weierstrass's theorem in the general case, can be entirely avoided. EXAMPLE VI: In order to prove that the straight line' 01 actually minimizes the integral fXl J= I / + y2 dx s/ao we draw from the point 0 to the point 1 any curve 6::' y =f(x), not coinciding with the straight line 01. We suppose for 2 simplicity that 6 is of class C'. / T ll~Through an arbitrary point 2: (x, Y2) of ( we can draw one / / _ and but one extremal of the set FIG. 18 of extremals through the point 0, viz., the straight line 02. We now consider the integral J taken from 0 along the straight line 02 to 2 and from 2 along the curve d to 1, that is, we form, in the notation of ~2, f), J02 + J21 1 the stroke always indicating' integration along the curve (. 1 For the notation compare ~ 2, e). 2 Notation according to WEIERSTRASS; KNESER, on the contrary, uses the stroke to indicate integration along an extremal. ~20] SUFFICIENT CONDITIONS The value of this integral is a single-valued function of x2, which will be denoted by S(x2), As the point 2 describes the curve 6 from 0 to 1, S(x2) varies continuously1 from the initial value S (0) = J0, (along C) to the end value S (X1) = Jo, (along 0) Hence the total variation A J= J 1 - Jo is expressible in terms of the function S(x) in the form AJ=-[S(x,)-S (x0)]; and we shall have proved that AJ 0 if we can show that S(x2) always decreases or at least does not increase as x2 increases from x0 to x1. For this purpose we form the derivative of S(x2). The integral J02 is the length of the straight line 02: Jo2 = (2- X)2 + (Y2 - yo); hence dJ _( )+ (Yx - ) +( )f' (x2) dx2 (x2- X)2 + (Y2- yo)2 since Y2 =f (x2) If we denote the slopes of the straight line 02 and of the curve G at 2 respectively by P2 and P2, i. e., 2 = y o 2 =f' (x2) X2 - Xo the previous result may be written d J2 _ 1 +P2P2 dx2 i l+p2 On the other hand, j,,1 S 1 + '2 (x)dx, 1 See the explicit expressions for Jo2 and J21 below. 86 CALCULUS OF VARIATIONS [Chap. III and therefore d J21 _ q-^ d V12P2 dx2 Hence we obtain the result dS(x,2) + 1/ i +-lP2~PI dx2 - (i/1+pl 1tpl} dax2 0 1 + P2 1/ + from which we easily infer that dS(x2) < 0 if p2 =P2, dx2 ( =0 if P2 = P2 ~ The latter alternative cannot take place1 all along the curve (S. Hence it follows that AJ>0. The reasoning can easily be extended to the case in which the curve d has a finite number of corners. It is thus proved that the straight line 01 firnishes a proper2 absolute2 minimum for the integral J. The preceding construction may be modified3 as follows: On the continuation of the line ~o beyond the point 0 4 2 choose a point 5, and replace ///e- - ^in the preceding construc-,/-^ ^^y, _ 1 tion the line 02 by the line FI/ 19 ~52. Accordingly the funcFIG 19 tion S(x2) is now defined by: S (x) - J52 + J21, and therefore S (xo) = Jo5 + Ju,, S (x1) = J51 = J5o + Jo1 Hence we have again A J =-[s ())- S (xo)] If P2 = p2 for every x2 in (X0Xl) it would follow that f(x) satisfies the differential equation f(x)-yo0= (X - XO)f(), and therefore 6 must be a straight line through 0, which could be no other than the line 0o, since g is to pass through 1. 2 Compare ~3, a) and b). 3 Compare p. 82, footnote 1. ~20] SUFFICIENT CONDITIONS 87 For the derivative of S(x2) we obtain the same expression as before, if we let, in the present case, pj denote the slope of the extremal 52. b) We now proceed to the general case. We suppose that for the extremal (o the conditions (II') and (III') are fulfilled. Then we construct as in ~ 19, c) a field 8k about (o by means of the set of extremals (15) through the point A, chosen as indicated in ~19, d) on the continuation of @0 beyond A. Since the extremal go is supposed to lie in the interior1 of the region A, we can take k so small that Ok is entirely contained in B. For our present purpose it will be convenient to use the numbers 0, 1, 5 to denote the points A, B, A respectively. Let now ( be any curve of class C' joining the two points 0 and 1 (see Fig. 19), and lying wholly in the field Ok, and let 2 be an arbitrary point of 6. Through the point 2 we can draw one and but one extremal of the field, i. e., one extremal of the set (15) for which Iy-o| 7c; let it be denoted by 2 ' y= - ( (x, Y2) We then consider the integral J taken from 5 to 2 along (, and from 2 to 1 along (, and denote its value by S (x2): (x2) = J52 + J21 = Fdx + f Fdx, (16) 5 Jx the arguments of F being x, y (X, y), y='= (X, ), ), those of F: x, Y =f(x), y =f(). For x2 - xo and x2 = x1, S(x2) takes the values' S (X0) = J50 + Jo, S (X) = J51, (17) 1 See ~11. 2Properly speaking, S(xj) is not defined for x2 = x. But in order that S (x2) may be continuous also at x2= x1, we must define S (xl) = S (xl - 0); and S (xl -0) is easily seen to be equal to J51. 88 CALCULUS OF VARIATIONS [Chap. 1II so that (18) The functio0 n S (x9) is continuous and admits in (xx1j) at dci irat re whtose value is S'(X2) -E(X2, Y2; P2, P2), (19) where j52 denotes the slope of G, P2 that of e2, at the point 2. WEIERSTRASS' reaches these results in the following way: Let 3 denote that point of G whose abscissa is X2 - h1, h being a small positive quantity; and let G3 Y (X,,Y2 + E) be the unique extremal of the field which passes through the ipoint 8. Then S (XI2 + — ) - IS (X2) - (J21,+ JuTl) - ('J)2 + J21) i5;3 - (J52 + J28) But this is precisely the difference which has been computed2 iln ~8, equation (80), the cnrves (~, (Y, 6 corresponding to the curves there denoted by (, Q, Q. Accordingly we obtain S (X + h) - I(x,)=- h [E (x2, Y2; P2, P2) + (h)J, (20) (h) denoting an infinitesimal. Similarly, if 4 be that point of G whose abscissa is --- h, we obtain S (X2 - h)- S (X2) = J54 + J42 - J52 which, according to the lemma of ~8, is equal to IS (x2 - h)- -S(x2)= + h [E(x2, Y2; P21, 22)+(h)J. (20a) Hence the derivative of S exists and its value is indeed given by (19). 1 Lectuioes, 1879; the proof here given is w e i e r s t r a s s's original proof with the necessary adaptations to the case where x is the independent variable, and with the substitution of the set of extremals through 5 for the set through 0. 2 In applying the lemma of ~8 to the present case, we have to make use of the remarks on p. 18 and p. 3I. The variation A y = - (X, y 2+ e) -'~ (x, yT) Is indeed a variation of the type (5a) of ~4, d). ~201 SUFFICIENT CONDITIONS 89 As the point 2 describes the curve d from 0 to 1, the function E(x2, Y2; P2, p2) varies continuously. For, on the one hand the E-function is a continuous function of its four arguments, provided that the point (x, y) remains in the region A, and the field Fk is contained in t; on the other hand, y2 =f(x2) and 52-= f' (x2) are continuous in (Xoxl) and the slope P2 of (2 at 2 is, according to ~18, b), a continuous function of X2, Y2. Integrating (19) between the limits x0 and xl, and remembering (18), we obtain therefore for the total variation AJ the expression1 AJf7= EF (X 2; p2, P2p2)dx2. (21) 0 We shall refer to this important formula as "WEIERSTRASS'S theoremt. The theorem remains true for curves @ of class D'. For, suppose the curve ( to have a corner at the point 2. Then (20) and (20a) still hold if we understand by -p the progressive and regressive derivatives of f(x2) respectively. The function S(x) is therefore continuous at x2 and admits a progressive and a regressive derivative. Hence it follows2 that (21) still holds when ( has a finite number of corners. c) Instead of first computing the increments S(x2 ~ h)S(x2), KNESER (Lehrbuch, ~20) and OSGOOD (loc. cit., p. 116) compute directly the derivative S'(x2) by applying the theorem on the diferentatation of a definite integral with respect to a parameter. Supposing for simplicity that 6 is of class C', it follows from the properties of the function (x, y) that S(x2) is continuous and differentiable in the 1 The theorem remains true also for the "improper field " Ok formed by the set of extremals through the point 0, and for a curve C which lies entirely in this field k'. For formula (19) holds also in this case at every point of C with the exception of the point0. Integrating (19) from Xo- h to x1, and passing to the limit h=O0 we obtain (21) since p2 approaches a determinate finite limit fcompare footnote 2, p. 83. 2 Compare E. II A, p. 100, and DINI, ~ 194. 90 CALCULUS OF VARIATIONS [Chap. III interval (xoxl) and that the derivative can be obtained by applying to the definite integrals J52 and J21 the ordinary rules' for the differentiation of a definite integral with respect to a parameter and with respect to the limits. Accordingly we obtain in the first place dJ2 = _ (,, F —(i Y2, ^ P2) (22) dx2 In differentiating the integral J52 =i (F (x (I 2),, (X, y2)) dx, we must remember that 72 is a function of x2 defined by the equation 4 (x, 7Y) =f(x2), (23) which expresses the fact that the curves ~2 and G both pass through the point 2. Accordingly we obtain: dJ52 dy, d= F (x2, Y2, 2) + F(It 1Y + F,,, ) d dx, the arguments of Ar, Oxy being x, 72. From our assumptions concerning O(x, y) it follows that Ozy (x, y2)- Gy (,X, Y2) Applying then to the second term under the integral sign the integration by parts of ~4, and remembering that the function y= (x, 72) is an integral of Euler's differential equation d, F1, —F,, dx we obtain the result: dJ- F2(x,, Y2,P2) y(2, Y P2)4 (X2, 7 2) a J2 = F (X2, y2, p2) + d x b -F, (x,, Y5, P5) Py(x,, 72)] where ps - bx (X5, 72). 1 Compare E. II A, p. 102, and J. I, No. 83. ~21] SUFFICIENT CONDITIONS 91 But since the extremals of the set (15) all pass through the point 5: (x5, Y5), we have y5= (x, Y) for every 7y; hence y(x5, y) - 0 for every 7, and therefore in particular Y(x5, Y2)=0 On the other hand, if we differentiate (23) with respect to x2, we get d72 f (x2, Y2) d x - 2-2; therefore d J52 d2 = F (x2, 2, p) + (p2 - 2) Fy (2, Y2, p2). (24) dx2 Combining (22) and (24) we obtain again the fundamental formula (19). ~21. HILBERT'S PROOF OF WEIERSTRASS'S THEOREM Weierstrass's theorem can be extended1 to aly set of extremals constituting a field about the arc (0, i. e., Whenever the extremal ~o can be surrounded by a field, the. total variation AJ J-J —J, for any admissible curve ( lying wholly in the field, is expressible by WEIERSTRASS'S formula: xl AJ= E (x, y; p, p) d, where (x, ~) is a point of (, A? the slope of ( at (x, y), and p the slope at (x, y) of the unique extremal of the field passing through (x, y). iThe extension seems to be due to H. A. SCHWARZ, who has given the generalized theorem in a course of lectures in 1898-99. 92 CALCULUS OF VARIATIONS [Chap. III The following elegant proof of the generalized theorem is due to HILBERT.' Suppose *k is a field of extremals about our extremal o0. In ik we draw any curve G of class D': y —f(x), joining A and B. Now let p(x, y) be an arbitrary function of x, y which is of class C' in ~k, and consider the integral J-= F (x, y, p (x, y)) + (y'-p (x, y)) Fy. (, y, p (x, y))] dx (25) taken along the curve ( from A to B. The value of J* will, in general, depend upon the choice of the curve d; we ask: How must we choose the function p(x, y) in order that the value of J* may be independent of the choice of the curve d and dependent only upon the position of the two end-points A and BY Our integral J* is of the form f l[M(x,y)+N(x,y)y'] dx, it has been seen in ~ ) that the necessary and suffiand it has been seen in ~7, d) that the necessary and sufficient2 condition that such an integral should be independent of the path of integration is that My —N. In the present case we have M(.x, y) = F(x, y, p) -pFy (, y, p), N(x, y) = F (, y, Yp) hence M = F, - p (F, + py F,1,) Nx = Fx I + p, FJ,.l 1 See GOttinger Nachrichten, 1900, pp. 253-297, and Archiv der Mathemnatic und Physik (3), Vol. I (1901), p. 231; also the English translation by MRS. NEWSON, in the Bulletin of the American Mathematical Society (2), Vol. VIII (1902), p. 473; further, OsGooD'S presentation in the Annals of Mathematics (2), Vol. II (1901), p. 121, and HEDRICK, Bulletin of the American Mathematical Society (2), Vol. IX (1902), p. 11. 2Notice that the region k, to which the curves a are confined, is simply connected. ~21] SUFFICIENT CONDITIONS 93 Hence, in order that the value of the integral J* may be independent of the path of integration 6, it is necess(ary and sufficient that the function p (x, y) satisfy the partial differential equation (Px + )Py) F11.Y, + p FyY + F,x - Fl = 0, (26) the arguments of the partial derivatives of F being x, yp(x, y). But this differential equation is identical with the differential equation (14) which is satisfied by the slope at (x, y) of the extremal of the field passing through (x, y). Hence the value of J* will be independent of the choice of the curve 6, if we select for the function p the slope just defined. In the sequel p will have this special meaning. The invariance of the integral J* being established, we select for the curve ( first the extremal (0; then we have all along 0o: y'=p(x, y), because (o is the unique extremal of the field which passes through a point of o0. Therefore (25) reduces to J*=- (x, y, y') dx = J.o axo On the other hand, if we select for ( any curve G of class D', different from o0, and joining A and B, we get XI *J* f, [F(x p)+( -p)F (, -, p)]d x where - y' denotes the slope of d at the point (x, j). Both values of J* being equal on account of the invariance of J*, we obtain an expression for J, in terms of a definite integral taken along (. This expression we use in forming the total variation A J — J -J. Then we obtain 94 CALCULUS OF VARIATIONS [Chap. III ^Jz f "[F(x, yT5p) -F(x, i,p) 1 -- Y - ( -p) F. (a, Y p)] x, which is the desired extension of WEIERSTRASS'S theorem, since the integrand is equal to E(x, I; p, p). ~22. SUFFICIENT CONDITIONS FOR A STRONG MINIMUM1 Weierstrass's theorem leads now immediately to sufficient conditions for a strong minimum: a) Suppose there exists a field Bk about eo such that at every point of Ok E (x, y; p (x, y), ) 0 (27) for every finite value of P, p (x, y) denoting again the slope at (x, y) of the extremal of the field passing through (x, y). Then it follows from W eierstrass's theorem that AJ O 0 for every curve (S of class D' drawn in 0k from A to B, and moreover that AJ> 0 unless E (x, y; p (x, y), y') = 0 (28) all along the curve (S. From the definition of the E-function it follows that (28) holds at a point (x, y) of d whenever Y -=p(x, y), i. e., whenever the extremal through (x, y) is tangent to; at (x, y). This can, however, not take place at every point of (, unless 6 completely coincides with o0. For2 the value of the parameter ry of the extremal of the field passing through that point of d whose abscissa is x, is determined by the equation f(x)= (X,), 1 Compare for this section also HEDRICK, Bulletin of the American Mathematical Society, Vol. IX (1901), p. 11. 2This proof is due to KNESER, Lehrbuch, ~22; see also OSGOOD, loc. cit., p. 118. ~221 SUFFICIENT CONDITIONS 95 from which we derive by differentiation dy f') = p (x, I) + ly (x, y) d or according to (13) dy Y - p (, )= - ~y(A, y) d~ But according to (11), y(x, 7y) *O; if therefore y'=- p(x, y) at every point of (, we should have -y = 0 throughout (xox,) dx or y = const., i,. e., ( would itself.be an extremal of the field, which could be no other than (0, since (S passes through the point (xl, Yl) and ~o is the only extremal of the field which passes through (x1, yl). Hence, if instead of (27) the stronger condition1 E (x, y; p(x, ), p)> (29) is satisfied at every point (x, y) of ~k and for every finite A, it follows that A J> 0 for every admissible curve ( drawn in the field ~k. In the terminology of ~3 we have therefore the result that whenever (27) is satisfied, (o furnishes a minimum for the integral J; if moreover (28) is satisfied2 the minimum is a "proper minimum." EXAMPLE III (see pp. 73, 78): F = y2(y+ 1)2 The set of straight lines y = mx + y parallel to the extremal AB furnishes evidently a field about (o, and for this field p(x, y) =m Therefore 1 Compare (6) and (6a). 2 It is even sufficient that (27) and (29) be satisfied in a neighborhood (p) of % inscribed in 1,.; the same remark applies later on to (Ilb'). 96 CALCULUS OF VARIATIONS [Chap. III E (x, y; p (x, y), ) = ( - m)2 [ 2 + 2 (m + 1) + 3mn2 + 4m + 1] When m> 0 or m <- 1, condition (29) is fulfilled, and therefore the straight line AB actually minimizes the integral 1= ( + 1)2 dx Xo in these two cases. b) The sufficient conditions thus immediately following from Weierstrass's theorem are, however, in general inconvenient for applications, and it is therefore important to remark that they can be replaced, under certain additional assumptions either concerning the curves ( or concerning the function F, by simpler conditions. From the relation (5) between the E-function and Fy,y,, it follows that both conditions (27) and (29) are always satisfied when Fy t(x, y, )> 0 (IIb') at every point' (x, y) of Ak and for every finite value of i). Hence if we remember the theorem concerning the existence of a field (~ 19, b)), we can state the following theorem: FUNDAMENTAL THEOREM V:2 If the extremal (:o AB does not contain the conjugate point to A, and if farther FY,', (x y, I) > 0 (lib,) at every point (x, y) of a certain neighborhood of %o for every finite value of ~p, then Go actually minimizes the integral X inJ=ra F(x, y, y')dx So Corollary: The minimum is moreover a "proper minimum," i. e., A J>0 for every admissible variation of the curve Go in a certain neighborhood of o0. It is even sufficient that (27) and (29) be satisfied in a neighborhood (p) of (o inscribed in Ok; the same remark applies later on to (IIb'). 2See OSGOOD, loc. cit., p. 118; compare, however, below, the remark on p. 99, footnote 1. ~ 22] SUFFICIENT CONDITIONS 97 For a so-called regular problem (compare ~7, c)) it is therefore sufficient for an extremum that the arc AB does not contain the conjugate to the point A. EXAMPLE VII: F=g(x, y) 11+y", g(x, y) being a function of x and y alone, of class C" in a certain region T. Here g (x, y) Fy (x, Y, ) - (-jl 2) Hence every extremal A B which lies in the interior of ~ and which does not contain the conjugate point to A, furnishes a minimum provided that g(x, y)>O along AB. For g(x, y),being continuous in a certain neighborhood of AB and positive along AB, will also be positive in a certain neighborhood of AB, so that (lib') is satisfied. This covers the case of Examples I and VI, in which g(x, y) =y, and 1 (1) respectively; and also the case of the "brachistochrone" in which Ogx, ty) - 1/yo- Yr All three functions are positive along the respective extremals. On account of the extension of Weierstrass's theorem given in ~21, Theorem V may be replaced by the following: If the extremal (o can be surrounded by a field and if Condition (lib') is fulfilled, then @o actually minimizes the integral J. Frequently the existence of some particular field about the arc @o is geometrically evident; in such cases the second form of the theorem is more convenient. 1 Geometrical Interpretation (ERDMANN): Let a straight line move perpendicularly to the x, y-plane along the curve y =f (x) from A to B. The area of that portion of the cylindric surface thus generated which lies between the x, y-plane and the surface: z=g (x, y) is equal to (x,(y) /l- +y"2dx. oX0 CALCULUS OF VARIATIONS [Chap. III EXAMPLE VIII:1 To minimize the integral J ---— l-:1 - y'2dx, o Y the admissible curves being confined to the upper half-plane (y > 0). Here the extremals are semi-circles having their centers on the x-axis. If / - 2 y = -({x- ao)s + TY is the particular semi-circle passing through the two given points, the set of concentric circles y =/-( - ao)2+y + -<4(, y) evidently furnishes a field about (Co. Moreover (lib') is fulfilled throughout the upper half-plane. Hence the semi-circle through the two given points actually minimizes the integral J. Remark: Though the above theorem is the one which is most important for applications, it should be observed that it assumes much more than is necessary. Indeed, the condition (fib') is by no means necessary, not even the milder condition Fy,(x, y, ) 0 (IIa) at every point (x, y) of @o and for every finite i). This is illustrated by Example III (see pp. 73, 78, 95). For here Fs we (a, yi, i) = 2 (6j2 + 6p + 1) can take negative as well as positive values at every point (x, y), and nevertheless, as we have seen above, a minimum takes place when m > 0 or n < -. c) Question of necessary AND sufficient conditions. From WEIERSTRASS'S results concerning the sufficient conditions for the problem in parameter-representation (see ~28), one is led to expect that the conditions2 (I), (III'), I Given by OSGooD, loc. cit., pp. 109, 115, where also a geometrical interpretation will be found. 2 The accent indicates the omission of the equality sign in conditions (III) and (IVa); compare pp. 68, 76. (II') may be omitted, since it is contained in (IVa'); compare ~18, equation (6a). ~22] SUFFICIENT CONDITIONS 99 (IVa') are sufficient for a minimum. Leaving aside the exceptional case when in one of the inequalities (III), (IVa) the equality sign takes place, we should then have reached a system of necessary and sufficient conditions. The analogy of the problem in parameter-representation is, however, misleading in this case. As a matter of fact the three conditions (I), ( IJV), (a') are NOT' sufficient for a minimum without some additional assumptions, not even if (IVa') be replaced by the stronger condition FJ,, (x, y,)> 0 (IIa') at every point (x, y) of Co for every finite value of 5p. To prove this statement it suffices to construct a single example in which the conditions in question are fulfilled and in which, nevertheless, no minimum takes place. Such an example is the following: EXAMPLE IX:' To minimize the integral J = [ay' 2 - 4byy' + 2bxy'"] dx, Li O a, b being two positive constants, with the initial conditions y = O for x =, and y = 0 for x =. Here Euler's equation reduces to - y"Fy, - 0 where'}, _, 2a - 24byy'+ 24bxy/'2 The only extremal through the two given points A(0, 0) and B(1, 0) is the straight line:: y=0. 1This statement seems to contradict directly the theorem given in OSGOOD'S article, loc. cit., p. 118. But it is to be remembered that OSGOOD makes (p. 108) the assumption that F,,',(x, y, p) =:0 in a certain neighborhood of %o. This assumption, together with (Ila'), is equivalent to (Ilb'). 2 See BOLZA, "Some Instructive Examples in the Calculus of Variations," Bulletin of the American Mathematical Society (2), Vol. IX, p. 9. 100 CALCULUS OF VARIATIONS [Chap. III The set of extremals through A is the pencil of straight lines through A; hence there exists no conjugate point, and condition (III') is fulfilled. Further El (, y; y', 5) (a - 8byy'+ 6bxy'2) +q- ( — 4by + 4bxy') + 2bx2; hence along 0o: E (x, fo(x); fo(x), j) -a + 2bxp2 > 0. (IVa') The three conditions (I), (III'), (IVa') are therefore satisfied, even the stronger condition F,,. (x, fo(x),?) x 2a + 24fbp2 0 ~. (IIa') Nevertheless the line eo does not minimize the inter- - -- -.-.-. —. gral J. I/j IA ~~ ^__ \k 10 For, if we replace the line A BI "Is' AB by the broken line A PB, ____________ _ the co-ordinates of P being FIG. 20 x-zh>O and y=-k, the total variation of J is easily found to be AJ =2[ h2[ - +j+ a +- 3bk2 + () where (h) is an infinitesimal. Now let p > 0 be given, as small as we please, then choose [k I<p and let h approach zero, keeping k fixed. Then since b > 0 it follows that AJ < 0 for all sufficiently small values of h, which proves that the line AB does not minimize the integral J. The complete solution of the general problem which we have considered in these three chapters would require the establishment of a system of necessary and sufficient conditions. The above example shows that it will be necessary,to add a fifth necessary condition before the complete solu ~ 22] SUFFICIENT CONDITIONS 101 tion of the problem is reached. We have therefore to conclude this chapter with the statement of a gap in the theory so far as it has been already developed.' d) We add a table of the various conditions which have occurred in the problem to minimize the integral J - F(x, y, y') dx, Xo the end-points being fixed: 1) The minimizing curve o: y -fo(x) must satisfy the differential equation d F1?- dF: D - 0. (I) dx (E ers equation, p. 22; assumptions concerning its general 2) F,,, (x, f/o(), /o (x)) O in ( 1) (II) (Legendre's condition, p. 47) F,,, ( x, f,(x), p) ~, (IIa) in (x0x1) for every finite _p (pp. '76 and 98). F,,J (X, y, j) 0, (IIb) 1If we modify the problem by the addition of a slope restriction, i. e., by subjecting the admissible curves to the further condition that their slope shall not exceed a finite fixed quantity, say Iy' I G, then the three conditions (I), (III'), (IVa') are sufficient for a minimum. For the function El(x, (x, y); z(x, y), P) is continuous in the domain Bk: XO <X x, 17 - I k, I l G, and positive for y = yo. Since the domain 37, is closed, it follows from the theorem on uniform continuity that we can take kc so small that E (x, (x, v); bX (x, t), m)>0 throughout the domain Elk, which proves the above statement. 102 CALCULUS OF VARIATIONS [Chap. II1 for every (x, y) in a certain neighborhood of (0 and for every finite 7 (p. 96). 3) xl Xo, (III) x0 being the conjugate of Xo.. (Jacobi's condition, pp. 58, 59, 67.) 4) E(x, fo(x);/o fx), ) 0, (IV) in (x0Xl) for every finite i. (Weieistrass's condition, p. 76.) El (, f,(x) fo (), ) ~0, (IVa) in (xox0) for every finite p (p. 76). The omission of the equality sign in (II)-(IVa) is indicated by an accent. Conditions (I), (II), (III) are necessary, conditions (I), (II'), (III') are sufficient, for a weak minimum. Conditions (I), (II), (III), (IV) are necessary, conditions (I), (IIb'), (III') are sufficient, for a strong minimum. ~23. THE CASE OF VARIABLE END-POINTS1 We have so far always supposed that the two end-points 1Three essentially different methods have been proposed for the discussion of problems with variable end-points: 1. The method of the Calculus of Variations proper: It consists in computing SJ and 62J either by means of T aylo r's formula or by the method of differentiation with respect to e, explained in ~4, b) and d), and discussing the conditions J-=0, 82J 0. The method was first used by LAGRANGE (1760); see Oeuvres, Vol. I, pp. 338, 345. He gives the general expression for SJ when the end-points are variable, viz.: xI SJ= J S P (F- F,,) dx + [Fax + F,S] ' and derives the conditions arising from 3J=0. The second variation for the case of variable end-points was first developed by ERDMANN (Zeitschriftfir Mathematik und Physik, Vol. XXIII (1878), p. 364). He finds 82 X R (u8y'- u'y)2 dx 6 J = 2 - + [F82X + FI'l 82Y + 2Fyxy8? + 2FyFx8Y'-+ d x+ (F + Fx 2 U) SY21, ere u is an iegral i' s d tial euation. By cons g suh s where u is an integral of J a c o bi' s differential equation. By considering such spe ~23] SUFFICIENT CONDITIONS 103 of the required curve are fixed. In this section we propose to consider the modification of the problem in which one of the end-points, say 0, is fixed, whilst the other, 1, is movable on a given curve (. Suppose the curve o'* y - =fo(x), Xo X X,, -which we suppose to be of class C' and to lie in the interior of the region S-minimizes the integral J with these cial variations for which Sy = Cu, he makes the integral vanish and thus reduces the question to the discussion of the sign of the remaining function of the variations x, Syi, i 2xi, i2Yi' These variations are connected by relations which depend upon the special nature of the initial conditions. For instance, for the initial conditions considered in the text the expression for 82J reduces to the expression (36) for J" (xl) multiplied by 8xx. For the general integral J= (xY, y, ~. ~. J,^,y l 2'. ~,Y l') dx, 0 0 where yl, Y2... y are connected by a number of finite or differential relations, the second variation in the case of variable end-points was studied by A. MAYER, Leipziger Berichte (1896), p. 436; for the integral in parameter-representation J=J F (x, y,x', y') dt to by BLISS, Transactions of the American Mathematical Society, Vol. III (1902), p. 132 (compare ~30). 2. The method of Differential Calculus: This method is explained in a general way in DIENGER'S Grundriss der Variationsrechnung (1867). It decomposes the problem into two problems by first considering variations which leave the endpoints fixed, and then variations which vary the end-points, the neighboring curves considered being themselves extremals. The second part of the problem reduces to a problem of the theory of ordinary maxima and minima. This method has been used by A. MAYER in an earlier paper on the second variation in the case of variable end-points for the general type of integrals mentioned above (Leipziger Berichte (1884), p. 99). It is superior to the first method not only on account of its greater simplicity and its more elementary character, but because-by utilizing the wellknown sufficient conditions for ordinary maxima and minima-it leads, in a certain sense, to sufficient conditions if combined with WEIERSTRASS'S sufficient conditions for the case of fixed end-points. For these reasons I have adopted this method in the text. 3. Kneser's method: This method, which has been developed by KNESER in his Lehrbuch, is based upon an extension of certain well-known theorems on geodesics. It leads in the simplest way to sufficient conditions, but must be supplemented by one of the two preceding methods for an exhaustive treatment of the necessary conditions. A detailed account of this method will be given in Chapter v. 104 CALCULUS OF VARIATIONS [Chap. II1 initial conditions. Then we must have AJ 0 for every curve ( of class D' which begins at the point 0 and ends at a point of the curve 6 and which lies moreover in a certain 2_ - neighborhood' 3 of o0. a) Among the totality of these "admissible curves" we consider in ( o ^^i ^- the first place those which end at the point 1. For these also the inequality FIG. 21 AJ,0 must hold, and therefore all the conditions which we have found to be necessary in the case of fixed end-points must be fulfilled in the present case. The arc Go must therefore be an extremIal, Legendre's condition F^1^J O (II) must be satisfied along Go, and the conjugate point 0' to 0 must not lie between 0 and 1. We suppose in the sequel that the arc (0 is an extremal, that the condition Fl:,(x, yz, () > 0 (IIb') is fulfilled at every point (x, y) of a certain neighborhood of 0o for every finite value of jp and that the arc (0 does not contain the conjugate point 0' (Condition III'). b) Further necessary conditions are obtained by considering variations which do vary the end-point 1. Various methods' have been proposed for this purpose. The following elementary method reduces the further discussion to a. problem of ordinary maxima and minima: If the extremal (o minimizes the integral J in the sense explained above, then ~o must, in particular, furnish a smaller 1 Compare ~3, b); we may for instance choose for t1 the special neighborhood (p) used in the problem with fixed end-points (~3, c)), increased by a semi-circle of radius p with the point 1 for center. 2Compare footnote 1, p. 102. ~23] SUFFICIENT CONDITIONS 105 value than (or at most the same value as) every extremal which can be drawn from the point 0 to the curve ( and which lies in a certain neighborhood of (0. And since under the above assumptions (lib') and (III') each of theso extremals-(when its end-points are considered ai fixed)-minimizes the integral J, it seems' selfevident that also the converse is true. Let then y = c (, y) (30) represent the set of extremals through the point 0, and let y7 denote again the value of 7y which corresponds to 0o. From the above assumptions (IIb') and (III') it follows that this set furnishes for I y — ol k an (improper) field2 1k about the arc @o if k is taken sufficiently small. Hence, if 2: (x2, Y2) be any point of the curve i in a certain vicinity of the point 1, then there passes one and but one extremal y = C (, 2) of the field through the point 2. The parameter r7 is a single-valued function of x2, y2 of class C': 72 = (x2, Y2). If =f (x) is the equation of the given curve, which we suppose to be of class C", then y2=f(x2) and y2- z= (x2,Jf(x2)). Hence the integral J taken along the extremal C2 from the point 0 to the point 2 is a single-valued function of x2, say 1 It will be seen under e) how far this conclusion is correct. 2Compare p. 83, footnote 2. In the present case the field Ok consists of all points (x, y) furnished by (30), when x, y are restricted to the domain xo XX, I y - Y| Ik, where X1 is some value greater than xl; kc is supposed to be taken so small that (IIb') holds throughout 0k and that Oy (x, y) = 0 throughout the domain xO<X<X,1, I-yo -I c. 106 CALCULUS OF VARIATIONS [Chap. TII X2 J(x2) = F(x, ( (x, 72), (x,,2))dx And this function J(x2) must have a minimum for x2-=x. Therefore we must have = 0, (31) J'(x,)=, J"(=) ~ 0. (31) c) The derivative of the integral J(x2) has already been computed1 in ~20, c) (equation (24)). Accordingly J'(x2) = F(x2, Y2, 2) + (i2 - P2) F5' ('.2, Y2, P2), (32) where p2 -x (x2, Y2) is the slope of the extremal 2, and p2zf '(X2) the slope of the curve (, at the point 2. Hence we obtain the result: The co-ordinates x, y of the movable end-point must satisfy the condition2 F (x, Yl, Y[) + (Y -Y; )F(x,1, Yl, Y[) =, (33) where y[ and y{ refer to the extremal @o and to the curve ( respectively. If this condition is satisfied we shall say that the curve E is TRANSVERSE' to the extremal Go at the point 1. Equation (33) together with the two equations f (Xo, a, f/) = YO, f (Xl, a, fl) =f- (xl), determine in general the two constants of integration a, / in the general solution of Euler's differential equation, as well as the abscissa xl of the point 1. We suppose in the sequel that condition (33) is fulfilled. d) We next proceed to the computation of J" (x2). From (32) we obtain iWe suppose that the co-ordinates of the movable end-point do not occur explicitly in the function F(x, y, y');. if they do occur, another term must be added to the expression of J'(X2). Compare for this case KNESER, Lehrbuch, ~12. An example of this exceptional case is the brachistochrone; compare LINDELOFMOIGNO, Calcul des variations, No. 113, and the references given in PASCAL, Variationsrechnung, ~31. 2 In accordance with ~8, end. 3 In the use of the word "transverse" I follow OSGOOD, loc. cit., p. 112. KNESER, who first introduced the term (Lehrbuch, ~10), used it with a slightly different meaning; he says: the extremal 10 is transverse to the curve Z if (33) is satisfied. ~ 23] SUFFICIENT CONDITIONS 107 1dx dx2 " \d dx2 rX,(x ) - 7^+ l 77 + z:,, + F2, ^ (dp, dp, + ( - pF2) LF + FJd,-j+ F, d 2 But Y2 = f(x2) = 4 (2, 72), P2 = (x2, 72), P2= (2) hence = (-2) - f "(x2) (34) dx2 dx2 dp2 dy. dp = zx (x2, 72) + xy (2, Y2) dx2 dx2 and 2 is determined by dX2 f'(x2) = (x (x2, 72) + -y (x2, 72) dx Substituting these values for dy2 dp2 dj2 dx2 ' dx dx2 and remembering that on account of Euler's equation Fy,x - F - ~^,-y Ox ~- F,',, O we obtain for xz x the following result:1 Let A1 and B1 denote the expressions: A1-, F + (2 1- Y[') F1J + Y1r F,. + (Y1 Y1 y)2 B FY, 1(35) B1 = (Y1 -_ y1/)2 Fyy,? the arguments of the derivatives of F being xl, yl, y?; then J )A k (Xi, Yo) J"''(X) = A - ( + B ) (36) ' (X1, 7o) For the further discussion of the inequality J"(xl) 0, we leave aside the exceptional case where y=' y, i. e., we suppose that the extremal o and the curve ( are not tangent to each other at the point 1. Then B > 0, since we have moreover already supposed that Fyy,> 00 1Given, in a slightly different form, by BLISS, Mathematische Annalen, Vol. LVIII (1903), p. 77. 108 CALCULUS OF VTARIATIONS [Chap. III According to equation (80) of 9 15, we have in the notation of 9913 and 14: ~r(xi, yo) C (XIl, x0) and therefore 4)7x (xi, yo)= CZX1, x5) ax, Now let H(xl, x) denote the function H (x1, x) = Aa (x1, x) + B. (x, x)(37) ax, then the expression for J" (xi) may be written J"'(x1) H ( x0) A (xi, x0) The function A (x1, x) r1 (x1) r, (x) - r2 (x1) r1 (x) is an integral of Jacobi's differential equation and vanishes for x x1. The function H(xl, x) is likewise an integral of Jacobi's differential equation, since it is linearly expressible in terms of r1 (x) and r-2 (X). Since B1> 0 and r, (x1) r' (x1) - r2 (x1) q (x1):# 0 (38) (see pp. 57, 58), H1 (x,, xi) t 0 (39) Hence if we denote by xj the root of the equation A (X1, X) O 0 next smaller than x, and by xj' the root next smaller than xj, of the equation H (x1, x) - 0 it follows from Sturm's theorem' that xi > xi At xzX', H(xl, x) changes sign. 'Compare p. 58, footnote 2. This remark is due to BLISS, Tr-ansactions, etc. p. 138. SUFFICIENT CONDITIONS 109 Again from (38) it follows that D A(Xi, x ) t (Xla- x) L (x1 - x) -a, A (1, x) = 1 =x1 axl/ and therefore L(Al+B1-A (ax )/A(x, ))=+-. a=a1-0 ax, Hence we infer that ( > 0 when x1' < x0< x1, J"( -)- =0 when x0= -=, | <0 when x' < xo < x' For reasons which will appear later on (under f), the point of the extremal (o whose abscissa is x[' is called, according to KNESER,' the "focal point" of the curve ( on the extremal o0. We have therefore reached the theorem: For a minimum it is necessary that the focal point of the curve ( on the extremal @o shall not lie between the points 0 and 1. e) It remains to consider the question of the sufficiency of these conditions. If in addition to (IIb') and (33) the condition x[ < xo (41) is satisfied, then J'(x) =0, J"(X) > 0, and therefore the function J(x2) has a minimum for x2-x1. Let now 6 be any curve of class D' which begins at the point 0 and ends at some point 2 of (, and which lies moreover in the improper field 8k about eo defined under b). Let e2 be the extremal of the field from the point 0 to the point 2 (see Fig. 21), then we have Jo < J% 2 1The discovery of the focal point (" Brennpunkt") is due to KNESER, see Lehrbuch, ~24. For the special case of the straight line, the focal point occurs already in ERDMANN'S paper referred to above. BLISS uses "critical point" for " Brennpunkt." 110 CALCULUS OF VARIATIONS [Chap. III On the other hand, since we have supposed k so small that (, (x, y) 0 for xo< x Xl, y - yo[ k, the region 7k is at the same time an (improper) field1 about the extremal @2 and therefore since (lib') holds throughout 'k, JC2 < Jh, according to ~22, b). Hence Jo, < J. The extremal @0 furnishes therefore a smaller value for the integral J than any other curve of class D' which can be drawn in the region ok from the point 0 to the curve (, and in this sense the extre)mal @o minimizes2 the integral J tf the conditions (IIb'), (33) an(d (41) are fulfilled. EXAMPLE VIa: To draw the curve of shortest length from a given point to a given curve. Here: F = 1/ + y 2; hence we obtain for the condition of transversality l + yl '-o, i. e., the minimizing straight line must be normal to the curve G at the point 1. Further we get easily H (X,, x) = y - (x1- ) + ( l; 1 1+ y'2 (i + yu 2 therefore + 2) X1 -x 1 I, In the discussion concerning the construction of a field about %( in ~ 19, we have for simplicity restricted y to an interval (y0 - k, Yo+ k) whose middle point is y = y0. We might just as well have taken an interval of the more general form (y0 - kel, y0 + k2). In the present case the term field must be understood in this slightly more general sense. 2 It should, however, be observed that the region ~k does not, strictly speaking, constitute a neighborhood (see ~3, b)) of the arc (, since its width approaches zero as x approaches the value x0. The proof that Q0 minimizes the integral J is therefore not quite complete. KNESER'S sufficiency proof, which will be given in chap. v for the problem in parameter-representation, is not open to this objection. ~23] SUFFICIENT CONDITIONS 111 Hence it follows that the center of curvature 1" of the curve C( at the point 1 must not lie between the point 0 and the point 1. Conversely: If this condition is fulfilled and if moreover 1" does not coincide with the point 0, then the straight line 01 actually furnishes a minimum. Entirely analogous results are obtained in the case when the point 1 is fixed and the point 0 movable on a given curve. The condition of transversality must be satisfied at the point 0. Again, if A0, B0 have the same meaning for the point 0 as the constants Al, B1 for the point 1, and if x0' denotes the root next' greater than x0 of the equation H (x,, x) - Ao (Xo,,x) + B, (x)=, (42) then x/' must not be less than xl. f) Geometrical interpretation of the focal point. Let us consider the problem to construct through a point 2 of the curve ( in the vicinity of the point 1 an extremal which shall be cut transversely at the point 2 by the curve (. Let y =f(x, a, 3) be the required extremal. Then we have for the determination of a and / the two equations M- (X2, a, P)- (x)= (43), (43) N F (xa, y2, 42) + (p)2- 2) F, (x2, xY, q2) =0, where Y2 = 7(X2) v )2 = I (), 2 2 = fx (2, a, 1) The two equations (43) are satisfied for x - x, a a=, /3 = 0, since 6 is transverse to (o at the point 1; the left-hand sides of the two equations (43) are functions of 2, a, /3 of class C' in the vicinity of x2=x1, a=ao, /3=/3o and their Jacobian with respect to a and 3 is different from zero for X2 = X, a - a0, / = /0, if -1 - y ~ 0 as we have supposed; for it reduces to ('t- y ) Fyy, (rl (x,) r (x) - r2 (xl) r (ti1)) Compare the Addenda at the end of the book. 112 CALCULUS OF VARIATIONS [Chap. III Hence the equations (43) admit, according to the theorem on implicit functions,' a unique solution: a=a (x), /A= (xz), which is of class C' in the vicinity of x2 =x and satisfies the initial conditions a (x,) = a, P (X1) = go If we denote f(x, a(x2), /(x2)) = g(x, x2) the required extremal is therefore y =(x ( X), (44) and if we consider x2 as a variable parameter, this equation represents a set of extremals each of which is cut transversely by the curve (; the extremal (0 is itself contained in the set and corresponds to x2 xl. The envelope ~ of the set (44) is defined by the two equations y -- (X, x), g x2 (x, X2) =, and the abscissae of the points at which the extremal C0 meets this envelope are the roots of the equation x2=x1 gX2 (X, x') =0. To obtain this equation we compute the derivatives da df dx2' dx, from the two equations dM/dxz -0, dN/dx2 = 0, substitute their values in the equation da fd/3 gaX2(, IX2) — f-d +f- = 0, and finally put x2-=xl, a= ao, A -=/o. Carrying out this process, we are led to the three eqlations 1 Compare footnote 2, p. 35. ~23J SUFFICIENT CONDITIONS 113 x1 (x) a (X1) + 1-2 (x) 3 (x1) =, (*- y) F,,''[x (xT) a (x,) + 'r; (x,) /' (xl) = - A1, from which, by eliminating a' (xi), 3' (x1), we obtain the result H (xl, x) = 0, i. e., The focal point1 is the point at whlich the extremal (o meets for the first time-counting from the point 1 towuard the point 0-the envelope of the set of extremals whiich are cut transversel y by the curve C. EXAMPLE Via: The set (44) consists of the normals to the curve (; the envelope? is the evolute of the curve C. g) Case of two movable end-points: We add a few remarks concerning the case when the point 0 is movable on a curve (S and at the same time the point 1 movable on a curve G1. The consideration of special variations leads at once to the result that the minimizing curve must be an extremal, that the condition of transversality must hold at both endpoints, and that the inequalities X0 Xt!, X1 C Xu must be satisfied. But still another condition must be added: If xj" denotes the root next greater than xl of the equation H (xl, x) = 0 then the following inequality must be satisfied:2 1 This geometrical interpretation of the focal point is due to KNEsE R see Lehrbuch, ~24. 2 This result is due to BLISS; see l71atheematische Annalen, Vol. LVIII (1903), p. 70. He also proves that for a regular problem the condition xl<xi'"<<x', together with the two transversality conditions and the condition that the minimizing curve is an extremal, are sufficient for a minimum.\ His proof is based upon Kneser's theory of the problem with one variable end-point. For the example of the curve of shortest length between two given curves, the inequality (45) had already been given by ERDMANN (loc. cit.). Another important example with both end-points variable (the special isoperimetric problem) has been completely discussed by KNESER (Mathematische Annalen, Vol. LVI (1902), p. 169). KY S<,i^f7/ ^-_^ ^^ / 114 CALCULUS OF VARIATIONS [Chap. I1I e pls on v le ed (45) The problems on variable end-points which we have discussed in this section are special cases of the problem: To minimize the integral J when the co-ordinates of the two end-points are connected by a number of relations:1 PI (X'o, yo, Xi, y1) =0 The "method of differential calculus" used in this section can be applied also to this case. The number of independent relations cannot exceed four; if it is exactly equal to four, we have the case of fixed endpoints. If both end-points are perfectly unrestricted, the vanishing of the first variation leads to the four conditions 11 0 1 F = O, F = 0, Fy, - Fy, - 0 which are in general incompatible. 1Compare KNESER, Lehrbuch, ~10. CHAPTER IV WEIERSTRASS'S THEORY OF THE PROBLEM IN PARAMETER-REPRESENTATION1 ~24. FORMULATION OF THE PROBLEM IN the previous chapters we have confined ourselves to curves which are representable in the form y-f(x), a restriction of a very artificial character in all truly geometrical problems. We are now going to remove this restriction by assuming henceforth all curves expressed in parameterrepresentation. a) Generalities concerning curves in parameter-representation.2 A "continuous curve" G is defined by a system of two equations (S: x= - (t), y-= (t), to t t, (1) q and J being functions of t, defined and continuous in (totl). As t increases from t0 to ti, the curve is described in 1 The treatment of the problems of the Calculus of Variations in parameter-representation is entirely due to WEIERSTRASS; he used it in his lectures at least as early as 1872. In order to avoid repetitions, we shall discuss in detail only those points in which the new treatment differs essentially from the old one. For the rest, we shall confine ourselves to an account of the results. As regards the relative merits of the twoo methods, one is inclined to consider the older method-in which x is taken for the independent variable-as antiquated and imperfect when compared with W e i e r st r a s s' s method; unjustly, however, for the two methods deal with two clearly distinct problems, and which of the two deserves the preference, depends upon the nature of the special problem under consideration. Generally speaking one may say that in all truly geometrical problents the method of parameter-representation is not only preferable, but is the only one which furnishes a complete solution. On the other hand, the older method has to be applied whenever a function of minimizing properties is to be determined (for instance, Dirichlet's problemn). For examples illustrating the relation between the two methods, see BOLZA, Bulletin of the American Mathematical Society (2), Vol. IX (1903), p. 6. 2 Compare J. I, Nos. 96-113. 115 116 CALCULUS OF VARIATIONS [Chap. IV a certain sense, called the "positive sense," from its origin, say 0, to its end-point, say 1. If we make the "parameter-transformation": t=X(T), (2) where x%() is a continuous function of T which constantly increases from to to t1 as T increases from T0 to sT, the equations (1) are changed into x= (X ())= (), Y= - (X ())) () (la) Vice versa, the equations (la) are again transformed into (1) by the inverse transformation = -(t) (2a) We agree to consider the two curves defined by (1) and (la) as identical, and conversely two curves will be considered as identical only' when their equations can be transformed into each other by a parameter-transformation of the above properties. The curve (E will be said to be of class C'(C") if the parameter t can be so selected that ~b(t) and +(t) have continuous first (cand second) derivatives in (tot,), and if moreover c' and ~' do not vanish simultaneously in (totl) so that '+ ' 2+ 2 0 in (tot,). (3) A curve of class C' has at every point a continuously turning tangent; the amplitude 0 of its positive direction is given by the equations cos = -, sin 0 =. (4) 1/P'2 + q'2 1/p'2+ + 2 " Every curve of class C' is rectifiable,2 and the length s of the arc tot is expressible by the definite integral 1 According to this agreement, a curve (more exactly "path-curve," E. H. MOORE) is not simply the totality of points defined by (1) but te toality of these points taken in the order defined by (1). 2 Compare J. I, Nos. 105-111. ~24] WEIERSTRASS'S THEORY 117.so= C1~ + 'dt. (5) By an."ordinary curve" will be understood a continuous curve which is either of class C' or else made up of a finite number of arcs of class C'. A point where two different arcs meet will be called a "corner" if the direction of the positive tangent undergoes a discontinuity at that point. A curve will be said to be regular at a point t= t', if for sufficiently small values of t- t', x and y are expansible into convergent power-series: x = a + a, (t -t') + a2 (t - t')2 + 9, y = b+ b (t -t') + b2 (t-t +, and if moreover al and bl are not both zero. b) Integrals taken along a curve; conditions for their invariance under a parameter-transformation. Let F(x, y, x', y') be a function of four independent variables which is of class C'" in a domain I which consists of all points x, y, x', y' for which a) x, y lies in a certain region I of the x, y-plane, b) x', y' are not both zero. We suppose that the curve ( defined by (1) lies entirely in A, and select two points 2 and 3 (t<< t3) on (. Then we consider the definite integral J=- f( xy, ', y)dt, ft2 in which x, y, x', y' are replaced by ((t), (t), +'(t), #'(t) respectively, and ask: Under what conditions will the value of the integral J depend only on the arc 23 and not on the choice of the parameter t The simplest example of an integral which is independent of the choice of the parameter is the length of the arc 23, which is always expressed by the definite integral r t3 J /X'2 + y'2 dt t2 118 CALCULUS OF VARIATIONS [Chap. IV no matter what quantity has been selected for the independent variable t, provided that t2< t3, so that if we pass from the parameter t to another parameter r by any admissible transformation (2), we must have )t2 dt 2 L XN 2 'xU +\ 27 dtxj\ a/ \d / diReturning now to the general case, our question may be formulated explicitly as follows: Under what conditions is f _ dx dy\ 73 dx dy1 /^ F(x, Y' (dt dt) ' (d ' d ' d ) d (6) with the understanding that this relation is to hold: a) For every transformation t= X() of the properties indicated above; 3) For all positions of the two points 2 and 3 on the curve (; y) For all possible curves g of class C', lying in i? On account of /) we may differentiate (6) with respect to r3; writing for brevity t, r instead of t3, r3, we obtain F xgy, t,, dt d' ( _( dx dy\edt ( - dx dy\ or since or sice dx dx dt dy dy dt dr dt d ' C dr dt dr F_/ dxdY ty dy t = dx dt dy dt (x'y, 'dt d' dr =F ' dt d-dr't dr (7) On account of a) this must hold for the special transformation t = k, k being a positive constant. Hence F(xl l dt dy\ F( dx dy\ F(xIy, kf k/ — kF xy, t,~ I dt dtI d ( ~24] WEIERSTRASS'S THEORY 119 But by properly choosing the curve (1) (see assumption 7)) and the parameter t, we can give the four quantities dx dy,' y' dt ' dt any arbitrary system of values in the domain G, and therefore the relation F(x, y, kx', ky') = kF(x, y, x', y') (8) must hold identically for all values of the independent variables x, y, x', y' in I and for all positive values of 7c, or as we shall' say: F(x, y, x', y') must be "positively homogeneous" and of dimension one with respect to x', y'. Vice versa, if this condition is satisfied, (7) holds since we suppose dt >0, dr and therefore also (6), as follows by integrating (7) between the limits T2 and r3. This shows that the homogeneity condition (8) is necessary and sufficient for the invariance of the integral J.' We shall in the sequel always suppose that the function F satisfies the homogeneity condition (8), and we shall denote the value of the integral Ji F (b(t), (t, ) t), ta (t))dt indifferently by J, or Jo,, and call it the integral of the function F(x, y, x', y') taken along the curve (. If we wish to reverse2 the direction of integration we must first introduce a new parameter which increases as the WEIERSTRASS, Lectures; also KNESER, Lehrbuch, ~3. This lemma has been extended to the case where F contains higher derivatives of x and y by ZERMELO, Dissertation, pp. 2-23; to the case of double integrals by KOBB, Acta Mathematica, Vol. XVI (1892), p. 67. 2 Compare KNESER, Lehrbuch, p. 9. 120 CALCULUS OF VARIATIONS [CMhap. IV curve is described from the point 1 to the point 0, for instance: t =-t. The equations gE-1 = (i ), I = _(-_t), y^ o i t - u where i0o — t1, t — to, represent the same totality of points as (1), but the sense is reversed. The integral of F(x, y, x', y') taken along (-1 has the value CF1 (/ dx dy\ J10 Fxy, x, )dz, 0 ' du di/ = ( (t ), - ), -'(t) If the relation (8) holds also for negative values of I;, as happens, for instance, when F is a rational function of x', y', then (x, y, -x', -y')- -F(x, y, x', y'), and therefore: J10 - Jo1. But the relation (8) need not hold for negative values of k; thus in the example of the length we have for negative values of k F(x, y, kx', ky') - kF (x, y, x', y'); hence in this case J10-oJo1. In other cases the relation is more complicated, for instance, when F = xy'-x'y + X /x'2+ yJ2 From the homogeneity condition (8) follow a number of important relations between the partial deriivatives of F. Differentiating (8) with respect to k and then putting k 1, we get x'Fx +y', - F. (9) Differentiating this relation with respect to x and y, we obtain F = — x'Fx, + y'.Fv,, F, = x)'Fi + y'F.,y. (10) ~24] WEIERSTRASS'S THEORY 121 Differentiating (9) with respect to x' and y' we get x'Fx,,- + y'Fy, =- 0, x'F,,,. + y'F,,Y = 0; hence if x' and y' are not both zero, 7'',: F',,:.Fi, = y 2 x'' x' 2; (' ) there exists therefore a function F1 of x, y, x', y' such that = Y2F1, F = - 'y'F, F?, = x 2F1,. (11a) The function F1 thus defined is of class C' in the domain I, even when one of the two variables x', y' is zero; but F1 becomes in general infinite when x' and y' vanish simultaneously, even if F itself should remain finite and continuous for x'= 0, y' 0. For instance: F ^=y'X1f2 + y72, F F = y I/x'2 + y'2 F =,- ( 1/x2 _+ y2) c) Definition of a Minimum:' Two points A(xo, Yo) and B(xl, Yi) being given in the region T, we consider the totality M of all ordinary2 curves which can be drawn in b from A to B. Then a curve ( of Mt is said to minimize the integral /it J= F (x, y, x', y') dt, if there exists a neighborhood 3X of ( such that J~ c J~ (12) for every ordinary curve (S which can be drawn in d from A to B. We may, without loss of generality, choose for 3i the strip3 of the x, y-plane swept over by a circle of constant radius p whose center moves along the curve (S from A to B. This strip will be called "the neighborhood (p) of (S." 1Compare ~3. The definition is due to WEIERSTRASS, Lectures, 1879; compare also ZERMELO, Dissertation, pp. 25-29, and KNESER, Lehrbuch, ~17. 2 An extension of the problem to a still more general class of curves will be considered in ~ 31. 3In case different portions of the strip should overlap, the plane has to be imagined as multiply covered in the manner of a Ri e m a n n- surface (WEIERSTItASS). 122 CALCULUS OF VARIATIONS [Chap. IV ~25. THE FIRST VARIATION We suppose that we have found an ordinary curve (E:: x = +(t), y = (t), to t t,, contained in the interior of A, which minimizes the integral J. We replace the curve ( by a neighboring curve: x - = x +e, y=y+ -, where | and V? are arbitrary functions of t of class D', which vanish at to and tl: e (to) = 0, 7 (to)= 0; = (tl) -= 0, (tl) 0. (13) The consideration of special variations of the form e=ep, rj,=eq, (14) where e is a constant, and p and q are functions of t of class D', which are independent of e and vanish at t0 and t1, leads as in ~4 to the result' that A J =- 8J+ e(e), (15) where (e) is an infinitesimal and rtl 8J-= J (Fxi + Fy + Fx.'+, -V') dt (15a) *to whence we infer again that 8J must vanish for all admissible functions ~, a. Considering first special variations for which v - 0, and secondly special variations for which -0, we see that we must have separately (Ft + F.') dt -= 0, J (F +,+, ) dt = 0. (16) 1The same results hold for variations of the more general type 1=4(t, ),.=.(t, e), where the functions t(t, e), 7(t, e), their first partial derivatives and the crossderivatives tt,, /te are continuous in the domain to < t c tl [ Ie I e, e being a sufficiently small positive quantity. Moreover (to, e)O, (to, e)_O, 0(tl, e) 0, (ti, e) 0. Compare ~4, d). ~25] WEIERSTRASS'S THEORY 123 To these two equations the methods of ~~4-9 can be applied with the following results: a) Weierstrass's form of Euler's equation: The functions x and y must satisfy the two differential equations d d -dt - dt — 0, Fy —F =0; (17) these two differential equations are however not independent; for, if we carry out the differentiation with respect to t and make use of the relations (10) and (lla) we obtain d d Fx -E Xy'rT, F - - F - x'-T (18) dt dt where T_ F,, -F' +F, (xy -" - y), (19) x", y" denoting the second derivatives of x and y with respect to t. Since x' and y' do not vanish simultaneously (see ~24, a)), the two differential equations (17) are equivalent to the one differential equation T - Fx.,- F, + F, (x'y"-x y')= ~. (I) This is WEiERSTRASS'S form of EULER'S differential equation.l Every curve satisfying (I) will again be called an extremal. The same result can also be derived from a transformation' of SJ which will be useful in the sequel. If we perform in the expression (15a) for 8J the wellknown integration by parts, and make use of (18), we obtain s &J= r[F + q eFn] + Tw dt, (15b) where w - y' -x'. 1 WEIERSTRASS, Lectures; compare ZERMELO, Dissertation, p. 37. If we introduce the curvature 1 x'y"'-x"y' e my a b +)3'w the differential equation may also be written I -1%.f? -y -FX ~r ( V'x'2 -Y')3' (Ia) 124 CALCULUS OF VARIATIONS [Chap. IV The differential equation (I) together with the initial conditions determines the minimizing curve, but not the functions x and y of t. In order to determine the latter, we must add a second equation or differential equation between t, x, y. This additional relation (which is equivalent to some definite choice of the parameter t) must be such that x and y come out as single-valued functions of t of class D' satisfying (3); otherwise it is arbitrary. The best selection depends largely upon the nature of the particular example under consideration (see the examples in ~26). If we add to (I) a finite relation between t, x, y we obtain as the general solution a pair of functions of t containing two constants of integration: x= f(t, a, i), y= (t, a, ). (20) The constants a, / together with the unknown values to and tl have to be determined from the condition that the curve must pass through the two given points: 0 =f(to a, P), Yo =g (to, a, ),21 X =f(tl, a, ), Y =g (tl, a,) () b) Extrenal through a given point in a given direction: In order to construct an extremal through a given point O(a, b) of i in a given direction of amplitude y, we select the arc of the curve measured from the given point for the parameter t and have then to solve the simultaneous system T =0 x'+y'2= (22) with the initial conditions x = a, y= b, x'cosy, y'= sin y for t-O. Differentiating the second differential equation we obtain the new system Fx 'x" x yy") -y v ' (, X'xf + y'yf = O0 ~25] WEIERSTRASS'S THEORY 125 Solving with respect to x", y" we obtain x", y" expressed as functions of x, y, x', y' which are of class C' in the vicinity of x - a, y = b, x' -cosy, y' sin y provided that F1i(a, b, cos y, sin y) 0. (23) Hence' there exists a unique solution x=:(t; a, b, y), y ==(t; a, b, y) of the system (22a) satisfying the initial conditions and of class C' in the vicinity of t= 0. This solution satisfies also the original system (22). For, by integrating the second equation of (22a) we get: x'2+ y'2_ const., and the value of this constant is found to be 1 from the particular value t=0. Thus we reach the result 2 If CF, (a, b, cos y, sin y): 0 one and but one extremal of class C' can be drawn through the point (a, b) in the direction y. Hence, if (23) is satisfied for every value of 7y, a unique extremal of class C' can be drawn from O in every direction. If (23) is satisfied at every point (a, b) of the region t for every value of 7, the problem will be called a regular problem (compare ~7, c)). c) "Discontinuous solutions:" As in ~9, a) we infer by the method of partial variation that every "discontinuous solution"3 must be made up of a finite number of arcs of extremals of class C'. Furthermore, the method of ~9, b) applied to the two equations (16) leads to the result:4 1According to CAucHY'S existence-theorem; compare p. 28, footnote 4. 2 See KNESER, Lehrbuch, ~~27, 29. 3I. e., a solution which has a finite number of corners; compare ~24, a). 4WEIERSTRASS, Lectures; compare also KNESER, Lehrbuch, ~43. 126 CALCULUS OF VARIATIONS [Chap. IV At a corner t t2 of the minimizing curve, the two conditions t2-0 t2+o t2-O t2+o F -,F, F, - F (24) must be satisfied, i. e., the two functions Fx, and Fy, must remain continuous even at the corners. We add here the following corollary, though its proof can be given only later (~28): At a corner (x2, Y2) of the minimizing curve, the function F1 (x2, Y2, cos 0, sin 0) must vanish for some value of the angle 0. Hence it follows: If at every point (x, y) of the region B T I(x, y, cos 0, sin l0) 0 for every value of 0, no " discontinuous solutions" are possible. ~26. EXAMPLES In applications it is frequently convenient to use one of the two equations (17) instead of (I), especially when F does not contain x or y, in which case one of the two-equations (17) yields at once a first integral. It must, however, be borne in mind that each of these two equations contains a foreign solution' (y const. and x = const. respectively), and that only their combination is equivalent to (I). a) Example X: To determine for a heavy particle the curve of quickest descent in a vertical plane between two given points (" Brachistochrone ' 2). This happens, for instance, in Example I: F= y1/Vx'y'2, where a first integral is obtained from (17): yx' 1/x 2+ 2 ' when a = 0, y = 0 is such a foreign solution. 2Compare LINDELOF-MOIGNO, loc. cit., No. 112; PASCAL, loc. cit., ~31; KNESER, Lehrbuch, p. 37. ~26] WEIERSTRASS'S THEORY 127 If we take the positive y-axis vertically downward and denote by g the constant of gravity, by vo the initial velocity, which we suppose different from zero, we have to minimize the integral J_ tl /' 2+ y2 'dt Vo y-yo+k where 2 V0 2g The curves are restricted to the region y —yo+ k >. Since F O0, we obtain the first integral x X~, _ -, / - -- = a. (25) Vx 2+Y2 y - y +k The theorem on discontinuous solutions shows that the constant a must have the same value all along the curve. If a O, we obtain x const., which is the solution of the problem when the two given points A and B lie in the same vertical line. If a= O0, we choose for the parameter t the amplitude of the positive tangent to the curve; then we have the additional relation xf - - = cos, 1//2 X+ y'2 which reduces (25) to y - yo + k r(1 + cos 2t), where 1 2a2 Hence y' - 2r sin 2t and x = - 4r cos2t. If we finally make the substitution 2t -rr, we get the result x - x+ h = + r (r- sin ),) Y - Yo+k= r (1 -cos~r) (2) Y — Y j k-k r (1 - COS T), 128 CALCULUS OF VARIATIONS [Chap. IV h being the second constant of integration. The extremals are therefore cycloids1 generated by a circle of radius r rolling upon the horizontal line y - yo + I 0. Among this double infinitude of cycloids there exists2 one and but one which passes through the two given points A and B and has no cusp between A and B, provided only that the co-ordinates of the two given points satisfy the inequalities X1 Xo, Y1 - Yo + k 0 b) Example XI: To determine the curve of shortest length which can be drawn on a given surface between two given points. If the rectangular co-ordinates x, y, z of a point of the surface are given as functions of two parameters u, v and the curves on the surface are expressed in parameter-representation u= b(t), v = (t), (27) the problem is to minimize the integral J = t 1/E Eu'2 + 2Fu'v'+ Gv'2 dt where E = Sx2, Fs =f SX v, G = xv, the summation sign referring to a cyclic permutation of x, y, z. The curves must be restricted to such a portion 0 of the surface that the correspondence between _ and its image a in the u, v-plane is a one-to-one correspondence. We further suppose that E, F, G are of class C'" in 2R and that _ is free from singular points, i. e., EG-F2>O. a) If we use Weierstrass's form (I) of Euler's equation, and denote by 4(F) the differential expression 1This result is due to JOHANN BERNOULLI (1696); see OSTWALD'S Klassiker, etc., No. 46, p. 3. 2 See HEFFTER, "Zum Problem der Brachistochrone," Zeitschrift fir Mathematikc unld Physik, Vol. XXXIV (1889), p. 313; BOLZA, "The Determination of the Constants in the Problem of the Brachistochrone," Bulletin of the American Mathematical Society (2), Vol. X (1901), p. 185; and E. H. MOORE, "On Doubly Infinite Systems of Directly Similar Convex Arches with Common Base Line," Bulletin of the American Mathematical Society (2), Vol. X (1904), p. 337. ~26] WEIERSTRASs'S THEORY 129 41 (F) Fxy Fv,, + F, (x'y" - xr"y') we obtain easily P D(VEu 12 + 2Fu'v'-H GV'2) (28) (V/ Eur2 + 2Fu f+ Gvt7 }2 where r = (E G - F') (u'v"- u v') + (Eu'[Fv') PF(, -2 E ) E)'2 + G,,u'v'+ G vv'1 (29) - (Eut'-] Gv') [ —Eu' I-Ev'It F, —~G~~1 The extremals satisfy, therefore, the differential equation' _0. (29a) This differential equation admits of a simple geometrical interpretation: The geodesic curvature of the curve (27) at the point t is given by the expression 12 F( E 2+v 3 (30) Hence the curve of shortest length has the characteristic property that its geodesic curvature is constantly zero, i. e., it is a geodesic. In passing we notice the relation IIE G - FE2 iC ( f/ wu12+ 2Fu'v'+ Gv'2 ) G (28a) which will be useful in the sequel. 13) If instead of (I) we use the two differential equations (17) and, moreover, select the arc s for the parameter t, we obtain for the extremals the two differential equations:3 d ( du dv\ _ du 2 du dv dv\ 2 2 - E -ds E- d 2F, —+ G -1 ds ds ds ds ds ds ds d I du dv\ Idu 2 dudv dv\ ( ds dS ds/ V \dsl 2 ds ds V \ds/ IThat (29a) is the differential equation of the geodesics might be taken directly from the treatises on differentiai geometry: KNOBLAUCH, Fltchentheorie, p. 140; BIANCHI-LuKAT, Differentialgeomsetrie, p. 154; DARBOUX, Thioi-ie des Surfaces, Vol. II, p. 403. 2See LAURENT, Tr-ait d'Analyse, Vol. VII, p. 132. For an elementary proof see BOLZA, " Concerning the Isoperimetric Problem on a Given Surface," Decennial Publications of the University of Chicago, Vol. IX, p.!3. 3 Compare KNOBLAUCH, loc. cit., p. 142; BIANCHI, loc. cit., p. 153; DARBOUX, lcc. cit., p. 405. DO;3 CALCULUS OF VARIATIONS [Chap. IVT They have likewise a simple geometrical meaning: From the definition of F, F, G it follows that du dv dx ds ds Uds dnd dv _ dx + G- ds ds ds Differentiating with respect to s we obtain Cli flu Fdv \N d2x I - - E ds ds ds+ yG ds2 d1(L'lLF dndv (dv\ 2 + -LE +F,, - + 1 G 2d ds ds 2 \dsJ hence on account of (31) d2X ds2z0, and similarly d2X X~ds2 Therefore d2x d2y d2Z dS2 d ds2 (Yd z - yYVZ?, ) (Itx - ) (LyV - Xvy,). (32) The geometrical meaning of this proportion is that at every point of the curve. the principal nornral coincides with the normnal to the surface, vhich is another characteristic property of the geodesic lines. 27. THE SECOND VARIATION Let eo x f (t, ao, )RO) f (t) tottj (3 ~~o: to t ~~~~~~ tl, ~~~(33) Y g (t, ao, 8o)= g (t) represent an extremal of class C" passing through the two given points A and B, derived from the general solution (20) by giving the constants the particular values a - ao, 8 - 83. We suppose that the functions f(t, a, 3) and g(t, a, /), their first partial derivatives and the following higher derivatives, ftt fta ftP Y ftta i fttp; 9tt, 9ta I 9tP I!7tt. I ttp I ~27] WEIERSTRASS'S THEORY 131 are continuous in a domain To t c T,, a-aoI-d, \ |- Po cd where To< to, T1 > tj, and d is a sufficiently small positive quantity. Then we infer, as in ~11, that in case of a minimum the second variation of J must be positive or zero. The second variation is defined by the integral rt 82j = 1 Fdt, where J 2F -= ~F 42 + 2FFy + Fy l ' + 2F+, e'+ 21FJ, p' + 2F -q 2Fx F+ '+' -+ 2F ' + F,, 2+ 2 + F,,12, (34) the arguments of the partial derivatives of F being x=f(t), y -- =f(t), '=- g'(t) a) WEIERSTRASS'S Transformation of the second vacriation:l This transformation proceeds by the following steps: 1. Express Fxx,, Fxy,, Fyy in terms of F, by means of (lla) and introduce the abbreviations wv =y -- x'r, L = EX, - y'y"F1 N -= F, - x'x"F,, (35) M- = tx,,- +x'y"F -- Fy= + y'x"F; the two expressions for M are equal since x and y satisfy the differential equation (I). We thus obtain 82F = 1 (d )2 + 2L' '+ 2M (4q'+ q +) + 2Nvq' + (Fxx - y" 2F).2 + 2 (Fx, + x"y'"F1) 4q + (Fyy - x" 2F) 7q2 2. Observe that 2L44'+ 2M(ey'+ r,') + 2N —'^ =,dt L + +N]_ [-2 dt+ dt - dt] 1 WEIERSTRASS, Lectures, at least as early as 1872. 132 CALCULUS OF VARIATIONS [Chap. IV and introduce the abbreviations dL dt dt M1 -= Fx!- + xtyt F1- dt ' (36) dN V = F,- x' -_ d — Then the above expression for 82F becomes 2F = F' (dt )+ L, 2 + 2M1 - + Ngq2 + [L2 + 2M$q + Nv2] 3. The three functions L1, M1J, N1 have the important property of being proportional to y'2, -x'y', x'2. Proof. From the definition of L, M11, N and the relations (10) follows Lx'+My'F =, Mx' + Ny'- F. Differentiating the first of these relations we get dL, + MdM +, Lx" + JI Y dt dt dx'+ t '-+Lx My" - Fx'+ Fyy' + F,,x"+.+ F, y" But Lxc"+My"= Fx,x" +- F1,Jy" and from (I) it follows that F,,, —F, = F, (x'y' -xy'). Substituting these values we obtain L1x'+ Mly' 0; similarly Mlx'+ Ny'= O; whence we infer that indeed L1: M1: N = y'2: -_- xJy': x1'2 There exists therefore a function F9 of t such that = y'2F2, M - xy'F, N = xF. (37) ~27] WEIERSTRASS'S THEORY 133 This reduces the expression for 82J to the final form (38) + [L2 + 2M:U + N21 (38) to If, as we suppose for the present, the two end-points are fixed, then ~ and V vanish at to and tl and the expression for 82J reduces to 82J-J [JI l k +j F2t ]dt (39) This definite integral must then be O, for all functions w of class D' which vanish at both end-points. From the assumptions made at the beginning of this section with respect to the functions f(t, a, /3), g (t, a, 3) together with our assumptions concerning the function F(see ~24, b)), it follows that F, and i2 are of class C' in the interval (ToT,); we suppose that they are not both identically zero. b) WEIERSTRASS'S form of LEGENDRE'S and JACOBI'S conditions: The second variation being now exactly of the same form as in the previous problem (~11), we can directly apply the results of Chapter II. Accordingly we infer in the first place, as in ~11: The second necessary condition for a minimum (maximum) is that F (F-0) (II) along the curve eo. We suppose in the sequel that this condition is satisfied in the slightly stronger form F,>0, along 0 (II') Again, Jacobi's differential equation (equation (9) of ~11) becomes ^() -F=. (- F(0 d-. 0(40) 134 CALCULUS OF VARIATIONS [Chap. IV Jacobi's theorem concerning the integration of this differential equation takes now a slightly different form. If we substitute in the differential equation F dt for x and y the general solution x =f(t a, 3), y g (t, a, I) and differentiate with respect to a we get FXXfo + FXlqg + FX Ifta + FX I't.a - (Fx'x + Fx'g1 +FX-x fta + Fxy gta) - 0 In this equation we express the second partial derivatives of F in terms of L,, N, F1, Fl by means of (Ila), (35), (36), (37) and obtain, after some simple reductions, dF dl dc1 l-dt ( )dtj] where o) gtffa t- tga If we operate in the same manner upon the differential equation d ct i.dt.Y,-'O we obtain Therefore, since ft and gt are not both zero, we find that *;-(- dF)I 0. An analogous result is reached if we differentiate with respect to /. Finally, giving a, / the particular values ao, Ae, we obtain Weierstrass's modification of Jacobi's theorem: ~27] WEEERS;TRASS'S- THEORY 135 The differential equation d d[u has the two pcarticular in/tegrals 01 (t) = g ),( f St (t) 9. (t) 02 (t) 9t (t) fp (t') - ft (t) 9p W (41) which are in general linearly independent. Reasoning now as in ~ 14 and 1 we obtain the result: Let ~(t, t() 01 (t) 02 (t0) - 02(t) 01 (t0); (42) then JACOBI'S condition takes the following form:' The third necessary condition for an extremunmn is th at ~ (t, to) #- 0 for to < t < tj. (III) If we denote by t' the zero next greater than to of the equation ~ (t, to) _0 Condition (III) may also be written: tit t,0 t' is the parameter of the "conjugate poi2t" to the point A. EXAMPLE X2 (see p. 126). We suppose that the tvo end-points A and B lie bet ween the two consecutive cusps 7 =0 and T- =2w of the cycloid (26), so that the values T =.To and T = Tj corresponding to A and B respectively, satisfy the inequality 0 < To < T,< 271r For the function F1 wve obtain 1 1 1 yy-Jo+k(V/x'2 ~y'Z)3 8l/2riV2rSin4 2 Hence F~1 is indeed positive along the arc AB. 1 wEIERSTRASS, Lectures; compare also KNESER, Lehrbuch, ~ 31. 2LINDELOF-MOIGNO, loc. cit., p. 231. 136 CALCULUS OF VARIATIONS [Chap. IV Again, we obtain from (26) 7 4 2O 7T0 o (T, o) = _- 4r2 sin - cos 2 sin - cos - 2 2 tan an - ro+22tan ] The parameter -' of the conjugate point A' is therefore determined by the transcendental equation r - t 2 tan T 2tan, 2 2 As r increases from 0 to ir and then from 7r to 2rr, the function r - 2 tan decreases continually from 0 to - co and then from +-c to +27r. Hence r = r0 is the only root of the equation between 0 and 27r. There exists, therefore, no conjugate point on the arc A.B. c) KNESER'S form of Jacobi's condition. As in ~ 15 the existence of a set of extremals through the point A can be proved,' representable in the form x=(k(t, a), yJ= (t, a), (43) 1 WEIERSTRASS obtains the set of extremals through A as follows (Lectures, 1882): Let (E: =f(t, a, ), =.g (t, a, ) represent the extremal passing through A and making at A a given small angle o with the extremal o: X=f(t, ao, P) J y g (t, ao, 13o). Let further to denote that valus of t which corresponds on a to the point A. Then we have for the determination of t, a, 3 the three equations: f(t, a, 3)-o0=0, g(t a,,P)=~, =0, v' —Y' ) 2 - a=0, -'2 /-'2 where the argument of x', y' is to, that of ', y': t, and where a=(x' +y'2) sin o. The three equations are satisfied for to = to, a = ao, 3 =; the functions on the lefthand side are continuous and have continuous partial derivatives in the vicinity of t = to, a =, a, /3 == 3o, and their Jacobian with respect to t0, a, 3 is different from zero at this point, since it is equal to 01 (to) 0' (to) - 02 (to) 01' (to), which is different from zero if, as we suppose, 01 (t) and 02 (t) are linearly independent. There exists, therefore, according to the theorem on implicit functions, a unique solution t~, a, / of the above equations, which leads to two functions 4 (t, a), 4 (t, a) having the properties stated in the text. ~27] WEIERSTRASS'S THEORY 137 where 9(t, a) and I(t, a) are continuous with continuous partial derivatives of the first and second orders-with the possible exception of aa,, aa —in the domain 1To t t T, [a -aO l do, a, being the value of a which corresponds to the extremal (o through A and B, and do being a sufficiently small positive quantity. Again, the Jacobian -— A(t, a) a (t, a) differs1 for a - ao from the function 0(t, to) only by a constant factor: A (t, ac) = C. (t, to), (44) where C 0. Furthermore the value t = to which corresponds on the extremal (43) to the point A, and which satisfies therefore the equations 0 (to, a), Yo = (to, a), (45) is a function of a, which is, in the vicinity of ao, of class C'. From (44) follows KNESER '2 form of Jacobi's condition (t, a,) * 0 for to < t < t (III) Further, if to denotes the value of t corresponding to the conjugate point A', we have A(to), ao) 0, (46) and at the same time At(to, ao)-+ 0, (47) provided that Fi, F2 are of class C' in the vicinity of to and Fi:Z 0 at to. The inequality. (47) follows3 from the fact that A(t, ao) is an integral of Jacobi's differential equation (40). From this second form of Jacobi' s condition it follows4 easily that the conjugate point A' has the same geometrical meaning as in the simpler case of ~ 15. l This follows either by direct computation from the equations which define to, a, P as functions of a, or else from the fact that A (t, a0) and o (t, to) are integrals of J a c o b i' s differential equation and vanish for t = to. 2 See KNESER, Lehrbuch, ~31. 3 Compare p. 58, footnote 2. 4See KNESER, Lehrbuch, ~24, and the references given in E. III D, p. 48, footnote 117. 138 CALCULUS OF VARIATIONS [Chap. IV ~28. THE FOURTH NECESSARY CONDITION AND SUFFICIENT CONDITIONS We suppose in the sequel that for our extremal 0o the conditions F > 0 (II') and ~ (t, to) =t 0 for to < t t,, (III') are fulfilled. a) These conditions are not yet sufficient for a (strong) minimum; a fourth condition must be added. Let E(x, y; x', y'; V', I') be defined' as the following function of six independent variables: E (x, y; x, y'; x', ') = F (x, y, ', M') - X FX, (x, Y( xc,,y') + 'F.(, x(, Y. x', y')], (48) or, as we may write on account of (9), E (x, y; x', y'; x, y)+ Y' [FF (x,, ', ') - F F (x, y, x, y). (48a) + (, y, X I Y ) - F. (x, y, x,y. (48a) Let further (x, y) be any point of the extremal (o, p, q the direction-cosines of the positive tangent to %o at (x. y), and p, q the direction-cosines of any direction. Then the fourth necessary condition for a minimum (maximum) is that E(x, y; p, q; X, q)-0 (-0) (IV) for every point (x, y) of (o and for every direction p, q. The proof follows2 immediately from Weierstrass's lemma3 on a special class of variations: Let This is WEIERSTRASS'S original definition; KNESER writes-E instead of Weierstrass's + E, Lehrbuch, p. 75. 2 Compare ~ 18, b). 3 The reasoning is the same as in ~8; compare also ~ 4, d). ~ 28] WEIERSTRASS S THEORY 139 C. x$=+(t), y =(t), t t t", be any extremal of class C" lying in the interior of the region A, and let 2:(t-t2) be an arbitrary point of (. Through the point 2 draw an arbitrary curve of class C': X — - () ~-, y = (T), the value of T-T2 corresponding to the point 2. Let 3: (x2 + 2, Y2 772) be the point of C corresponding to T -2 + h, where h is a sufficiently small positive quantity. Finally; from a point 0: (t= t < t2) of C to the point 3 draw a curve ( representable in the form G: x = x +, Y- = +, where ~ and r are functions of t and h which vanish identically for h 0, and which satisfy the following conditions1: 1. f, 7V themselves, their first partial derivatives and the cross derivatives,th, lt.7, are continuous in the domain tow tC t2, I c ho, to ^ t h t2? | ^ | ^ ^0, ho being a sufficiently small positive quantity. 2. $(to h)=O, 7q(to, h)=0, 0 e(t2. h) = e2 77(t2, h) =r, X for every 0- h-c ho. Then the difference2 /24 J03 - (J2 + J23) FIG. 22 has the following value: o Jo3 -(Jo2 + J23) -h [E (2, Y2; 2, Y 2 ) + (i)] (49) Similarly, if we denote by 4 the point of ( corresponding to 1 Functions,?1 satisfying these conditions are, for instance, the following: t = 2U, r = 7.2V, if u, v are two functions of t of class C' which vanish for t = to and are equal to 1 for t= t2. 2For the notation compare ~~2, f), 24 a), and 8. 140 CALCULUS OF VARIATIONS [Chap. IV T7-2 -h and draw a curve ( from 0 to 4 of the same character as 6, we obtain: J04- J42-J02- + h [E(x2, Y2; x2, y;; T', 2') +(h)]. (49a) By the same method and under analogous assumptions I L we further obtain the following 2 s results, which are sufficiently explained by the adjoining diaFIG. 23 gram: J23 + J31-J21 = h [E(x2, Y22 x; X2, ' 2 ) + (h), (50) J41 - (J,2 + J21) = - [E (2, 2; x2', y2'; 2, Y2 ) + (h)]. (50a) From the relation (8) it follows that E (x, y; kx, ky'; x, kg') = kE(x, y; x', y; ', y'), (51) if kC>0 and k>0. Hence if we set p - = - - -- = cos0, q = Y = sin 0, V x ^2 + y12 Cs0 2 + y12 f -~r — r ~~~~~~~(52) p = = =cos 0 q= ~ Y _ -= sin 0, I/ + '2 + +'c2 we get E(x, y;x y; ', y') = /2+ E,; p,; p, ), (53) which reduces the second and the third pair of arguments of the E-function to direction-cosines. If we choose for the parameter T on the curve ( the arc, we may replace in the above formulae x2, Y2 and ', y by the direction-cosines P2, q2 and p2, q2 of the positive tangents at 2 to ~ and to 6 respectively. b) Relation between the E-function and the function F]': If the angles 0 and 0 are defined by (52), we have, according to (48), ~ 28] ~28] ~WEIERSTRASS' S THEORY11 141 = Cs [x, x ~y, Cos, sin0 -Fx,(x, y, cosO0, silo0)1 + sin 0 FE5 (x, y, Cos 0,sin 6)-Fy (x, y, Cos 0, Sinl 0)] But Fx,(x, y Cos 0,sill 0) F(r, y, Cos 0, sin.0) *fjtFx' (x, y, Cos (0 + T), sin (0 + T)) cd-i where wo - 0; and an analogous formula holds for F5,. If we perform the diff erentiation with respect to Tr and. then make use of the relations (Ila), we get F x y, cos (O +ir), sin (0 + -i)) sin (w- r) dr By adding to 0 a proper multiple of 2w, we can always cause co to lie in the interval -I- (0 ~< w T so that sin (a) - -r) does not change sign between the limits of integration. We may then apply the first mean-value theorem and obtain the following relation' between th-e E-function and the function F1: E (x, y cos 0,sinO0; cosO sin) -I (- Cos (-0)) F, (x, y, cos 0*, sin 0*),(54) where 0* is a mean value between 0 and 6 From this theorem follow a number of important consequences: 1. If we. let 0 approach 0, we obtain W== 1- Cos(0-0) Hence it follows that Condition (II) is contained in Condition (IVT). 1 WEIERSTRASS, Lectures, 1882. 142 CALCULUS OF VARIATIONS [Chap. IV 2. Condition (IV) is always satisfied when Fi (x, y, cos y, sin y) - 0 (IIa) for every point (x, y) on (o and for every value of y. 3. The E-function vanishes whenever - 0 ("ordinary vanishing")1; for a value 0 0 it can only vanish2 ("extraordinary vanishing") if Fi(x, y, cosry, sin y) vanishes for some value y- 0* between 0 and 0. c) EXAMPLE XII:3 To minimize the integral C yy dt xJt5 2 + y '2 The value of the E-function is easily found to be E~x, y p q;y (P~_ p q)2 [(p2 _ q2) + 2pqP] (E(,,+ q2)2 (iq2 + q2) = y sin2 (0 - 0) sin (20 0). Apart from the exceptional case when both end-points lie on the x-axis, E can be made negative as well as positive by choosing 0 suitably; and therefore no minimum can take place. More generally, whenever the homogeneity condition (8) holds not only for positive but also for negative values of k, as happens, for instance, when F is a rational function of x', y', no extremum can-in general-take place. For in this case (51) holds also for negative values of k, so that E(x, y; p, q; -X, -) -= -E(x, yt; p, q; +, + ~) Condition (IV) can therefore be fulfilled only if E(x, y; p, q; 0, q) = 1 KNESER'S terminology, Lehrbuch, p. 78. 2 Hence follows the corollary on discontinuous solutions stated on p. 126. For from (24) follows E(x, y;P, q; P, 0)=0. 3 To this definite integral leads NEWTON'S celebrated problem: To determine the solid of revolution of minimu nresistance. Compare PASCAL, loc. cit., p. 111; KNESER, Lehrbuch, ~~11, 18, 26; the above expression for E was given by WEIERSTRASS (1882). ~28] WEIERSTRASS'S THEORY 143 along (0 for every direction j, q, which, on account of (54), is possible only in the exceptional case when F1 = 0 along o0. d) Sufficiency of the four preceding conditions:' The four conditions which so far have been shown to be necessary for a minimum of the integral J, are-apart from certain exceptional cases2-also sufficient. Let us suppose 1. That @0 (or AB) is an arc of an extremal of class C" without multiple points, lying wholly in the (I') interior of the region3 TB; 2. F, (x, y,, q) > 0 along'4 0; (II') 3. The arc @0 does not contain the conjugate point A' of the point A. (III') 4. E(x, y; p, q; 7, >)>0 along4 @o (IV') for every direction j, q different from the direction p, q of the positive tangent to @0 at (x, y). Moreover we retain the assumptions made in ~27 concerning the general integral of Euler's differential equation. We propose to prove that under these circumstances the extremal @o actually minimizes the integral ^ti J-= (xx, x', y') dt From the assumptions (III') follows the existence of a field of extremals about the arc (0, i. e., there exists5 a neighbor1WEIERSTRASS, Lectures, 1879 and 1882; ZERMELO, Dissertation, pp. 77-94; and KNESER, Lehrbuch, ~ 20. 2 The exceptional cases are 1. (S0 has multiple points or corners, or meets the boundary of LR; 2. F5 - 0 at certain points of (; 3. A' coincides with B; this case will be considered in ~38. 4. E= 0 at points of Q0 for certain directions jA, q not coinciding with p, q. 3 Compare ~24, b). 4That is, for every point (x, y) of (0, p, q denoting the direction-cosines of the positive tangent to oe at (x, y). 5 Compare ~19. A sharper formulation and a detailed proof of these statements will be given in ~ 34 in connection with K n e s e r' s theory. 144 CALCULUS OF VARIATIONS [Chap. IV hood (p) of C0 such that to every point P of (p) there can be drawn from the point1 A a uniquely defined extremal which varies continuously with the position of the point P and coincides with @0 when P coincides with B. Let now C-: x = (s), s = +(s), So s S, be any ordinary curve drawn from A to B and lying wholly in the neighborhood (p) of ~0, s denoting the arc of the curve ( measured from some fixed point of S, and let AJ denote the total variation A d- J-J-Ji. Then a reasoning2 analogous to that employed in ~20 leads to the following expression for A J (Weierstrass's Theorem): ' i _ — _-U J where (y;, y;, q;p,s a ds, (56) where (x, y) denotes a point of (, p, q the direction-cosines of the positive tangent to (S at (x, y), and p, q the directioncosines of the positive tangent to the unique extremal of the field passing through (x, I). It now only remains to show that, as a consequence of our assumptions (II') and (IV'), the integrand in (56) is never negative3 along the curve (S. Let (x, y) be any point of the above defined neighborhood (p) of @0 and let, as before, p, q denote the direction-cosines of the positive tangent at (x, y) to the unique extremal of the field passing through (x, y), and j, q the direction-cosines of any direction 0, and define 1 Or better from a point A in the vicinity of A on the continuation of e0 beyond A, as in ~19, c). 2The lemma of ~8 must be replaced by the lemma of ~28, a). Other proofs of We i e r s t r a s s's theorem will be given in ~ 37 in connection with K n e s e r' s theory. 3 It is in this last conclusion that the problem in parameter-representation differs essentially from the problem with x as independent variable; compare ~22, c). ~28] WEIERSTRASS S THEORY 145 El(ax, y; p, q; X, q) E r(x, y; p, q;, ~) i -E -( + P.q i when 1-(p + q) 0, (57) F1 (x, y, p,q), when 1 - (pp + qq) = 0 i. e., -p =, c = q. The direction-cosines p, q are single-valued and continuous1 functions of x, y in the neighborhood (p) of o0. Hence it follows, on account of (54), that E1 is a continuous function of x, y, 0 in the domain (xy) in (p), 0 c 2-, and since, according to our assumptions (II') and (IV'), El is positive along (o for every value of 0, it follows from general theorems on continuous functions that E1 is positive throughout the domain (x (,y ) in (, 0 27r, provided that p has been taken sufficiently small. The integrand of (56) is therefore positive at all points of ( at which the direction A, q does not coincide with the direction p, q, and zero where these two directions do coincide. Hence AJ> 0 unless it should happen that-p -p, - q all along (, in which case we should have AJ- O. But the latter alternative is impossible2 unless ( be identical with o0. This proves that the arc (o actually minimizes the integral J if the four conditions enumerated at the begiinnig of ~28, d) are fulfilled. EXAMPLE VII (see p. 97): F = g (x, y) 1/x'2 + y12 Here Eli(X, y; p, q; p, q)=g(x, y), 1 Compare ~ 34, Corollary 4. 2The proof is similar to that given in ~22, a); for the details compare KNESER, Lehrbuch, ~ 22 146 CALCULUS OF VARIATIONS [Chap. IV and therefore Condition (IV') is satisfied if g (x, y) > 0 along (So. This shows that in the problem of the brachistochrone an are AB of the cycloid (26) actually furnishes a minimum if it contains no cusp (compare p. 136). Corollary. If the condition F (x, y, cosy, sin y) > 0 (IIa') is satisfied for every point (x, y) of Co and for every value of y, then (II') and (IV') are a fortiori satisfied, the latter on account of (54). EXAMPLE XI (see p. 128): The Geodesics. Here EG-F2 (V Eu'2 + 2Fu'v'+ Gv'2)3 Hence under the assumptions made on p. 128 concerning the nature of the portion of the surface to which the geodesics are restricted, Condition (IIa') is always satisfied. e) Existence of a minimum "im Kleinen": We add here an important theorem which has been used, without proof, by several authors1 in various investigations of the Calculus of Variations, viz., the theorem that under certain conditions two points can always be joined by a minimizing extremal, provided only that the two points are sufficiently near to each other. An exact formulation and a proof of this theorem have first been given by BLIss.2 His results are as follows: We suppose that in addition to our assumptions concerning the function F (see ~24, b)) the condition F (x, y, cosy, siny ) >0 (58) 1WEIEISTRASS (Lectures, 1879) in his extension of the sufficiency proof to curves without a tangent, see ~ 31: HILBERT in his existence proof (see the references given in chap. vii); OSGOOD in his proof of the identity of Weierstrass's and Hilbert's extension of the meaning of the definite integral J to curves without a tangent (Transactions of the American Mathematical Society, Vol. II (1901), p. 295). 2 Transactions of the American Mathematical Society, Vol. V (1904), p. 113. His proof is based upon an extension of a theorem of PICARD'S concerning the existence of an integral of a differential equation of the second order, taking for two given values of the independent variable two arbitrarily prescribed values (Traits d'Analyse, Vol. III, p. 94). ~281 WEIERSTRASS' THEORY 147 is fulfilled for every point (x, y) in a finite closed region No contained in the interior of E, and for every value of a. Since Fl(x, y, cos, sin Y) is continuous at every point (x, y) of I and for every value of a, a finite closed region, i,, contained in E and containing No in its interior, can be determined such that the inequality (58) still holds for every point (x, y) of T1, and for every value of y. Under these circumstances, if a positive quantity e be assigned arbitrarily, a second positive quantity p, can be determined such that from every point P1l(xi, yi) of io to every point P2(x2, Y2) in the circle (P1, p), where 0 <p pe, an extremal of class C' can be drawn which lies entirely in the circle (PI, p), and which has the property that at every one of its points the slope with respect to the direction P1P2 is numerically less than e. Moreover the circle (Pi, p) lies entirely in the region iL. This extremal is at the same time the only extremal of class C' which can be drawn from P1 to P2 and which lies entirely in the circle (PI, p). Let this extremal be represented by x = - (t; xI, 1; xY2, y2), yT = i (t; x, Y,; xx, Y2), Then there exists a positive quantity I, independent of xl, yl, x2, y2, such that the functions I,,, t,, 't are continuous and have continuous first partial derivatives with respect to t, x1, yI, x2, Y2 throughout the domain tf; (x,, y,) iin o; 0 < (, - )2 + (y 2- l)2 P Finally also the value t = t which corresponds to the point P2 is a continuous function with continuous first partial derivatives of.x1, yl1, Ix2, y2 for all positions of the two points P1, P2 here considered. For the parameter t of a point P of the extremal we may choose the projection of the vector P1P upon the vector PiP2. This unique extremal P1P2 furnishes for the integral J a smaller value than any other ordinary curve ( which can be drawn from P1 to P2 and which lies entirely in the circle (P1, p). If in addition to the inequality (IIb') the further condition F(x, y, cos y, sin y) > 0 148 CALCULUS OF VARIATIONS [Chap. IV is fulfilled for every point (x, y) of the region Ao and for every value of y, and if both points P1 and P2 lie in N0, then the unique extremal PiP2 furnishes for the integral J even a smaller value than any ordinary curve, different from the extremal P1 P, which can be drawn from P1 to P2 and which lies entirely in Io0, provided that I P21 2- Po, where po is a certain positive quantity less than p and independent of the position of P1 and P2. ~29. BOUNDARY CONDITIONS' a) Condition along a segment of the boundary: If the minimizing curve 0231 has a segment 23 in common with the boundary of the region W to which the admissible curves are confined (see Fig. 7), we obtain the condition which must hold along the boundary as follows: i \ s In order to fix the ideas, we sup~..%,........ js/^// pose that as we go along the boun" — / —'^ dary G from 2 to 3, i. e., in the 4"/////.. positive direction of the minimizing curve, the region & lies to our left. FIG./// FIG Let the curve ( be represented by C: = (s), y= (s), s denoting the arc, and suppose that the first and second derivatives of 0(s) and (s) are continuous along 23. Then if we construct at a point (a, y) of 23 a vector of length u, normal to 23 and directed toward the interior of I, the co-ordinates of its end-points are = - +, y = + v, where N _ A_ u= U i/'+ t2 '2 + g r2 Hence if we substitute for u a function of s of the form u = -2, 1 Due to WEIERSTRASS, Lectures, 1879; compare ~ 10 and KNESER, Lehrbuch, ~ 44. ~29] WEIERSTRASS'S THEORY 149 where e is a positive constant and p a function of s of class D' which is 0 in (S2sa) and vanishes at s2 and S3, the preceding formulae represent for sufficiently small values of e a curve which remains in the region X and which is therefore an admissible variation of the arc 23. For this variation we obtain, if we apply (15a), for AJ the expression A J = - 3J Tp 1 2 + 2 ds ()(59) from which we infer, by the method of ~5, that in case of a minimunm we must have T 0 along 23, (60) where T is the expression (19) in which x, y are replaced by 5, y. If F1 is positive not only along the arcs 02 and 31 but also along 23, the preceding condition admits of a simple geometrical interpretation:1 For, if we introduce in the expression for T the curvature 1/ri of ( at a point P, and denote by 1/r the curvature at the same point P of the extremal which passes through P and is tangent to ( at P, then (60) may be written, according to equation (Ia) of p. 123, footnote 1, 1_1 r ' r^~~. ~ (61) r r Hence if ~> 0, i. e., if the vector from the point P to the center of curvature M of ( lies to the left of the positive tangent to ( at P, also r must be positive and the center of curvature M of the extremal must lie between P and Mif or coincide with M. If, on the contrary, r < 0, i. e., if the vector PM lies to the right of the positive tangent, IM must lie either on the iThis is an extension of the results given for the special case F= V22+-y'2 by KNESER, Lehrbuch, p. 178. 150 CALCULUS OF VARIATIONS [Chap. IV opposite side of the tangent to M1 (when > 0), or else on the same side as, but beyond, if (or coincide with MI). If, as we go along the boundary from 2 to 3, the region I lies to the right, the condition becomes: T 0 along 23 (60a) or 1 1 - _. (61a) r r b) Conditions at the points of transition: An additional condition must hold at the point 2 where the minimizing curve meets the boundary, and likewise at the point 3 where it leaves the boundary. To obtain the first, let h be a positive infinitesimal and let 4 be the point of 6 whose parameter is s -s2 -h; join the points 0 and 4 by a curve ( of the type defined in ~28, a), and consider the variation 0431 of the minimizing curve. For this variation we obtain, according to (49) and (53): A J - J04- (J02 + 24) = - h[E(x2, Y2; P2, q2; p2, q2) + (h)], where p2, q2 and P2, q2 are the direction-cosines of the positive tangents at 2 to the curves 02 and 23 respectively. Similarly, if we join the point 5 (s s- h) of C with the point 0 by a curve (, we get, according to (49a), AJ=Jo5+ J52- J2= + h[E(x2, Y2; P2, q2; P2, 42) (h)], whence we infer in the usual manner that at the point 2 the following condition must be satisfied: E (x2, Y2; P2, q2; P2, ~2)-= 0 (62) Applying similar reasoning to the point 3 and making use of (50) and (50a), we reach the result that at the point 3 the analogous condition E (x3, y3; p3, q3; p3, q3) = (63) must be satisfied, where p3, q3 and p3, q3 are the directioncosines of the positive tangents at 3 to 31 and 23 respectively. ~29] WEIERSTRASS'S THEORY 151 The two conditions (62) and (63), together with the condition that the minimizing curve must pass through the given points 0 and 1, determine in general the constants of integration of the two extremals 02 and 31. If the problem is a "regular" one, i. c., if the condition F,(x, y, cosy, sin 7) 0= 0 is satisfied at every point (x, y) of the region i6 and for every value of y, it follows from (54) that (62) and (63) can only be satisfied if p2 2, q2 - q2; P3 —3, (3 =3 - This means geometrically that the arcs 02 and 31;mu1st touch the boundary at the points 2 and 3 in such a manner that their positive tangents coincide with the positive tangents of the boundary. c) Case where the minimizing curve has only one point in common with the boundary: Sup- pose that the minimizing curve has only the point 2 in common with ~ - / the boundary (. Then the arcs 02... and 21 must be extremals. To find the point 2, let 3 be the point of 6 I/ll//i FIG. 24 whose parameter is s s + 7h, and consider a variation 031 of the curve 021 (see Fig. 24). For this variation we obtain A J = J03 + J, - (J02 + J21) = 03 - (J02 + J23)] + [J23 + J31 -J211 which, according to (49) and (49a), is equal to: AJ=h[E(x2, y2; Pl, q2; P2, )2) -E (2, Y2; p2, q2; 2, q2) + (I], where P2, q2; P2, q2; 2z, q2 are the direction-cosines of the positive tangents to the arcs 02, 21, 23 respectively at the point 2. 152 CALCULUS OF VARIATIONS [Chap. IV Similarly, if 4 be the point of 6 whose parameter is s =s2 h-, and we consider a variation 041 of the curve 021, we obtain A J = [Jo4 - J02 + J,42 + [J41 - (J2 + J2)] - h [E (x2, Y2; P2, q2; P2, 2) -E(x2, Y2; P;2,q; 2, P12)+(h)] Hence we infer that at the point 2 the condition ~_ __ q-+ + E(x2, Y2; p2, q2; p2, q2)- E(x2, Y2; P2 q2; p2, q2) (64) mtzst be satisfied. d) EXAMPLE VI1 (see p. 84): F = 1/'2 + y'2 Suppose the region I to be the whole plane with the exception of the interior of a simply closed curve of class C", and suppose that the straight line joining 0 and 1 passes through the excluded region. /' The minimizing curve must be com FIG. 25 FIG. 25 posed of segments of straight lines and segments of the boundary, the latter turning their convex side outward since in this case 1/r = 0 and therefore 1 9F~;~ or 0, according as 23 is described positively or negatively with respect to BL. The lines 02 and 31 must touch the arc 23 positively at 2 and 3 since F1 (x, y, cos a, sin '-) = 1. Again, E (x, y; cos 0, sin 0; cos 0, sin0) = 1 - cos (0- 0) Hence if the minimizing curve is to have one point 2 in common with the boundary, the condition/ cos (0o-,) cos- ( 2 - ) must be satisfied at 2. This means that the lines 02 and 21 must make equal angles with the tangent to the FIG. 26 FIG. 26 boundary at 2. 1 Compare KNESER, Lehrbuch, p. 178. ~301 WEIERSTRASS'S THEORY 153 e) EXAMPLE I (see p. 1): F = J y/v + xy'2; the region a is the upper half-plane: y 0. The extremals are here a) The catenaries t-px=t, y = a cosh -; a 3) The straight lines =-a, y t. Since the catenaries never meet the x-axis, the only possible solution containing a seg- 2 3 Inent of the boundary consists of the ordi- / nates of the two given points: FIG. 27 x = Xo and x = x1, together with the segment 23 of the x-axis between them. Since along the x-axis T=-l, condition (60) is satisfied along 23; and since E (x, y; cos 0, sin 0; cos, sin) (1 - cos (0 — 0))y, conditions (62) and (63) are satisfied at 2 and 3. ~30. THE CASE OF VARIABLE END-POINTS The methods explained in ~23, slightly modified, can be applied to the case when all curves considered are expressed in parameter-representation. In one respect the treatment of the problem in parameter-representation is even considerably simpler, viz.: the variation of the limits of the integral J can be completely avoided. For let 0: x = (t), y = (t), to-tt,, (65) be the minimizing curve, and x = (), y (), T T 71, (66) 154 CALCULUS OF VARIATIONS [Chap. IV a neighboring curve. If we then apply to E the "parametertransformation" (see ~24, a)) t - 0- (to) (T — T0) t = to -+- -— _ 71 - To we obtain for ( a representation in terms of the parameter t for which the end-values are to and t1, the same as for o0. We consider briefly the case where the point 1 is fixed and the point 0 movable on a given curve of class C": xga =(a) (a) (67) The minimizing curve (65) must again be an extremal; it begins at a point 0 of the curve ( whose parameter on 6 we denote by ao. Let 2: (ac ao+-e) be a point of d in the vicinity of 0, x0 -+ o0, Yo + r0 its co-ordinates; then =o-e [+(ao) + (e)] 'o 0= e [l(ao) + (e)] An admissible variation ( of sufficient generality which passes through 2 and 1, can easily D oG8~^^, be constructed analytically in the 2 form - on< (^: =x+, - =y +j, where FIG. 28 ')j ~.2s== -- t, r- = ov, t, v being two arbitrary functions of t of class C' which vanish for t- t and are equal to 1 for t =to. For this variation of the curve (E we obtain, according to (15b), AJ- [Fx, + q 7V + J q (y'e - x',) Tdt + E(E) Substituting the values of a, r at to and tl and remembering that T=O along the extremal o0, we get' AJ=[-whee( +Y') where 1 WEIERSTRASS, Lectures, 1882. WEIERSTRASS'S THEORY IL55. d, dgj da ' da We obtain, therefore, the condition of transversality in the form V'F.x(X, Y, X',y')+'F, (X, yx', y')I2=0 (68) where x', y' refer to the extremal G o ', F, to the given curve G. EXAMPLE XI (see p. 128): The Geodesics. The condition of transversality is it"(Eu' - FvT') + - I(Fu)+ GOl = 0 (69) its geometrical meaning' is that the geodesic must be orthogonal to the given curve. The focal point is determined by the following formulae:2 Let AO and Bo denote the following two constants x Fx, + "F, L5z + 2M,,~, _N~' ' A0X 2+ (70) B - (X'~'- Y'zj)2F1 0 Bo - (X Yll where the arguments of Fx-, Fy,, F1 are xO, yo, x0, Yo and L, MI1, N are defined by (35). Bo is different from zero if we suppose, as in 923, that eo and ( are not tangent to each other at the point 0. Let further Do(t0, t) H (to, t) = A0~ (to, t) + Bo (71) U to the function 0 being defined by (42). Then the parameter t~' of the focal point is given by the equation H (to, t) = (72) if x=(t a) y =~t a) 1 compare BIANCHI-(LUKAT), Differentialgeomnetrie, p. 65. 2 See BLISS, Transactions of the American Mathematical Society, vol. III (1902) p. 136. CALCULUS OF VARIATIONS [Chap. IV is the extremal which passes through the point a of the curve ( and is cut transversely by 6 at that point, and if A (t, a) denotes the Jacobian of the two functions b, - with respect to t, a, then' (t, a)= CH (to, t) (73) which proves the geometrical meaning of the focal point. The question of sufficient conditions will be discussed in detail in connection with Kneser's theory in chap. v. ~31. WEIERSTRASS'S EXTENSION OF THE MEANING OF THE DEFINITE INTEGRAL F(x, y, x', y') dt We have confined2 ourselves in all the preceding investigations to "ordinary" curves. This limitation was indeed necessary for most of our proofs, but it is not implied in the nature of the problem. The most general class of curves for which the problem has a meaning would be the totality of curves for which the integral J= F (x, y, x', y') dt is finite and determinate. In many problems of a geometrical origin, however, a still further generalization is desirable. a) Example of the length of a curve: Thus, for instance, the problem to determine the curve of shortest length between two given points A and B, is not exactly equivalent to the problem to minimize the integral J= I x '2 + y'2 dt t0~1 because the length of a curve cannot in all cases be expressed by this integral. The length of a continuous curve iSee BLISS, loc. cit., p. 140. 2 Compare ~24, a) and c). WEIERSTRASS'S THEORY 157 i: =X = — (t), y=+(t), t, t tl (74) is defined1 as follows: Consider any partition nI of the interval (tot,) into n subintervals by points of division 1T, T2,.., r,_ 71 where to < 7 < 2.* * *. <,-1 < tl and denote by A, Pi, P2, * *, P,,-, B the corresponding points of 2, by XO, Yo; xI, Y1; x2, y2; *. *; X-I Yn,,-1; X1, Y their co-ordinates. Then the length of the polygon $3n inscribed in the curve E whose successive vertices are these points, is 1?-1 Sn- Z I/(A x)2 + (A g)2, V=0 where2 A x = x+1 - -x, A = yV+ - y. If Su approaches a determinate finite limit3 J as all the differences (v+-, - TV) approach zero: J -L S, AT=O the curve 2 is said to have a finite length whose value is J. If the first derivatives ' (t), +'(t) exist and are continuous in (tot,), the above limit always exists and can be expressed by the definite integral4 I/X'2 + y2 dt b) Extension of the meaning of the general integral. In an entirely analogous manner WEIERSTRASS5 has generalized the meaning of the definite integral 1 See JORDAN, Cours d'Analyse, Vol. I, Nos. 105-111. This is the definition which is most convenient for our present purpose; compare also ~44, a), end. 2 With the understanding that 0 = to, xo =XO, Y = YO and Tn=tl, X =Xi], ye= Y l 3That is, corresponding to every positive e, another positive quantity se can be assigned such that I J - Sn I <e for all partitions n in which all the differences (iV+ - TV) are less than 8e. 4Compare JORDAN, loc. cit., No. 111, and STOLZ, Transactions of the American Mathematical Society, Vol. III (1902), pp. 28 and 303. 5Lectures, 1879; compare also OSGOOD, Transactions of the Anerican Mathemat. ical Society, Vol. II (1901), pp. 275 and 293. 158 CALCULUS OF VARIATIONS [Chap. IV t1 J -- F(x, y,x', y')dt, taken along a continuous curve S (defined by (74)) which lies entirely in the interior of the region TR of ~ 24, b). Consider as before a partition II of the interval (tot1) and denote by 1VWn the sum ~ —1 IWn = E> FB(x, v Av, Av) - V=0 (75) Then, if the curve 2 is of classl C', this sum Wn approaches a determinate finite limit as all the differences (r,+i - rT) approach zero, viz., the definite integral2 Js (A B): tl L Wn = F (x, y, x', y') dt. At=0 This remains true when 2 has a finite number of corners. We now agree to define the definite integral (x y, x', ') it F(x, y, x, yi)dt ) (76) 1 This implies that '2 (t) -+ '2 (t) 0 in (totl); compare ~24, a). 2 For the definite integral may be written - (,, ') (-t-1 j = q F (x, u, rJ )dt= A ( (x',(T '(TV)t = ( (T,,-iTV) v= V v=O where TV is some intermediate value between TV and T+1. On the other hand AXV= 4'(T') (TVy+i-T), Ayv= '(T"')(Tv+l -T), where rv' and T'," are again intermediate values between TV and tv+.l Hence we have, on account of the homogeneity of F, n F (vl- I) TV1 = ^ F(4(TV ( ),(P), '(T') 'P{ (TV)) ^-V). v==0 From the theorem on uniform continuity applied to the function F (x, y, x', y') on the one hand, and to the functions Pb (t), 4 (t) and their derivatives on the other hand, it follows that corresponding to every positive quantity e another positive quantity aI can be determined such that for v = 0, 1, 2, *, n- 1, provided that all the differences (T+1 - TV) are less than d%. Hence I W — J I <e (ti- to), which proves our statement. ~31] WEIERSTRASS'S THEORY 159 taken along the curve ~, as the limit of Vrin in all cases in which this limit exists and is finite: and we denote its value by J (AB): J*(AB) L Wn. (77) AT=0 This is a natural extension of the definition of the definite integral since it coincides with the ordinary definition for all "ordinary" curves. c) First modification of Weierstrass's definition: Various modifications of this definition will be of importance in the sequel: Since the curve S is supposed to lie in the interior of the region 1, the rectilinear polygon whose vertices are the points A, P1, P2,., P,,_, B will likewise lie in the interior of T, provided that the differences (Vr+1 - r) have been taken sufficiently small. Let Vn denote the value of the integral J taken along this polygon from A to B. If, then, the curve S is rectifiable, and if one of the two sums VI and Wn approaches for LAr= 0 a determinate finite limit, the other approaches the same limit,1 so that we may also define Jf(AB) =L- V. (78) AT=O d) Second modification of Weierstrass's definition: If the curve S is rectifiable and lies in a finite closed region To (contained in the interior of the region R) in which the condition F,(x, y, cos y, siny) > 0 (58) is fulfilled for every value of Y, then the preceding extension of the meaning of the definite integral J may be modified as follows: Let a positive quantity e be chosen arbitrarily. Then determine for the region 0o the quantity p, defined in ~ 28, e) and choose a positive quantity p a pe arbitrarily. Further select, according to 1 See OSGOOD, Transactions of the American Mathematical Society, Vol. II (1901), p. 293. If V1,+ and vy+l denote the length and the amplitude of the vector PYPv+l, the difference V1- - WI may be written in the form n1-1 VII -1nVII = [F (y, Yv+1, cos Y1, sin vl) v=o. -F(xv, Y, F V y Cos yv+1, sin yv+)] ds, where Xv+l = X + -s cos yO 1, Yv+i = Yv + s sin YVv+l * The above statement follows, then, from the theorem on uniform continuity applied to the function F (x, y, x', y'). 160 CALCULUS or VARIATIONS [Chap. IV the theorem on uniform continuity, another positive quantity a so small that (t)- (t") < p//2, (t')- (t") < p// for every two values t', t' of the interval (tot,) for which It"- t' l < Finally choose the partition rr so that T+1-T V < < for v=0, 1,2,.., n -1. Then the distance I PP,,+~I is less than p, and therefore we can, according to ~ 28, e), inscribe in the curve s a ut ique polygon of minimizing extremals with the points A, P1, P2, * -, P,,-1, B for vertices, i. e., we can draw from Pv to Pv+, a unique extremal,+11 of class C' which lies entirely in the circle (P,, p) and which furnishes for the integral J a smaller value than any other ordinary curve which can be drawn from P, to P,+i and which lies entirely in the circle (Pv, p), Moreover, at every point of 4+1 the slope with respect to the direction PvP,+1 is less than e. We denote by Un the value of the integral J taken along this polygon of extremals, i. e., n-i UnI= Jfi (PPv+1). (79) v=0 Then if we pass, as before, to the limit L Ar 0, and if one of the two sums Un and Wn approaches a finite and determinate limit, the other approaches the same limit,' so that we may also define 1 First remarked by OSGOOD, Transactions of the American Mathem7atical Society, Vol. II (1901), p. 295. The statement can be proved as follows: Let the extremal (v,+ be represented by iv+l-: x = -fVl1 (t) Y = v+_1 (t) 0 - t- lv +_1, where, as in ~28, e), the parameter t of a point P of (,+l is the projection PvQ of the vector PvP upon the vector PVPV+I, and lv+1 is again the distance I PvPv+l I. If we denote by Tvyl the amplitude of the vector PvPv+l and by u the perpendicular QP with the sign + or - according as the point P lies to the left or to the right of the vector PVPV+i, then we have (v+, (t) = xv+ t cos YV+ - u sin "v+~, v +1 (t) = yv + t sin YV+1 + u cos,^v+,;+ -(t) = cos v+l - u' sin yv-1, v+1 (t) = sin yv+l-+ ' cos yv+l Hence if we write v+1- (t)= +v = xV+, v1 = y1+,V -q+-1 (t) = cos Yv+I + v, vl (t) = sin yv-+-+ Ov, ~31] WEIERSTRASS'S THEORY 161 J*(AB)= L Un. (80) AT=0 We shall call the totality of rectifiable curves for which the sum Wn approaches a determinate finite limit, " the class (K)." e) Extension of the sufficiency proof to curves of class (K): After these preliminaries, let @o denote an extremal of class C' drawn from A to B and lying wholly in the interior of the region TS. We suppose that &o does not contain the conjugate A' to the point A, and that for every point (x, y) of &o and for every value of - the condition we have for every t in the interval (O lV+1) I tv 1 c P, I v\ Ic P since Fi+l lies in the circle (Pt, p); and I l<e, I <e, since the slope u' of v,+l at the point P with respect to the direction PVPv+l is numerically less than e, Applying now to the integral Jv+l the first mean-value theorem we obtain JCV. 1 (Pv PV+ ) = 1v+1 iF (xV + Tv, * + NV X cos Y^l + v, sin y,+1 + ), where the argument of v, Iv, v, ov is some value of t between 0 and lv+1. On the other hand, we have on account of the homogeneity of F, F (xy, Yv, A xv, AA yv) = lv+l F (xv, yv, cos Yv+l, sin yv+_) The extremal of v+1l -though it need not lie entirely in the region i% —certainly lies in the larger region 1il defined in ~28, e). Further, the function F (x, y, x' y') is uniformly continuous in the domain: (x,y ) in E1, 1-a / |x'2+- -+-, where a is any positive quantity less than 1. Hence if a positive quantity a- be assigned arbitrarily, the quantities e, p and 8 can be chosen so small that I F (XV + e~, Yv + -v, cos YV+l +?Sv, sin YV+i + Ov) - F (x, YV, cos Yv+i, sin vy^+) I < a, frr v=, 1, * * -, n-1, and therefore n-i I U11- WI I <-E v+l v=O But if, as we suppose, the curve b has a finite length I, we have a-i ' I+1 I v=O and therefore IUn - Wv, I < al which proves the above statement. 1 Without multiple points. 162 CALCULUS OF VARIATIONS [Chap. 1V F1 (x, y, cos y, sin y) > 0 (IIa') is fulfilled. Then we can construct, according to ~28, d) and ~ 34, about the extremal (o a field s which lies in the interior of 2A; and if we take k sufficiently small the inequality (IIa') will be satisfied through — out the region k. Now let 2 be any curve of class (K), not coinciding with (So, beginning at A and ending at B, and lying entirely in the interior of Hk; let it be represented by (74). We propose to prove that J70< J<, (81) Jo being defined as in b). Proof.1 We may apply to the curve 2 the results of d), the field -. taking the place of the region there denoted by So. Accordingly we can choose a partition II of the interval (totl), whose points of division Pv do not all lie on 0o, so that the distance PVPV+{l! < p/3, (v=0,' I.., n- 1), and that at the same time the arc PvPv+i of 2 lies entirely in the circle (Pv, p/3), where p has the same signification as in d), and is, moreover, chosen so small that the circle (Pv, p) lies entirely in the interior of 6.k We may then, on the one hand, inscribe in 2 a polygon of minimizing extremals with the vertices A, P1, P2, ~ ~ *, Pn-, B. This polygon is an ordinary curve; it lies entirely in the interior of ak, and it does not coincide with (l. Hence we have, according to ~28, d), Un > Je, say UnI-JF = p>0. (82) On the other hand, let nI' be a partition derived from I by subdivision of the intervals, and so chosen that Un —J I <p, (83) which is always possible on account of (80). Let Q1, Q2, * ~. Q, -I be the points of division interpolated between the points Pv and IThe outlines of this proof were given by WEIERSTRASS in his Lectures, 1879 Another proof has been given by OSGOOD, Transactions of the American Mathematical Society, Vol. II (1901), p. 292, by means of the theorem given in ~36, c). ~ 311 WEIERSTRASS'S THEORY 163 P,+1 of the partition H. These points lie in the circle (Pv, p13) and therefore Q & Qj~lj 2'p/3 I (i = 0, 1,~~~, m - 1; Q0 = P,, M =- P~,+I) Hence the minimizing extremal from Qj to Qi~1 lies in the circle (Q!, 2p/3) and therefore also in the circle (P,, p). Hence it follows, according to d), that the minimizing extremal from P5 to P,+1 fnrnishes for the integral J a smaller valne than the polygon of minimizing extremals P5 Q19Q2... m-IP,+j, or at most the same value.' Therefore Un' Un. (84) But from (82), (83) and (84) follows (81), since we may write iz - when the) + (Ucv - Usa) r iden - tic I Viz., when the two curvos are identical. CHAPTER V KNESER'S THEORY ~32. GAUSS'S THEOREMS ON GEODESICS KNESER has given, in his "Lehrbuch der Variationsrechnung" a new theory of the extremum of the integral s i J= (x, y,x', y',) dt, essentially different from Weierstrass's theory and reaching farther in its results, inasmuch as it furnishes sufficient conditions also for the case when one end-point is movable on a given curve. Kneser's theory is based upon an extension of certain well-known theorems on geodesics, of which we give-by way of introduction-a brief account in this section. a) Suppose on a surface there is given a curve (o whose points are determined by a parameter v. At a point M(v) of (S we construct the geodesic E normal to (S and lay off on E an arc MP — u.1 The position of the end-point P is uniquely determined by the two c\ s e_ quantities u, v. h"il 'P If we restrict ourselves to such FIG. 29 a region > of the surface that also 60so ~ conversely P determines uniquely the values of u and v, these two quantities may be introduced as curvilinear co-ordinates on the surface ("geodesic parallel-co-ordinates"). According to a well-known theorem due to GAUSS,2 the lines =- const. are orthogonal to the geodesics v -- const. 1. e., the length of the arc is I u I, its direction is determined by the sign of u. 2 GAUSS, Disquisitiones generales circa superficies curvas, art. 16. 164 KNESER' S THEORY 1.65 b) Hence it follows that the square of the line element takes, for this special system of co-ordinates, the form' ds2 = di2q + md2 dv2 We consider now a particular geodesic, C0, of the set v= const., say v 0o, and on it two points 0: (u0, vo) and 1: (ut, v0), where 0o < u1. We join the points 0 and 1 by an arbitrary curve 2: tU -= u (r), v = v (r), (To C r, ). Then the length of the arc 01 of ( is given by the definite integral - r 4d)(/Vdu\ + M? 2.12 On the other hand, the length of the arc 01 of the geodesic @o is J = 1 — Uo. This may be written J l d r J dr, and therefore the total variation becomes2 A J- - rJ = iY\(d\) +' d\l d ru The integrand is never negative, and can be zero throughout the whole interval (Tro1) only when G coincides with 0o. Hence it follows that among all curves which can be drawn in X between the two points 0 and 1, the geodesic (o has the shortest length.3 It should be noticed that the assumption that the geodesic @o belongs to a set of geodesics satisfying the condi1 GAUss, loc. cit., art. 19. 2Compare DARBOUX, Theorie des surfaces, Vol. II, No. 521. 3 The conclusion can easily be extended to the case where the point 0, instead of being fixed, is movable on a given curve orthogonal to the set of geodesics. 166 CALCULUS OF VARIATIONS [Chap. V tions imposed upon the region A, is equivalent to Jacobi's condition. c) The necessity of Jacobi's condition follows from a well-known1 theorem on the envelope of a set of geodesics. If the set of geodesics through the point 0 has an envelope, and 02 and 03 are two geodesics of the set touching the 2 envelope at the points 2 and 3, then Q0 FIG. 30 1 arc 02 + arc 23 marc 03 The point 3 is the conjugate to 0 on the geodesic 03. Now, if 2 be taken sufficiently near to 3 on the envelope;, the compound arc 023 is an admissible variation of 03 for which AJ 0. And since the envelope; is never itself a geodesic,2 the arc 23 can be replaced by a shorter arc 23, and therefore AJ can even be made negative. Hence the arc 03 does not3 furnish a minimum, still less an arc 01 of the same geodesic whose end-point 1 lies beyond the conjugate point 3. The method whose outlines have just been given applies with only slight modifications to the case where only one of the two end-points is given, while the other is movable on a given curve on the surface. ~33. KNESER'S THEOREM ON TRANSVERSALS AND THE THEOREM ON THE ENVELOPE OF A SET OF EXTREMALS We consider in this section KNESER'S extension to any set of extremals of the two fundamental theorems on sets of geodesics given in the preceding section. 1 DARBOUX, Theorie des surfaces, Vol. II, No. 526, and Vol. III, No. 622. 2S'ee DARBOUX, loc. cit., Vol. III, p. 88. 3 Apart from a certain exceptional case; see ~38. 5~33] KNESER's THEORY 167I a) Construction of a transver-sal to a set of extremtals: Let X (ta), y qtf(t, a) (1) be a set of extremals for the integral J - fF(x, yJ, x', yJ') dt containing the particular extremal e = X (t, ao) I y (~ t, ao) tot, t, tj whose minimizing properties are to be investigated.' A and B are again the end-points of %.We suppose that the functions (t, a) and k(t, a) are of class C" in the domain T t -E~ T, + E ao where to - To, T - t1, c and d are positive quantities. We suppose further that for the extremal ~ ~(t, ao) + ~'t (t, ao) # 0 in (tt).(2) ~t (to tj ry/ o cg -i) 2 It follows, then, from the continuity of pt (t, a) and *tt (t a), that the quantities to - To, T1 - tj, c, d can be chosen so small that also 4' (t, a) + tfr it, ca) # 0 (2a) Wt tO>Lj- / O~e~ t aL throughout the domain iii. We denote by ]Ak the rectangle TktO T1, a - ao k < d in the t, a-plane, and by 13'k its image in the x, y-plane defined by the transformation (1). To every point (t, a) of T& corresponds a unique point (s, y) of tk which we shall call "the point [t, a]." To a continuous curve in tgcrre aspndr) a uniqu cuv i in Sk cor~respondss a unique curve in 'Bk: 168 CALCULUS OF VARIATIONS [Chap. V which we call' the curve [t= g (), a- h(r)]. The point * of 6 coincides with the point t =g(r) of the extremal a=-h(T) of the set (1). If for every value of T the curve 6 is transverse' to the extremal a -h(r) at their point of intersection, we shall say that ( is a transversal to the set of extremals (1). We write for brevity F(0(t,a), ~(t, a), (t,(t, a), t (t, a)) =F(t a), (3) and use the analagous notation for the partial derivatives of F and the function Fl. Then the condition of transversality may be written F d dF(t 7 +,(t )d+rJ(ta)d =0. (4) But d, dt da d y dt da dr dr + d ' d dr= dr dr hence, remembering the relation (9) of ~24, we get F (t a) d [ a (t, (t a) + F a) a), a) d= (5) This differential equation for the functions t and a of T is the necessary and sufficient condition that the curve G may be a transversal to the set (1). We now introduce the further restricting assumption' that F(t, ao):t0 in (tot,). (6) 1For the deductions of this section it is not necessary to assume that also conversely to every point (x, y) of _k corresponds a unique point (t, a) of 21C, provided that we consider the points and curves of - only in so far as they are the images of definite points and curves of ok, and this is what our notation is to indicate. Accordingly two points t', a'] and [t", a"] of ik are considered as distincteven if they should have the same co-ordinates x, y-if the points (t', a') and (t", a") of jk are distinct. 2 Compare ~30. 3 We shall free ourselves from this restriction in ~ 37, c). KNESER'S THEORY 169 It follows, then, from the continuity of F(t, a), that we can take To, T1 so near to to, tj and k so small that F (t,a): 0 (6a) throughout the region sk. If the condition (6a) is satisfied, it follows from CAUCHY'S existence theorem' on differential equations that through every point [t', a'] of the -lee- Ak a uniquely' defined transversalodi the set (1) of extremals can be draZwn, representable in the form yx a(t, a) X(a) being single-valued and of class C" in the vicinity of a-c a', and taking for ca-a' the prescribed value t =t'. The curve a may degenerate' into a point, viz., when the functions 0(r), (vr) reduce to constants, say x~, y0. For such a degenerate curve the condition of transversality (4) is evidently always satisfied. Conversely, if any point (x~, y0) in the interior of the region I of ~24, b) is given for which F, (x~, y0, cos y, sin y) = 0 for every y, and if we construct by the method of ~~15 and 27, c) the set of extremals through the point (x~, y0), this point may always be considered as a degenerate transversal to the set of extremals. For there exists, according to ~27, c), a function t~(a) of class C', such that for every a within certain limits ~ = (t~, a), y~ = (t~, a); the point (x~, y~) is therefore indeed the image of the curve t=t0(a) in the t, a-plane. 1 Compare p. 28, footnote 4. 2 Compare footnote 1, p. 168. 3 See KNESER, Lehrbuch, p. 47. 170 CALCULUS OF VARIATIONS [Chap. V b) The function u(t, a): Let A0 be a point on the continuation of eo beyond A, corresponding to an arbitrary value t= to between To and to, and let1 t = t~(a) be the transversal S~ passing through the point [to, ao]. We suppose k taken so small that in the interval (ao —k, ao + k) the function tO(a) is of class C' and To < t (a) < T1. The curve t= tO(a), interpreted in the t, a-plane,divides the rectangle Rk into two regions; we denote [a=aO+k by Bk that one for which \ t~(Cta), / a X and by k its image in the v//>A >< ^ \ ^x, y-plane.;Q"~~~~\lj^~\- a 0, - k 0//~\ 0\ t -- t T, We consider now any point \'I^^ ' P: [t, a]of k. Theextremal t= FIG.31 ~: x= (t,a), y=:(t,a) of the set (1) which passes through P, meets the curve 5~ at the point P~: [t~, a]. Now denote by u or u (t, a) the value of the definite integral u= F (t, a) dt u (t, a). (7) The function u (t, a) is single-valued and of class C' in the domain ek; moreover it represents,3 in 1, the value of our integral J= f F(x, y, x', y')dt taken along the extremal E from the point P~ to the point P: u(t, a)= JP(PoP) 1When the transversal 3~ shrinks to a point, the function t~(a) becomes identical with the function so denoted at the end of a). 2 In Fig. 31 _fc is the non-shaded part of ek' 3 Only in Sj), since we always suppose that the lower limit of the integral J is less than the upper limit; compare ~24, b). ~33] KNESER'S THEORY 171 The partial derivatives of u(t, a) are: Du =F (t,a), (8) au _ Daa F (to, a) da + ( a ) dt But DF(t, a) Da a) = F +- F7+C + F,, ta + F yi/ta t [F+ {a + F Aa] + a F - at F,] + I/ [F,-1 F,] since t ta= - at, 'tc- = at. Now 9 9 Fx- F =O and Fy- -F,, =0 at xat since f (t, a) and (t, a) satisfy Euler's differential equation. Hence we obtain a = (Fx, ~C + FL, ql) -(F d + Fa + F )a But the second term disappears since t t~(a) represents a transversal and therefore satisfies the differential equation (5). Thus we finally obtain u= F (t, a)c(t, a)( + F,1 (t, a). a(t,). (9) If the point P [t, a] moves along a curve ( defined byl t = g (), a = h (), i. e., -x = (g (T), h(T)) = (T), =' (g (Tr), h (T)) - ('), u becomes a function of T whose derivative is, according to (8) and (9): 1 The functions g (r) and h (r) are supposed to be of class C' and to furnish points (t, a) in Lk so long as ' is restricted to a certain interval (T'T") to which we confine ourselves in the following discussion. 172 CALCULUS OF VARIATIONS [Chap. V d= t, ad d a '/,7, or du d, d r=F,(t, a) +F,(t, a). (10) The extensions of the two theorems on geodesics of ~32 follow immediately from this formula by specializing the curve (. c) KNESER'S Theorem on Transversals: In the first place we suppose that the curve 1( is a transversal to the set (1). Then it follows from (4) and (10) that du = O dr and therefore u- const. Thus we obtain the Theorem I: Two transversals ~o and V1 to the same set of extremals intercept on the extremals arcs along which the integral J has a constant value. (7? -C_ ",' ' / ) More explicitly: If G' and G" are two extremals of the set (1) meeting the transversals ~0,;1 at the points PO, P{! and PO', P' respectively, then p-^ 1PJ;, (Po P )= Jp,, (Po'P * ) ____________P Conversely: If along the curve. " 'FIG. 32 \ 1 the function u(t. a) is constant, ~~ FIG. 32 -;' then A1 is a transversal of the set (1). In the special case of the geodesics, transversality is identical with orthogonality,1 and therefore Kneser's theorem is indeed a generalization of Gauss's theorem on geodesic parallels. The theorem remains true if one or both of the two transversals shrink to a point;2 thus we obtain the following corollaries: 1 Compare ~30, a). 2 Compare the remark at the end of a). __ ~33] KNESER'S THEORY 173 Corollary I:' If Z1 is a transversal to the set of extremals through a point P0, then the integral J has the same value if taken along the different extremals from the point P0 to the curve $1, and vice versa. Corollary II: If $~ is a transversal to a set of extremals passing through a point P1, then the integral J has the same value if taken along the different extremals from the curve ~O to the point P1. Corollary III: If the extremals passing through a point PO all pass through a second point PI, then the integral J has the same value if taken along the different extremals from PO to P1. d) Theorem on the envelope of a set of extremals: In the second place, we suppose that the curve 6 is tangent to all the extremals of the set (1), and therefore is the envelope of the set; More, explicitly: As it has been remarked before, the point r of 6 coincides with the point t q (7) of the extremal a -h(7) of the set (1); we suppose that for every value of T, at least in a certain interval (r'r") in which (d + \d1 A 0, the curve 6 and the corresponding extremal are tangent to each other at this common point, so that dx d~r Ot d-y dr It follows, then, that there exists a function m of r such that Ot-M dr dy d7m, ^=m_; 1Applied to geodesics, this is GAUss's theorem on geodesic polar co-ordinates, GAUSS, loc. cit., art. 15. 174 CALCULUS OF VARIATIONS [Chap. V m is continuous in (r'r") and can not change sign.' We may without loss of generality2 suppose that m > 0 in (r'r"), i. e., that the positive directions of the tangents to the two cuarves coincide. From the homogeneity properties of F it follows, then, that Fx (t, a) - F= x Y d, d)' and nd F,(t, a) -= F,,y d ' d and therefore, according to (10), du d., d dy dT ( aI' #dT ' d) Hence, integrating from T -T' to T 7' (rT' T'") and remembering the meaning of t (t, a), we obtain the Theorem 1I:3 Let ~o be a transversal to the set of extremals (1) and ~ the envelope of the set; let, further,; P'Q', P"Q" be two extremals Ip%~ I -of the set starting from the points P', P" of QO and touch~ i\ ^ ^ -^ ing; at the points Q, Q', then4 ',FIG. 33.F JIG, (P" Q) - J,' (P'Q'), a +< J(Q'Q"), (12) 1This follows from (2a) and the assumption that )( +( d7+ 0 in (T") XdTi\ dr i 2 If m is negative, introduce a new parameter T=-or on. 3 The theorem in the special case when 2~ shrinks to a point is due to ZERMELO, who proves it by means of Weierstrass's expression for iAJ in terms of the E-function (Dissertation, p. 96). The theorem in its general form and the above proof are due to KNESER; see KNESER, Lehrbuch, ~25, and also idem, Mathematische Annalen, Vol. L (1898), p. 27. The simplest case of the theorem is the theorem on the evolute of a plane curve. 4By a limiting process it can be shown that the theorem remains true if the assumption ~341 KNESER'S THEORY 175 with the understanding that the positive direction Q'Q" on W has been chosen as indicated above. The theorem remains true if the transversal ZO shrinks to a point, in which case we obtain / the corollary: Ji (Po Q")= J (Po Q') p.o + J(Q' "), (13) FIG. 3 PoQ', PoQ" being two extremals of the set through Po, and H the envelope of the set.1 ~34. CONSTRUCTION OF A FIELD Before we can extend to the general case of extremals the results given in ~32, b) concerning geodesic parallel co-ordinates, it is necessary to impose upon the set of extremals (1) such further conditions that the correspondence between the two EgmS4 ik and F defined in ~33, a) becomes a one-to(dT)+ (dT 0 ceases to be satisfied at Q", i. e., if the curve I has a "cusp" at Q", provided that there exists a positive quantity tk such that d / (.- )C and (T"-T dr d T approach, for LT= T"-0, finite determinate limiting values not both zero (a condition which is, for instance, always fulfilled if Z and y are regular in the vicinity of T"). The proof follows immediately from the homogeneity property of the function F; see ~24, (8). 1The two theorems on sets of extremals proved in this section can be d3rived by still a different method indicated for the case of the geodesics by DARBOUX (Theorie des Surfaces, Vol. II, No. 536). Let %: x=f(t, ao, l3o), Y=g(t, aO I,0o) be a particular extremal derived from the general solution of Euler's equation, and let Mo(t-=to, x=ao, y=bo) and Ml (t=tl, x=al, y=bl) be two points on qo which are not conjugate in the more general sense that ~ (t1, to) O0. Then it follows from the theorem on implicit functions that if we take two points PO (xo, yo) and P1 (x1, yl) sufficiently near to Mo and M1 respectively, a uniquely defined extremal can be drawn through PO and P1: x: =f(t,a,,3), y=g(t,a,3) The constants a, 3, the two values of t which correspond on 15 to the two points 176 CALCULUS OF VARIATIONS [Chap. V one correspondence, or in other words that the set of extremals (1) furnishes a field about the are (0. The proof' of the existence of a field is based upon the following Theorem: Let x = ~(t, a), y= ~(t, a) (15) be a one-parameter-set of curves satisfying the following conditions: A) The functions q and 4 are of class C' in the domain To- E t TT + E, I a - aol l, e and d being two positive quantities. B) The particular curve xs- (t, ac), y = i (t, a) (16) has no multiple points for To-e-t - T1 +e. C) If we denote by A(t, a) the Jacobian a (4) - 4( ' ~') a (tt a) then (t ) (ta) (t, ao) 0 in (To-, T,+ ). (17) Po and P1, and consequently also the value of the integral J taken from PO to P1 along 5 are single-valued functions of x0, yo, x1, y, which are continuous and have continuous partial derivatives in the vicinity of ao, b,,, al, bl. We denote this integral J6. (P0 P1) considered as a function of x0, yo, xl, y1, by J (x0, OY, 1 1); it is a generalization of the geodesic distance between two points (see DARBOUX, loc. cit.). The total differential of this function can be obtained by precisely the same method as that which DARBOUX applies to the geodesic distance, and the result is dJ(x(,, xl, yl) = Fx'(x y,, x l, ')dl-+F(X+, (xl, X1, xl', y1') dy -Fx'(xo, 0, Xo', Yo') dxo-F5,(x5, yJ, o', yo') dyo, (14) the derivatives x0', yo' and xl', y1' referring to the extremal Q. Now suppose that Po and P1 move along two curves o0 and 1, whose co-ordinate; are expressed in terms of the same parameter a. Then the extremals joining corresponding points of Q0 and s1 form a set of extremals with the parameter x, and J(ox, Yo, x1, Y1) changes into a function of T whose derivative is obtained immediately from (14). By specializing the curves o0 and e1 the two theorems I and II are obtained. 1KNESER'S proof (Lehrbuch, ~14) must be supplemented by a lemma such as that given below under a) and b). Compare also OSGOOD, Transactions of the American Mathematical Society, Vol. II (1901), p. 277, and BOLZA, ibid., Vol. II (1901), p. 424. ~ 34] KNESER'S THEORY 177 Under these circumstances a positive quantity k < d can be taken so small that the transformation (15) establishes a one-to-one correspondence between the domain To- t0 T1, a-a0I k in the t, a-plane, and its image Ak in the x, y-plane. t,,+ k r 4 B' -i ----- a+k, i 2, — a-k I X- t T, to t, T, FIG. 35 FIG. 36 Proof: We suppose it were not so; that is, we suppose that however small k may be taken, there always exists in 2k at least one pair of distinct points (t', a'), (t", a") whose images coincide at a point (x, y) of Bk, and we show that this hypothesis leads to a contradiction to our assumptions. a) We first select a sequence of decreasing positive quantities k > k > k2 > * * k, > * * > 0, beginning with k and approaching the limit zero, subject to the following rule: After kl has been chosen, we select in the rectangle i, a pair of distinct points Pi(t, as) and P '(t,1' a"') whose images coincide; this is always possible according to our hypothesis. According to B), a' and a,' cannot both be equal to a0; we may therefore choose k2 smaller than at least one of the two quantities la — aol, a1 — aol, so that at least one of the two points P', P' lies outside of T2. Next we select in.k a pair of distinct points P (t', a') and P ' (t2', a') whose images coincide. As before, we can 178 CALCULUS OF VARIATIONS [Chap. V choose k3 smaller than at least one of the two quantities a. -ao, I a' ao, etc., etc. Proceeding in this manner, we obtain corresponding to the sequence 7k, an infinite sequence of distinct pairs of points P (tl, a; ), P'(t, a ), v = 1, 2... ); the two points P', P'' lie in %k, and their images coincide at a point (x,, y,) of -k. We consider now the set of points Z = (t', ac; t:', a') ) = z^, in the four-dimensional space (t', a'; t", c"). The set Z contains an infinitude of distinct points all lying in the finite domain ~ ' To t' T,; -k a'- ao k; To t" T,; — ka -aock; it has therefore at least one accumulation point1 T=, a; Tr, a), which belongs itself to B since ~ is closed ("abgeschlossen "). b) We are going to prove that a -a0, a "a0,? -T = Out of the sequence zv} we can select2 a subsequence {zv. (i = 1, 2,; v+l> vi) such that L z =, i.e., Lt=, L a - =a', L t,, t L a" =a". i=: i =co i=oo i — But since L kV. 0 and, it ao' - I a. o l^i h it follows that 1 Compare E. I A, p. 185, and II A, p. 45; J. I, No. 27. 2 See J. I, No. 28. ~34] KNESER'S THEORY 179 a -= ao, a 0; besides r' and r" are contained in (ToT1). On the other hand, let D(t', a'; t", a") denote the distance between the two points (x', y') and (x", y") corresponding to (t', a') and (t", a"). Then we have D(t:, a; t', a ') 0 But since D(t', a'; t", a") is a continuous function of its four arguments, we have D(r', a,; r, ao) L D(ti., a'.; t, ) a-;) 0, that is, the images (c',.') and (o", a") of the two points (r', ao) and (r", ao) coincide. According to B), this is only possible if -= T, say =. There exists therefore a point (r, ao) in 1k, in every vicinity of which pairs of distinct points (t, a'), (t", a") can be found whose images in the x, y-plane coincide. c) The theorem on implicit functions1 leads now immediately to a contradiction. For, let (~, r) denote the image of the point (7, cao); take (x, y) in the vicinity of (I, r) and consider the problem of solving the system of equations x = (t, a), yj =(t, a) with respect to (t, a). Since A(r, ao) 0 it follows from the theorem on implicit functions that after a positive quantity e has been chosen arbitrarily but sufficiently small, a second positive quantity 8e can be determined such that, if (x, y) be taken in the vicinity (8e) of (:,;), the above two equations have one and but one solution (t, a) in the vicinity (E) of (7, ao). Further, we can determine, on account of the continuity of q and J, a positive quantity e'ce such that the image 1 Compare p. 35, footnote 2. 180 CALCULUS OF VARIATIONS [Chap. V of every point (t, a) in the vicinity (e') of (r, ao) lies in the vicinity (8e) of (:, q). Hence if (t', a') and (1", a") are any two distinct points in the vicinity (e') of (7, ao), their images (x', y') and (x", y") must lie in the vicinity (8,) of (I, V) and can therefore not coincide, according to the definition of Be. But this is contrary to the result reached under b); the hypothesis from which we started must therefore be wrong and our theorem is proved. Corollaries: 1. From the continuity of the functions c(t, a), A (t, a) and the one-to-one correspondence between Tk and ek it follows that the image 2' of the boundary 2 of the rectangle Bk is a continuous closed curve without multiple points (a so-called "Jordan-curve'"). It divides, therefore,l the x, y-plane into an interior and an exterior. According to a theorem due to SCHOENFLIESS2 the set of points Sk is identical with the interior of 2' together with the boundary 2'. 2. Let to, t1 be two values of t satisfying the inequality T < to < t, < T1, and let %0 denote the are of the curve (16) corresponding to the interval (to, tl). Since the line: a ao, to t- t lies in the interior of 1k, its image Go lies in the interior of 8k and has, therefore, no point in common with the boundary 2'. The two curves Go and 2' being continuous, it follows,3 therefore, that a neighborhood (p) of the arc o can be construtcted which is entirely contained in Ok. 3. Since A(t, ao) 0 in (ToT,) and A(t, a) is continuous in ^k, it follows from the theorem on uniform continuity4 that c can be taken so small that 1Compare J. I, No. 102. The interior as well as the exterior is a " continuum." 2 Gottinger Nachrichten, 1899, p. 282; compare also OSGOOD, ibid., 1900, p. 94; and BERNSTEIN, ibid., 1900, p. 98. 3 Compare p. 13, footnote 4. 4Compare E. II A, pp. 18 and 49; P., Nos. 21 and 100; J. I, No. 62. ~35] KNESER'S THEORY 181 a (t,a) = 0 in. (18) We suppose in the sequel that k has been selected so small that Jk and *k are in a one-to-one correspondence, and that at the same time (18) is satisfied. Under these circumstances the region Hk is called a field about the arc o, formed by the set of curves (15). 4. The one-to-one correspondence (15) between Nk and Sk defines t and a as single-valued functions of x and y which are of class C' throughout ~k; we denote these inverse functions by t —t(x, y), a= a(x,y). (19) Their derivatives are obtained by the ordinary rules for the differentiation of implicit functions, according to which at Da at Da t1 -+ = - y, + a t, a (20) 8t 8a 1 t 3a o At at + a Dx, 1 = t a + Oa Dy ~35. KNESER'S CURVILINEAR CO-ORDINATES1 Our next object is to extend to the general case the results given in ~32, b) concerning the introduction of geodesic parallel co-ordinates. a) Curvilinear co-ordinates in general: Let us introduce, instead of the rectangular co-ordinates x, y, any system of curvilinear co-ordinates = U(x,y), v= V(x,y) (21) where the functions U(x, y) and V(x, y) are of class C" in a region X contained in the region u of ~24, b); in the same region their Jacobian is supposed to be different from zero. We interpret u, v as the rectangular co-ordinates of a 1Compare KNESER, Lehrbuch, ~16. 182 CALCULUS OF VARIATIONS [Chap. V point in a ut, v-plane and denote by E the image in the u, v-plane of the region A. We suppose, further, that the correspondence established by (21) between ~ and E is a one-to-one correspondence. The inverse functions x =X(u, v), y-= Y(u, v) (22) will then likewise be single-valued and of class C" in the region [ and moreover their Jacobian Da(X, Yl) D-a (X, v) 0 in i. (23) a (tl, v) We consider now the integral J= F '( Y dx 'Y d-T (24) taken along an ordinary curve C: x = (r), y = (r) from a point A (To) to a point B(rT), the curve ( being supposed to lie in the interior of the region A. If we introduce the new co-ordinates iu, v into the integral J, it will be changed into J' — G, v, -' dr; (25) s \ cl-i- </2 the function G of the four arguments zu, v, z', v' being defined by G(u,, v, ', v') = F (X, Y, X,,uI'+ Xvv', Y'It'+ Yv'). (26) The integral J' is taken along the image G' of G in the,u, v-plane:. U= U- v ( T),, (T)), v = ( ), (T)) from the point A' (image of A) to the point B' (image of B). From the equality J'= J (27) it follows that if the curve G minimizes1 the integral J, its 1 With the understanding that only such curves are admitted as lie in the regions > and i respectively. ~ 35] KNESER'S THEORY 183 image t5' necessarily minimizes J', and vice ver-sa. Hence the problem to minimize the integral J and the problem to minimize the integral J' may be called eqjaivalent problems. The following properties of the function G(in, v, 'i', v') can immediately be derived from its definition (23): 1. G(u, v, it', v') is positively homogeneous' of dimension 1 in it', v'. 2. By differentiation we get Git, = F~,, ~ + Fu,~ Y,,, G,V = Fi~~, Xv FU, Y, Hence if x - X~u'-H 27,v', ~ = X,,tc + XIt b zj'= u,, +S Y'u' V` Y,, +- Y,6 the following identity holds: AC G,, (,, v, Wi', v') -+ - G,, (un, v, ui', v') - NF(x, yX, X', Iy') + ~JF~(X, y, X', y'), (28) from which we infer that the E-fnnction is an absolute invariant for the transformation (21), i. e., if we denote the new E-function by E'(u, v; tt', v'; Ah, ') we have E'(U, v; 'It, v'; il, ) )=E(x, y; xc', y'; 3, j). (29) 3. Also F1 is an invariant; if we, denote the corresponding function derived from G by G1, we obtain easily GI D2F1, (30) where D is defined by (23). 4. Also the left-hand side of Enler's equation is an invariant; after an easy computation, we obtain GitV, - G,,,v + G1 (u'' - 'a'"v) — D - Fx, 1 + F1 (x'y"- x"y')]. (31) The image of an extremal of the old problem is therefore an extremal for the new problem; and the same relation holds for the transversals, as follows from (28). 1 Compare ~ 21, equation (8). 184 CALCULUS OF VARIATIONS [Chap. V All these results are in accordance with, and can partly be derived a priori from, the equivalence of the two problems. b) Definition of Kneser's curvilinear co-ordinates: To the assumptions concerning the set of extremals (1) enumerated in ~33, a), we add the further assumption that A (t, aco) 0 in (toti), (32) where A(t, a) denotes again the Jacobian a(t, a) It follows, then, from the continuity of A(t, a), that the quantities to — To, T1 - t, k can be taken so small that A (t, a) 0 (33) throughout the region Ek. According to ~34, the correspondence between the domains ik and Ak defined by (1) is then a one-to-one correspondence, and the inverse functions t=t(x, ), a=a (x, y) (34) are single-valued and of class C" in the domain 'k. We now combine with the transformation (34) the transformation U = u(t, a), v = a (35) between the t, a-plane and the ut, v-plane, u(t, a) being defined by (7). Since, according to (6a) and (8), t = F(t a) *t0 in k, it follows that the correspondence between the region g1 and its image 5i in the u, v-plane, defined by (35), is a one-toone correspondence and moreover that the Jacobian 8(?'v) i= 0 in ~. a(t, a) ~35] KNESER'S THEORY Hence, if we combine the two transformations (35) and (34), we obtain a transformation of the form (21) which establishes a one-to-one correspondence between the region #k in the x, y-plane and the region 1k in the n, v-plane, and /which satisfies all the conditions imposed under a) upon the transformation (21). For every point (x, y) in the region IC defined in ~33, 6), the function t- U(x, y) represents, according to the definition of u(t, a) given in ~33, the value of the integral J taken along the unique extremal of the set (1) passing through the point (x, y), from the transversal of reference $~ to the point (x, y). c) Properties of Kneser's curvilinear co-ordinates: For KNESER'S curvilinear co-ordinates, the images of the extremals are the lines v const.; the images of the transversalsl the lines u= const. Moreover, the function G(u, v, ui', v') has the following characteristic properties: G (u, v, u', 0) = Z', G.(i, v, u, 0)= 1, G, (, vG, i', 0) 0= 0, which hold for every ut, v and for every u' which has the same sign2 as F(t, a). For the proof of these statements it is convenient to represent a curve ( in the region Fe of the x, y-plane in the form x = (t, a), l t = g (), y =- (t, a), a= h(T), which is always possible on account of the one-to-one correspondence between 8kI and ~k. The image (S' of ( in the uz, v-plane is then represented by u = (t, a, l t = g (r) v = a, a =h( r), and on account of (26) the following identity holds: 1 Again with the restriction that the transversal must lie in the region r. 2 Since F (t, a) * 0 and is.nnf.;nuous in El, it has a constant sign in 4k. 186 CALCULUS OF VARIATIONS [Chap. V d / d F a, (t, a), (t,a), ) I (t, a), - )(t, )) =G (t, a),, d (t,o), ). If ( is an extremal of the set (1), it can be defined by the equations t =-, a = a', a constant.l Hence the above formula becomes: F(r, a') G((u (, a'), a', uT(T, a'), 0), and therefore, on account of (8): CT(r, a') = G (u (r, a'), a', 'ar (T, a'), O) Since T and a' are arbitrary and, moreover, G(u, v, put', 0) = p G (i, v, u', 0) for every positive p, the first of the three equations (36) is proved. The second follows immediately by means of the identity tu'G+, v'G,^ G To prove the third, let t = ({o-), C = odefine a transversal; then, according to ~33, c): ut (q ((r), or) = const. Hence the condition of transversality, which must be satisfied at the point of intersection of this transversal with the extremal t-T, a a', reduces to d V d Gv ( ( 'r, C), aC, (r, a'), 0) -0, from which we infer the third of the equations (36), since d75 don lIts image is the line 6': u==u(r, a'), v-=a' and the angle 0' which the positive direction of (' makes with the positive u-axis is 0 or rr, according as the constant sign of F (t, a) is + or -. ~36] KNESER'S THEORY 187 The relations (36) lead to two important consequences: In the first place, we obtain immediately from the definition of the E'-function on applying (36): E'(', v; '', 0; it, v) = G(,, v,, u, i) -. (37) In the second place, we get by Taylor's theorem: G(u, v, it, i)- G (Z, v', 0) = ( - u') Gi, (t, I, I'a, 0) + v Gv, (u, v, u', 0) + ) [(,i, - -j2 G + 2 (i, -,t) G. + v S G], where the arguments of G,^, etc., are u,v,, '='+ 0 (i' - Z'), -'= O, and < 0 < 1. If we simplify the remainder-term by the introduction of G1, and make use of (36), we obtain: G (i, v, u, v) - i = 1"2 t2 V1 (38) From the preceding equation we see that whenever G1 and t are both positive (negative), also G(it, v, it, i) is positive (negative). Hence, if for a given point (it, v), the functions G ((t v, i,, ) and Gi (t, v, it, v) are different from zero (and therefore do not change sign) for all values of il, iv (except possibly Ai 0, i= 0), they must both have the same sign. Remembering now the relations (26) and (30), we obtain the following result,l which will be useful in the sequel: If at a point (x, y) the functions F(x, y, cos 7, sin ) and F1 (x, y, cos y, sin y) are both different from zero for all values of y, then they must both have the same sign. ~36. SUFFICIENT CONDITIONS FOR A MINIMUM IN THE CASE OF ONE MOVABLE END-POINT The introduction of Kneser's curvilinear co-ordinates leads to a number of important consequences: a) Kneser's sufficient conditions: Through the point A 1 See KNESER, Lehrbuch, p. 53. 188 CALCULUS OF VARIATIONS [Chap. V (x0, yo) of the extremal (o (compare Fig. 31, p. 170) we construct the unique transversal ': [tx=(a)]; and from an arbitrary point A of $ we draw any ordinary curve 6, joining the points A and B and remaining in the region by: a - ^ = - (r), y -= (T), To 71 The image of (o in the i, v-plane is the line v -=o; the images of 0o and Z are the lines t 0 and Z = 0- U(xo, Y0); the image of the curve ( is an ordinary curve (': 't U= (7' ), =V-V(r); 0 T C 71 The abscissae u0 and uI of the images A' and B' of A and B are v=a,+k ^i,, A,_^ I '0/ o=U(x,0 Yo) Iu1= U(Xl, Y)=,; A i \~ / a and according to the defiA'i B' nition2 of U(x, y) we have __/X1____ _ i_ _ _ u=a9 JFo(AB) = ul- tu / 'I JI,~v= -k u- = O u" =-t FIG. 37 On the other hand J,(AB) = Ji,.(A'B') - G (,U, d d d.r But since3 It(7T) t, u(71) =z,, we have fdu I dr d = u - UZo, Sdr and therefore the total variation J - J-(AB) - Jo(AB) may be written: SnsL\ dv ctuf ~cT J- - G ut', V, dr,Xtd —d dr. (39) 1 dr d 7rj drj^ The relation (38), together with (30), leads now to the following result: 1The arc of z corresponding to the interval (ao- k, ao+ k) of a lies entirely in the interior of i~>; for A lies in ~iG since to > t~, and T and 3o do not intersect in Ok. The image 5j of O is that part of 57 in which u - 0 or u-o 0 according as the constant sign of F(t, a) is + or -. 2 Compare ~35, b). 3 Compare, for this important artifice, ~32, b). ~ 36] KNESER'S THEORY 189 If the conditions A(t, ao) 0, F(t, ao) O 0 -tre satisfied for to t t c tI, and 'if, moreover, F1 (x, y, cos y, Sill y)> 0 (IIa') along the extremal 0(o for every value of y, then the extremal (0 furnishes for the integral J a smaller value than every other ordinary curve which can be drawn in -k from the transversal $ to the point B, provided that k be taken sufficiently small; and therefore the extreimal eo minimiTzes1 the integral J if the end-lpoint B is to remain fixed while lthe other end-point is movable on the curve $. b) Weierstrass's theorem for the case of one valilable end-point: Still another important conclusion can be derived from (39). On account of (37) we obtain from (3(9) A J = E' (u, v; ',; ' dr, \dr ' dr where u' is any quantity having the same sign as F(t, a). We may therefore' write the last equation: El, dit c9\ AJ= E'(t, v; cos 0', sill ', d' ) dr (40) V'O \dr dr/ d where 0' is the angle defined on p. 186, footnote 1, and whose value is 0 or 7r. But since the E-function is, according to (29), an absolute invariant for the transformation (21), we obtain, by returning to the original variables x, y, the extension of WVeiersttrass's theoremt to the case of one movable end-point: rll AJ= E ( y; ', y'; X, y)dr (41) 1To make the connection with the problem: To minimize the integral J by a curve joining a given curve C with the point B, the following remark is necessary: After an extremal %0 of class C' has been found which passes through B, is cut transversely by Z at A, not touched by Zs at A, then it is always possible, according to ~ 23,f) and ~30, to determine a set of extremals which has the properties assumed in ~33 of the set (1) and to which the curve Z is a transversal. The transversal 2 of the preceding theory will then coincide with the given curve Z. 2 Compare ~28, equation (51). 190 CALCULUS OF VARIATIONS [Chap. V where (x, y) is a point of the curve (; ', y' refer to the curve (; x', y' to the unique extremal of the set (1) passing through the point (x, y). Reasoning now as in ~28, d), we infer that in the above enumeration of sufficient conditions the condition (IIa') may be replaced by the milder condition E(x, y; p, q; _7, ) >0 along e, (IV') understood in the same sense as in ~28, d). c) Osgood's theorem concerning a characteristic property of a strong minimuim: The introduction of Kneser's curvilinear co-ordinates leads to a theorem due to OSGOOD1 concerning the character of the minimum of the integral J, in case the stronger condition (IIa') is satisfied. If we denote by 0 the angle which the positive tangent to d' at the point (i, v) makes with the positive n-axis, and introduce on (' instead of the parameter T the arc s of 6', we may write (40) in the form2 s1 ' _ _ AJ== E'(i, v; cos 0', sin 0; cos, sil 0) ds Applying the theorem3 on the connection between the E-function and F1 to E' and G1, we get E' (, V; cos 0', sin 0'; cos, sin 0) = (1 - cos (0 - 0')) G1 (i, v, cos 0*, sill n), where 0* is some intermediate value between O' and 0. Since 0'=0 or wr, the first factor on the right is 1 t cos 0. But if we suppose that (IIa') is satisfied, we can always take k so small that F1(, y, cosy, sin ) > 0 for every x, y in 7j and for every y. 1See Transactions of the American Mathematical Society, Vol. II (1901), p. 273. For the following proof see BOLZA, ibid., Vol. II (1901), p. 422. 2 Compare ~28, equation (51). 3 Compare ~28, equation (54). ~36] KNESER'S THEORY 191 From the relation (30) between F1 and G1, and from the continuity of G1, it follows, then, that a positive quantity m can be assigned such that G1 ('t, cos (, sill ) n m for every u, v in 37 and for every w. Accordingly we obtain A J m4r 1 ( cos 0) ds, or, since - do os 0 = - ds A J m [I F (1 - uo)], 1 being the length of the curve &' from A' to B'. Now suppose that the curve (E in the x, y-plane passes through a point P of the extremal a -= c + h of the set (1), where 0 < h < k. IQh k t=(t,,+/ (' will then pass through a point P' whose ordinate is A'i v= a-= u. FIG. 38 v - no + h. Let Q' be the foot of the perpendicular from P' upon the line u =u0. Then I Q1'P' + IP'B', that is, l I Vh2 - (zq- u0)Ih, and therefore a J m [7, h2 + (, - tu)2 T (u, - to)] > 0. (42) Hence, if we use the symbol,' in the sense analogous to that of A, we may formulate the result as follows: Under our present assumptions concerning the extremal @o and the functions F and F1, it is always possible to determine, corresponding to every positive quantity h numerically less than k, a positive quantity e1 such that A J = J (A B)- Jo (A B) El, (43) 192 CALCULUS OP VARIATIONS [Chap. V for every ordinary curve G which joins the transversal with the point B, and remains within 4uk BUT NOT WHOLLY IN THE INTERIOR OF h]. OSGOOD1 derives from his theorem a simple proof of WEIERSTRASS'S extension2 of the sufficiency proof to curves without a tangent: Let, in the notation and terminology of ~31, d), 2: =- (), y= () T, 0 C C 71, be a curve of class (/K), not coinciding with 0o, joining the points A and B, and lying wholly in the interior of the region By. Let HI be a partition of the interval (0roT) whose subintervals are chosen so small that the corresponding rectilinear polygon %, inscribed in 2, lies in the interior of IB. The polygon being an ordinary curve, we have, if Kneser's sufficient conditions of ~36, a) are fulfilled for the extremal 0o, VI> J%, if VnI denotes, as in ~31, c), the value of the integral J taken along the polygon %Pn. Hence if we pass to the limit and remember equation (78) of ~31, we obtain It remains to show that the equality sign cannot take place. Let Q be any point of 2 not situated on the extremal 0o, and denote by a0 + h the value of the parameter a of the extremal of the field passing through Q. Then: 0 < l h < k. Now consider in the above limiting process only such partitions HI for which Q is one of the points of division. There exists, then, according to OSGOOD'S theorem, a positive quantity el such that VnI- JO p Eh. Loc. cit., p. 292. 2 Compare ~ 31, e). ~37] KNESER'S THEORY 193 Hence if we pass to the limit, J~ — J~, > It> 0 and therefore Ji > Jo, Q E. D. 37. VARIOUS PROOFS OF WEIERSTRASS'S THEOREM. THE ASSUMPTION F(t, a) * 0 The function 'a U(x, y) introduced in S3o, b) was derived from it(t, a) by substituting for t and at the inverse functions (34): t t(X1 Y) I a — a (xy Hence tlhe par-tial derivatives of U(x, y) with respect to x and y are, on account of (8) and (91): Du F-at + F" a+] rIql)D DX ax DX DU- at aa.: F -+ (Fx, 4),, + FY, qF~a) ay ay a Remembering that F (At Fx, + it Fx and that by definition (t (X~, Y) I a (x, y)) --- x (t (x, yJ), a (x, y)) y~ we obtain the important result:1 __ 9u ax au where P(x, y) and Q (x, y) denote those functions of x and y into which Fx (t, a) and Fy (t, a) are transformed when the variables 1, a are replaced by their expressions in terms of x, y. From these expressions of the partial derivatives of U KNESER, Lehrbuch, p, 47; compare also p. 175, footnote 1. 194 CALCULUS OF VARIATIONS [Chap. V two further proofs of Weierstrass's theorem for the case of one variable end-point, can be derived. a) Knlse"r's proof:1 We repeat the construction of ~36, a), denoting, however, the points A0, A, A, B by num-G — ~ bers: 5, 0, 0, 1 respectively. ~^ X^~ ^\ ~Then we apply Weierstrass's 7.> ~1 construction2 slightly modified: ';\ \ Through an arbitrary point 'to ' 2 (r-T2) of ( we draw the 2 FIG 39 unique extremal of the set (1). It meets the transversal Z~ at a unique point, 7. Now we consider the integral J taken from 7 along the extremal 72 to 2, and from 2 along the curve (G to 1, and call its value S(2): S (r2) = J72 + J21, using the same notation as in ~~20 and 28. In particular we have (see Fig. 39): (TO) = J6 + Jl, S (T1) = J50 + J01 But according to Kneser's theorem (~33, c)) J60 = J50 hence A J 1 - o01 = - S (-1) - S (TO) According to the definition of the function U(x, y) given in ~35, b), we have J72 = U (X2, 2); on the other hand j21 -s F(4, y, ', ys )dT 2T Hence, making use of (44), we get as in the case of fixed end-points: 1 KNESER, Lehrbuch, ~20. 2 Compare ~~20 and 28. ~ 37] 1KNESER'S THEORY 1 9 5 -r,) - I - I P dS'(E (X2,!1,; X2', Y21; X2, Y2 (40)r> dr2 Integrating with respect to T2 from ro to m1, we obtain WeVeierstrassis theoremt (41). The above deduction leads to the following geometrical interpretation of the E-fnnction, due to KNESER: Let 3 be the point of G corresponding to T - -r2+ h, and draw the extremal 83 through the point 3, and the transversal 24 through the point 2 (see Fig. 40). Then S(Tr2-h) - S( (2) JS4~ J43- J72 - J23 and since JSI4 ''2. ~2 S(r2 +-I h) - 7-2 - ) J23 FIG. 40 Hence we obtain, on account of (45), the result:' J23 - J43 h [E (x2,i2; x2, y2; X,2 )+(h)J. (46) b) Proof by means of uilbert's invariant integral: The important formula (44) leads immediately to HILBERT'S invariant integral2 for the case of parameter-representation. The integral [P( + Q(-, y dr (47) taken along G from 0) to 1 is, according to (44), equal to {. Tc d hence T - J*_ U (XI, Iy) - U (XO, I7s) wo, /o denoting the co-ordinates of the point 0. The value of the integral J* is therefore independent of the curve G and depends only upon the position of the endI KNESER, Lehrbuch, p. 79; compare footnote 1, p. 138. 2Compare ~21. Another proof of the invariance of the integral J*, following more closely the reasoning of HILBERT'S original proof, is given by BLIss, Transactiolns of the American Mathematical Society, Vol. V (1904), p. 121. 196 CALCULUS OF VARIATIONS [Chap. V points; it even remains invatriat t when the point 0 moves along the transversal S, since U(x, y)- const. along every transversal. Hence, by letting 0 coincide with 0 and (S with (0 we obtain J*- J01 The integral Jo1 can therefore be expressed by an integral taken along the curve S, viz., Joi= zT ) \ x, y, x, yt) r+ Eyr, (R x, y,x y) Y d. Substituting this value of Jo0 in the difference: A J= JJ1- Jo we obtain immediately Weierstrass's theorem. c) The assumption F(t, a) = 0: It is important to notice that in the preceding two proofs of Weierstrass's theorem no use has been made of the assumption (6) that F(t, ao) = 0 at all points of the interval (tot1), but only of the two special assumptions' F (to, ao) =0, F (to, a,) = 0 (6b) which, according to ~33, a), are necessary for the construction of the two transversals $~ and Z. Hence, also in the sufficient conditions derived from Weierstrass's theorem, the condition (6) may be replaced by the milder condition (6b), whereas, in the former deduction of sufficient conditions by means of Kneser's curvilinear co-ordinates, the assumption (6) was essential. This apparent discrepancy2 between the two methods can be removed as follows: 1The first of these may be replaced by F(t, ao) O0, because for to any value of t between To and to may be chosen. Only in very exceptional cases can F vanish all along an extremal, since the differential equation F= 0 is, in general, incompatible with Euler ' s differential equation. 2The discrepancy is still more striking in KNESER'S own presentation, since he makes, instead of (6), the stronger assumption F(x, y, sin y, cos y) = 0 along %l for every y (compare Lehrbuch, pp. 49 and 53). ~37] KNESER'S THEORY 1.97 Compare the two problems: (I) To minimize the integral 0tl J7=- ~F (x, y, x', y')dt, and (II) To minimize the integral J(o) = ''(O(x, y, x', y') dt, where t F(0)(x, y, x', y') = F(x, y, x', y') + -(x, y) '+ (x, ) y', (48) ( (x, y) being a function of x, y alone, of class C' in ok. Since J') = j + ~d (x,, Y1) - ~ (on, y,), (49) we obtain A J()_ = A-J for all variations which leave the end-points fixed. If, on the other hand, the integrals are to be minimized with one end-point, say (xl, yi), fixed, while (x0, y0) is movable on a given curve i, the same result holds, provided that D (x, y) remains constant along this curve. With this condition imposed upon >), the two problems are equivalent; that is, every solution of the one is also a solution of the other. Hence it follows that every extremal for the one is also an extremal for the other.1 In particular, our set of curves x= (t, a), (t,a)(1) is a set of extremals also for J(O). We now suppose that the function F satisfies the two conditions (6b), but not (6), and we propose to show that it is always possible so to select the function q (x, y) that F(~ (t, a) >0 throughout the region ik defined in ~33, a). The analogous statement for transversals is, in general, not true. 198 CALCULUS OF VARIATIONS [Chap. V Let m be the minimum of F (t, a) in the region Atk, and let K be a positive constant greater than Iml. Further let, as before, t = t(X, ) a -- a (x, y)) denote the inverse fnnctions defined in 935, eqnation (34). 1. Case of fixed end-p,)oinits: In this ease we select D (x, zJ) = Mt (X, Y) ( (0) Then F'0' (t, a) F(t, a) -- M t (c (t, a), tf(t, (a)) But by the, definition of the inverse fnnctions we have t (4) (t, a),.0 (t, a)) tt hence p(O) (t, a) = F (t, a) + M, which is positive in T~. 2. Case of one variable end-point: Suppose (xi, yj) fixed and (x0, Yo) movable along the cnrve Z, w hich is a transversal of the set (1) for the problem (I) and represented, as in ~36, a), in the form X (t, (t, a) t xX(a) In this case we select +(Dyp) rM~t(x,y)-X(a(xy))1; (51) then (D (x, y) 0 along Z, and ((t, a), q1 (t, a)) - Mn/ (t - X (a)) Hence we obtain, as before, F(0) (t, a)=F(t, a)-HM>>0 i k It follows, fnrther, that Z is a transversal of the set (1) also for problem (II). For ~ 38] KNESER'S THEORY 199 F ( + + d+ ()Fi(' + tFJ'(~),The first term on the right vanis for (), since (, The first term on the right vanishes for t —x(a), since Z is a transversal of the set (1) for problem (I); the second term vanishes likewise for t=X(a), and therefore also the lefthand side, which proves our statement. The assumption (6), upon which the introduction of Kneser's curvilinear co-ordinates depends, may therefore be made without loss of generality; for, if it should not be satisfied, we can always replace the given problem by an equivalent problem for which it is satisfied. ~38. THE FOCAL POINTS The assumption A (t, ao) = 0 in (tot1) (32) was indispensable in the previous sufficiency proofs for the construction of a field; but our deductions give no indication whether it is at the same time a necessary condition for a minimum. We are going to prove, according to KNESER,1 that at least in the milder form A (t, a0) t 0 for to < t < t, (32a) which corresponds to Jacobi's condition in the case of fixed end-points, the condition is indeed necessary for a minimum. We retain all the assumptions of ~33 concerning the set of extremals (1), and we suppose moreover that, in the notation of ~33, ca), F1 (t, a) > 0 in (tot); (52) 1KNESER, Mathenmatische Annalen, Vol. L, p. 27, and Lehrbuch, ~~24, 25. 200 CALCULUS OF VARIATIONS [Chap. V but we drop the assumption (32) and suppose, on the contrary, that A (to, ao) = 0, (53) where to < to < t, and, moreover, that to is the smallest value of t, greater than to, for which (53) takes place. The corresponding point A'(xt, y0) of (o is then the focal point1 of the transversal $ on the extremal o,. a) Existence of the envelope: We propose to find all points2 [t, a] of the x, y-plane in the vicinity of [to, ao] for which - (t, C)= 0. (54) For this purpose we notice in the first place that the function A(t, ao).is an integral of Jacobi's differential equation ci I ciu\ d dt( dt This is proved exactly as the similar statement in ~27 b) and c) by substituting in Euler's differential equation x- (t, a), y-= r(t, a), differentiating with respect to a and then putting a= a0. Since Fi-= F (t, ao) is continuous in the vicinity of t- to, and, according to (52), different from zero for t-t, it follows that3 \t (to, ao),=. (55) Hence it follows, according to the theorem4 on implicit functions, that there exists a unique solution t = t(a) of (54) which is of class C' in the vicinity of a -ao, and takes for a a= the value t —to. The curve5 [t- t(a)] in the x, y-plane, i. e., the curve 1 Compare ~~23 and 30. If s shrinks to the point A, the focal point A' becomes the "conjugate" point to A. 2 For the notation compare ~33, a). 4 Compare p. 35, footnote 2. 3 Compare p. 58, footnote 2. For the notation, see ~33, a). ~38] KNESER'S. THEORY 201 A: x = (t(a), a) = (a), = ((), a = (a) is the envelope' of the set of extremals (1). For, since dx dt dt d dt d= t + qCG = at + ma a da da+ da da it follows that T h apat - ft h- - (i (a), at) 0(5) This shows, apart from the points at which (da) (da) that the curve; touches all the extremals of the set (1) for which a is sufficiently near to ca, and therefore ~ is indeed the envelope of the set. b) Application of the theorem on envelopes: We must now distinguish two cases: Case I: The envelope ~ does not degenerate into a point, i. e., + (a) and t(a) do not both reduce to constants. Let us suppose that the functions + (a) and *(a) are of class C(?) in the vicinity of a -ao, that for a —a0 their derivatives up to the order r —1 vanish, but that the rth derivatives do not both vanish. Then we obtain by Taylor's formula Ud = (a - a))-I [A + a], da = (a - ao)-1 [B + /], (57) da ' da where A and B are constants which are not both zero, and a and / approach zero as a approaches a0. Substituting these values in (56) we get A = n (to', ao), B = nit(t', ao), (58) where n is a factor of proportionality which is different from. zero. 1 Compare E., III D, p. 47, footnote 117. 202 CALCULUS OF VARIATIONS [Chap. V We now introduce on ~ a new parameter r by the transformation a - a, = c, where E= ~ 1 will be chosen later on. Since, according to. (2) and (2a) the functions Ot(t, a) and kt(t, a) do not both vanish at a c-(0, it follows from (56) that we may write dx= md, dg = mlV, ' (59) dr dr where m is a function of?, which is continuous in the vicinity of T —0, and, on account of (57) and (58), is representable in the form = ETl -( + v) where L v- _0. T=0 Whenever it is possible so to select the sign e that m is positive for all sufficiently small negative values of r, we can construct, according to the theorem II of ~33, d), an admissible variation of the arc AA' of (0 for which AJ 0. |. Subcase A): r odd.1 If we p <_-. ' choose e equal to the sign of n, A A A I nm is positive for all sufficiently 1 FIG. 41 small values of |r; see Fig. 41. Subcase B): r even. m has the same sign as nr, no matter how we choose e. " Therefore PA' 1. If 1 <0, mI is positive for nega- A five values of T; see Fig. 42. FIG. 42.~ 2. If n>0, mn is negative for 'pv —, L. negative values of T;2 see Fig. C\, - 43. FIG. ' In subcase A) and subcase B1) FIG. 43 \ a we have lThis covers the "general" case in which S has no singular point at A'(r= 1). 2If we draw a straight line 2 through the point A' not tangent to e%, then. crosses the line 2 in case A); it lies all on one side of 2 in case B), on the same side as the arc AA' in case B1), on the opposite side in case B2). This follows easily from (57). ~38] KNESER'S THEORY 203 J = JF (P Q) + J. (QA')-JFi (AA') = 0. according to theorem II of ~33, d), and therefore the arc AA' of the extremal (o certainly furnishes no properi' inilimum, and still less the extremal (o (or A B) itself. But it furnishes not even ac improper minimcm. For2 the envelope ~ cannot at the same time be itself an extremal, and therefore the integral J(QA') can be further diminished -and consequently AJ can be made negative-by a suitable variation of the arc QA'. The statement that: itself cannot be an extremal can be proved most conclusively by substituting in the left-hand side of Euler's differential equation for x, y the functions x = ( (t, a), a), = (t, C), and making use of the characteristic property (59) of the envelope. If we remember the homogeneity properties of F and its derivatives, and the fact that ((t, a), r(t, a) as functions of t alone satisfy Euler's differential equation, we obtain after an easy reduction: d d Fx -- -Fx EF,Att, d. The arguments of Fx, etc., are da dy 'Y ' dr ' dr; those of Ot, *t, F1, At are t, a. Since, according to our assumptions, F1(t, a) and At(t, a) For the distinction between "proper" and "improper" minimum, compare ~3, b). 2 Compare DARBOUX, Th6orie des Surfaces, Vol. III, No. 622, and ZERMELO, Dissertation, p 96. 204 CALCULUS OF VARIATIONS [Chap. V are different from zero for t= to, a=ao, they remain different from zero in a certain vicinity of this point. Moreover, At and At are not both zero. Hence the envelope a does not satisfy Euler's differential equation.l In subcase B2) the same construction cannot be applied, and therefore the question cannot be decided by this method. Case II:; degenerates into a point. In this case all the extremals of the set pass through the point A', and we can directly apply Corollary II of the theorem on transversals, ~33, c). \ c i*^A Accordingly, we have for every -A e '- extremal C of the set: FIG. 44 aJ = JF(PA') - Je (AA') = 0, and therefore the arc AA' of the extremal ~o certainly furnishes no proper minimum. Summing up the different cases, we may state the result: If the end-point B of the extremal AB coincides with the focal point A' (and a fortiori, therefore, if B lies beyond A': t1 > to) the arc AB ceases to furnish a minimum, except in the following two cases: 1. When the envelope has at A' a cusp of the special kind defined under subcase B1), the present method fails to give a decision.2 2. When the envelope degenerates into a point, the arc AA' furnishes no proper minimum, but it may furnish an Another more geometrical proof can be derived from the fact (see ~25, b)) that only one extremal can be drawn through a given point in a given direction if F1 (x, y, x, y') 40 for the given point and direction; compare Darboux's proof (loc. cit.) for the case of the geodesic. 2 Under the restricting assumption that F(x0o, yQ', cos y, sin y) 40 for every y, OSGOOD has shown that the arc AA' actually furnishes a minimum, if the other sufficient conditions of ~ 36 are satisfied, Transactions of the American Mathematical Society, Vol. II (1901), p. 182c ~38] KNESER'S THEORY 205 improper minimum.l If, however, B lies beyond A', the arc AB furnishes not even an improper minimum.2 Thus the necessity of the condition A (t, a0) O for to < t < t, (32a) is proved for all cases with the one exception just mentioned.3 1 The set of geodesics on a sphere which pass through a point affords an example of this kind. 2 For, from F1 (to', a) i O it follows that if a is sufficiently near to co, the " discontinuous solution" PA'B (see Fig. 44) cannot satisfy the corner condition (24) of ~25, c) (compare footnote 2, p. 142), and therefore a variation PMNB can be found for which A J < 0. 3 This agrees with the result derived by BLISS from the second variation (compare ~ 30); the latter method proves the necessity of (32a) also in the exceptional case. CHAPTER VI ISOPERIMETRiC PROBLEMS1 ~39. EULER'S RULE THE special example which has given the name to this class of problems has already been mentioned in ~1. More generally, we understand by an isoperimetric problem one of the following type: Among all curves joining2 two given points 0 and 1 for which the definite integral 't K= G(x, 8, x', ')dt takes a given value 1, to determine the one which minimizes (or maximizes) another definite integral /rt J (x, y, x', y')dt. to Concerning the two functions F and G we make the same assumption as in ~24, b) concerning F alone. The "admissible curves" are here the totality of ordinary curves which join the two points 0 and 1, lie in the domain 4 of the func. tions F and G, and for which the integral K has the given value 1. Aside from this one modification, the definition of a minimum is the same as in the unconditioned problem, ~24, c). We suppose that a solution has been found: (: x= (t), y=-(t), tIt-tl; and we replace the curve ( by a neighboring curve C: x=-x+, yy+-v, 1 This chapter is based chiefly on WEIERSTRASS'S Lectures of 1879 and 1882, and on chap. iv of KNESER'S book. 2 Or: joining a given point and a given curve, etc. 206 ~39] ISOPERIMETRIC PROBLEMS 207 where ~ and V are functions of t of class D' satisfying the following conditions: 1. They vanish for t-=to and t ti; 2. In the interval (toll), they remain in absolute value below a certain limit p. 3. The integral K taken along (5 from to to ti has the same value as if taken along I (viz., =1/), or, as we write it, AK = K01K-K- = 0; (1) a) Admissible variations: Our next object is to obtain an analytic expression for functions ~, X satisfying these conditions, not necessarily the most general expression but one of sufficient generality for the purpose of deriving necessary conditions for the minimizing curve. Such an analytic expression can be obtained, according to WEIERSTRASS, as follows: Let p1, p2, ql, q2 be four arbitrary functions of t of class D' vanishing at to and tl. Then we consider the functions = Elp1 +, r22 =- 1 ql+ E2q2, (2) where el, 62 are constants, and propose so to determine e. as a function of e1 that the condition (1) is satisfied for every sufficiently small value of e1. For this purpose we notice that the integral Kol is a function of el, e2 which is of class C' in the vicinity of e = 0, e2 = 0, and which is equal to Kol for E1=0, e2 —0. Further, for -=0, 2 =0 its partial derivative with respect to ei has the value t Ni = (Gcp, + Gy q + GQ.pi + GY. qi) dt Hence if we introduce the assumption' that the curve Q is not an extremal for the integral K, the functions P2, q2 can iIf C were an extremal for the integral K, the curve ( (or at least sufficiently small segments of it) would in general minimize or.maximize the integral K, and it would therefore be impossible to vary these segments without changing the value of K. 208 CALCULUS OF VARIATIONS [Chap. VI be so chosen that N2 = 0, and the conditions of the theorem on implicit functions are fulfilled for the equation (1) in the vicinity of the point e= 0, E2 -0. Accordingly, we obtain a unique solution E2 of the form1 N1 E2 - -- 1 + (1E) E, (3) N2 where,(e) denotes, as usual, an infinitesimal. Substituting this value in ~, V we get E = N (P1 — n )+ (EI) P2, (4) El - -)E+ (El),q2~E These functions ~, V have all the required properties for sufficiently small values of e, 1. The same argumentation applies to "partial variations" which vary the curve only along a subinterval (t't") of (tot,). It is only necessary to take the functions pi, P2, ql, q equal to zero in the whole interval (totf) with the exception of the interior of the subinterval (t't"). b) Ezler's rule: According2 to ~25, the total variation AJ for the variations (4) may be written ^J= (M,- NI, + (E,), where t] Mi = j (Fipi + Fi, q + F.p'i + F-u q ) dt For an extremum it is therefore necessary that M2 M1- N1 O.. N2 After a definite choice of the functions P2, q2 has once been made the quotient M2/N1 is a certain numerical constant which we denote by -X: x =-. (5) N2 1 Compare p. 35, footnote 2. 2 Compare, in particular, the footnote on p. 122. ~39] ISOPERIMETRIO PROBLEMS 209 We have then the result that the equation Ml + xAl = (6) must be satisfied for all functions pi, qt of class D' which vanish at to and tl. This shows at the same time that the value of the constant X is independent of the choice of the functions P2, q2. If we put H=F+ G, (7) equation (6) becomes (HP + yq1 + Hxp1 +.H,, q ) dt=0. Hence we infer exactly as in ~25 by the method of ~6, that x and y must satisfy the differential equations d d H — - =H,, 0, H1 —,H = 0, (8) which are equivalent to the one differential equation H,, - H:, + HI (x y" - Xy) = O, (I) where Hi is defined by: y ' - x' ' '2, v y x y x2 We call, again, every curve which satisfies (I) an extiremal for our problem (KNESER). The above deduction applies to so-called "discontinuous solutions"' as well as to solutions of class C', and shows that the isoperimetric constant X has the same constant value along the d'ifferent segments of a "discontinuous solution." Moreover we obtain, exactly as in ~~9 and 25, at a corner t t2, the "corner-condition:" 1 Compare ~9, in particular footnote 3, p. 37. 2 This important remark is due to A. MAYER, Mathematische Annalen, Vol. XIII (1877), p. 65, footnote; and WEIERSTRASS, Lectures. Even if the minimizing curve contains unfree points or segments, all those segments of the curve whose variation is unrestricted (apart from the condition A K= 0) must satisfy the differential equation (I) with the same value of the constant A. 210 CALCULUS OF VARIATIONS [Chap. VI /2-0 t 2- -+ t 2 -- ) HX. H,,%, Hy. (10) -I" t2 t", I fa-o 'n -t +H All these results may be summarized in the statement that, so far as the first variation is concerned, our problem is equivalent to the problem of minimizing the integral f (F+XG) dt, the curves being subject to no isoperimetric condition. This simple rule, which is the analogue of a well-known theorem in the theory of ordinary maxima and minima, is usually called Edler-'s rule, according to EULER,1 who first discovered it. The rule still holds in the case where the point 0, instead of being fixed, is movable on a given curve: *t = f (- ),, - (T) For, a reasoning similar to that employed in ~30, combined with the remark that for all admissible curves AJ —J+ XAK, leads2 to the condition t=t0 Hxa'+ H,. aT-T =0 (11) c) EXAMPLE XIII: Among all curves of given length joining two given points A and B, to determine the one which, together with the chord AB, bounds the maximum area. Taking the straight line joining A and B for the X-axis, with BA for positive direction, we have to maximize the integral3 ftl J= 2 J (xy' — x'y) dt 1 EULER, Methodus inveniendi lineas curvas maximni ninimnive proprietate gaudentes, 1744; see STACKEL'S translation, p. 101. The first rigorous proof is due to WEIERSTRASS, Lectures, and Du BOIS-REYMOND, Mathematische Annalen, Vol. XV (1879), p. 310. The proof given in the text is due to WEIERSTRASS. 2 For details of the proof we refer to KNESER, Lehrbuch, ~33. 3 We substitute this analytical problem for the given geometrical one, without entering upon a discussion of the question how far the two are really equivalent. Compare J. I, Nos. 102, 112, and II, Nos. 129-33. ~39] ISOPERIMETRIC PROBLEMS 211 while 2 + y/ 2 dt Jto has a given value, say 1, which we suppose greater than the distance A B. Since H= I (xy' - 'y) + X 7 'x + y2 we get HI = + ( - x,-,"), ' (12) and therefore the differential equation (I) becomes xy"-x"y' _ 1 X"Y1"~ - - '(13) (1/x'_ + y')- X Hence the radius of curvature of the maximizing curve is constant and has the value X, while its direction is determined by the sign of X. Again, since Hl never vanishes, there can be no corners,' and therefore the curve must be an arc of a circle of radius X1. The center and the radius of the circle are determined by the conditions that the arc shall pass through the two given points and shall have the given length 1. There are two arcs satisfying these conditions, symmetrical with respect to the x-axis. d) EXAMPLE XIV: To draw in a vertical plane between two given points a curve of given length such that its center of gravity shall be as low as possible.' Taking the positive y-axis vertically upward, we have to minimize the integral y = 1/x2 + y, 2 dt while at the same time K = 1/ 2 + y,2 dt has a given value, say 1. Here H = (y + X) /x'2 + y2 1 Compare ~25, c) and ~28, b); in particular footnote 2, p. 142. 2 Position of equilibrium of a uniform cord suspended at its two extremities. 212 CALCULUS OF VARIATIONS [Chap. VI Using the first of the two differential equations (8), we obtain at once a first integral x'(y+X) - I/ x,2 + y12 On account of (10), c must have the same constant value all along the curve, If c =0, we obtain the solution x = COnSt. which is possible only if the two given points lie in the same vertical line. If c O 0, we obtain as general solution of Euler's equation two systems of catenaries: x-=a+ ot, y + X = - -3 cosht. Determination of the constants. If we suppose Xo<xi, the constant P must be positive in order that we may have to < ti. Since the curve is to pass through the two given points, the following equations must be satisfied: x0 = a + Pto, Y + X = /3 cosh to, Xl = a + tl, Y1 + A = + A cosh t Moreover, the curve must have the given length 1; this furnishes the further equation 3 (sinh ti- sinh tot) I From these five equations we have to determine the five constants a, /3, X, t, t l. If we introduce instead of to and tl the two quantities2 t, + to X + X - o2a i- 2 = 23 tl -to X l — X v = 2 2/3 we derive from the above equations the following: 1 y + x = 0 is not a solution, since it does not satisfy the second differential equa, tion (8). 2WEIERSTRASS, Lectures, 1879. ~40] ISOPERIMETRIC PROBLEMS 213 Yi - yo = 2/ sinh / sinh v, I 2/3 cosh / sinh v. Hence we get tanh u =+ Yl (16) Since we suppose I > 1/ (1x, x)2,)+ (y, - YO)2 > I Y- Yo, each of the two equations comprised in (16) has a unique solution g. Further, we obtain from (15): 12- (i - y0)2 = 432 sinh2, and therefore sinh v i/12' - (Y - Yo)2 -- -, v 2 -say= k. (17) ~ X1 - OX0, Since k > 1 the transcendental equation (17) has one positive root v. After u and v have been determined, the values of a, P, X, to, tl follow immediately. Each of the two systems of catenaries (14) contains, therefore, one catenary satisfying the initial conditions. ~40. THE SECOND NECESSARY CONDITION We suppose that the general solution' of the differential equation (I) has been found: x =f(t, a,l, ), y = g(t, a, f, x) (18) It contains, besides the two constants of integration a, 3, the isoperimetric constant X. Moreover, we suppose that a particular system of values of these constants a = a0, - = /0, A -= X has been determined2 so that the extremal 1 Compare the remarks in ~25, a). 2 There are five equations for the determination of the five unknown quantities a,/3, t, tO, tl. 214 CALCULUS OF VARIATIONS [Chap. VI = - f(t, ao, Po, Xo), e0'. to = t-: t1, (19 y - g (t, I0,/a 0, XO), passes through the two given points 0 and 1 (for t-to and t-tl respectively), and furnishes for the integral K the prescribed value 1: K01i= We suppose that the functions f; g, ft, gt, ftt, gtt and their first partial derivatives with respect to a, /3, X are continuous functions of their four arguments in a domain To C to C, a a-ao | d, i P - )o | cl d * I -xo l c ^ where To < to and T1 > tl. Further, we assume that for the particular extremal (,, ft + gt o0 in (TT), ito (20^ ftgx - fxgt i, and that fta — f gt and ft.p -fgt are linearly independent.1 Finally we retain the assumption introduced in ~39 that ~0 is not an extremal for the integral K. a) A lemvac on a certain type of admcissible variations: In ~39 the existence of admissible variations of the form:= (t), E), =r (t,e ) (21) has been established, satisfying the conditions enumerated on p. 122, footnote 1, and besides the isoperimetric condition AK =O for every sufficiently small value of iel. From the latter condition it follows that also o o Hence we obtain in particular for e 0: 1 Compare ~ 13, end. ~40] ISOPERIMETRIC PROBLEMS 21I5 (Gxp + G~q + Gx.p'+ G,1 &') cit 0, (22') where If we transform the left-hand side of (22) by integration by p~arts, and remember that, as in ~25, a), Gx-d Gx,U G-dG cit Y 'dt Ut where U ~Gx'' G'Y Y we obtain t fUwdt-0 where w= - y'p - Since pand q vanish at to and tj, the same is true of wv. Vice ver-sa, the following lemmac' holds: Let wv be any fnnction of class D' which satisfies the conditions w (t,) _0, w (t,)0,(24) J;tl Uwdt -0;(25) then it is always possible to constrnct an admissible variation of type (21) for which Proof: Since (% is not an extremal for the integral K, it follows that U*~ 0; it is therefore always possible so to select a fnnction wl, of class D', and vanishing at to and [j, that 'Due to WEIERSTRASS; see KNIESER, Mat hemnatische Anonalen, Vol. LV, p. 100. 2163 2CALCULUS OF VARIATIONS C [Chap. VI Uwl-l dt#* O Now let W - EW + E1 WI and choose =m fy'o) _____u' 4 _ _ _ _ - x These functions vanish at to and tj for all values of the constants E, e1; they represent admissible variations if, moreover, the condition AK=O (1) is satisfied. But by the same process as above, we find: ____ 0 UwdtzO, (26) to DAK 0= O Uwdt4~O. (26a) On account of (26a) we can apply the theorem on implicit functions to the equation (1), and obtain for e1 a unique solution which, on account of (26), is of the form' El- (E) E Hence Y'$ - X'-q = - cIEW + (C) c which proves our statement. b) Weierstrass's expression for the second variation: Since ZAK-O, we may write AJ -AJ+x, aK. (27) Hence if we apply to the increment AF+ XOAzG Taylor's formula, we obtain for every admissible variation of type (21) AJ = f (HI4 + HIrj + Hx, '+ Hyrl') dt +tJ 1 tl 1~ (HXX$2 + +H /2 (,E) C2~c)$ I Compare p. 35, footnote 2. ~{40] ISOPERIMETRIC PROBLEMS 217 where H=F+XG. The first integral is zero since C0 is an extremal. To the second integral we apply the transformation of ~27, a). We thus obtain the result: A 2 'Io / /d/\2 \i 2 (HI(HIt d)+ H2W2) c2t + (E) 2, (28) where H1 and H2 are derived from H in the same manner as F1 and F2 from F; see ~24, b) and ~27, a). We shall denote the first term on the right-hand side by 82J. For a minimum it is therefore necessary that Jt I dt)+ H2W2 dt 0;(29) and on account of the lemma proved under a) this condition must be fulfilled for every ficwtion w of class D' which satisfies the equations (24) and (25). c) The second necessary condition: Since we can construct admissible variations1 which vary the arc (0 only along any given subinterval (t't") of (totl), we can apply to the above integral the reasoning of ~11, b). Hence the second necessary condition for a minimum (maximum) is that ^H1O ( O) (II) along the arc @o. This is the analogue of Legendre's condition. Also the second necessary condition for the isoperimetric problem coincides, therefore, with the second necessary condition in the problem to minimize the integral tl (x, y, x, y') dt without an isoperimetric condition. 1 Compare ~ 39, a). 218 CALCULUS OF VARIATIONS [Chap. VI ~41. THE THIRD NECESSARY CONDITION AND THE CONJUGATE POINT We assume in the sequel that (II) is satisfied in the. stronger form H,> along 0o. (II') It follows, then, by the method of ~11, b), that (29) is satisfied, provided that the point 1 is sufficiently near to the point 0. We have next to determine how near the point 1 must be taken to the point 0 in order that the inequality (29) may remain true. And it is at this point that the equivalence of the two problems, which we have been comparing, ceases.' In the unconditioned problem the inequality (29) must be fulfilled for all functions w of class D' which vanish at to and tl; in the isoperimetric problem only for those which besides satisfy the equation (25). It is therefore a priori clear that the condition (29) is certainly fulfilled for the isoperimetric problem if it is fulfilled for the unconditioned problem. Hence if we denote by T' the upper limit of the values of tl for which the inequality (29) remains true in the isoperimetric problem, by T" the corresponding upper limit for the unconditioned problem, then T' is at least equal to T", but it may be greater, and in general it actually is greater, as will be seen later. a) Determination of the conjugate point: The point T' can be determined by a proper modification, due to WEIERSTRASS, of the method for the determination of the conjugate point in the unconditioned problem:2 Since we consider only those functions w for which This has first been discovered by LUNDSTROM, 'Distinction des maxima et des minima dans un probleme isop6rimetrique," Nova acta Xreg. soc. sc. Upsaliensis, Ser. 3, Vol. VII (1869); compare also A. MAYER, ilMathematische Annalen, Vol. XIII (1878), p. 54. 2 Compare ~~12, 13,16, 27, b). ~41] ISOPERIMETRIC PROBLEMS 219 Uw dt, we may write 82J in the form SJ - 2 t (H, W'2 H2w2 +- Iw U) dt, pu being an arbitrary constant. Transforming the first term by integration by parts (see ~12) and remembering that w vanishes at to and tl, we obtain, if w' is continuous in (t0ot), rth 8J= E2 w [ (w) + U] dt, (30) where d ( (31) (w) _ H2 - - (Hw ). (31) To obtain the general integral of the differential equation r (w) + A U = 0 (32) we substitute in the differential equation' dt for x and y the general integral (18), differentiate with respect to a, I, X respectively, and finally put a -=ao, /3-= 30, X X0. If we denote 0 (t) = gtfa - ftag 02'(t) = gt -ftg (a = ao, X -= A, x = go), 03 (t) - gtfA f- gA the result2 is as follows: 1H means here: F-+- G. 2For the computation compare ~27, b). In the differentiation with respect to A an additional term appears on account of the factor A which occurs explicitly in F+,AG. The immediate result of the differentiation is Y'* (03(t))+(G, -dt G,)=O; but according to ~25, equation (18), Ghe t-t G,='Uabove hence the above result. 220 CALCULUS OF VARIATIONS [Chap. VT * (01(t)) - 0, * (02 ()) = 0 ~ * (0(t)) + U = (33) Hence we infer that the function Iv = C, (t) +,O,2(t) + 1f03(t) in which el and e2 are arbitrary constants, is the general integral of the differential equation (32). Now if it were possible to find values for cl, c2, ~k and a value t' such that W (to) - C1 01 (4o) + C2 02 (to) + /A01 (to) 0, W C1 0, 1 (t') + C1 02 (t') + kt~l (t') 0 Y i Uw dt JUO1 dt +eC2JUO2dt+-F/J, U03 dt 0 to< t'; t', the second variation could be made equal to zero (and therefore presumably AJ< 0) by choosing w equal to zero in (t't1), and equal to this particular integral in (tot'). In order that 82J> 0 for all admissible functions w, it is therefore necessary' that 01 (to) 02 (to) 03 (to) D (t, to) O (t) 02 (t) 03(t) 0 0 (34) f UO1dt, F, 4 UO3dt forlt f07.' t,~~~~~~~~~t < t:- t,: 'wEIERsTRASS, Lectures, 1872. This condition, together with F!, #0 in (t,,tj), is also sufficient for a permansent s ign of 62J (MAYER, Matheleratische Annalen, vol. XIII (1878), p. 53). -The proof is based upon the following extension of Jacobi's formula (14) of ~ 12 for the unconditioned problem: (pu + qv) * (puq + v) = H,(p' + qv)2 '-2q (p'n + q'n) d [HI (pu + qv) (p',u q'v) - (p + qn) q] where u, v, m, n are the functions introduced below, under b), and p and q are two arbitrary functions. Compare BOLZA, "Proof of the Sufficiency of Jacobi's Condition for a Permanent Sign of the Second variation in the So-called Isoperimetric Problems," Transactions of the American Mathemva'ictl Sociely, vol. III (1902), p. 305, and Decennial Publications of the University of chicago, vol. IX, p, 21. ~41] ISOPERIMETRIC PROBLEMS 221 If we denote by to the root next greater than to of the equation1 D (t, to) O, the above inequality (34) may also be written tl< to The point to of the extremal ~o is again called the cojtjugate of the point to. b) The third necessary condition: The preceding result makes it highly probable2 that the minimum cannot exist beyond the conjugate point. And indeed it can be provedc by a modification of the method employed by WEIERSTRASS for the analogous purpose in the unconditioned problem,4 that if to< tl, the second variation, and therefore also A J, can be made negative. For the proof it is convenient to throw the determinant D(t, to) into another form in which its properties can be more easily discussed. Let -= e,(to) 02 (t) - (to) 01 (t) - (t, to) v C1,e(t) + C202(t) - 03,(t) v(t, t), where the constants C1, C2 satisfy the equation 1 01 (t0) + C2 02 (t) - 03 (t) = 0 These two functions5 satisfy the two differential equations 1 D (t, to) cannot vanish identically; see below, under b). 2 Compare remarks in ~ 14, p. 59. 3 The proof has been given by KNESER, Mathematische Annalen, Vol. LV (1902), p. 86. From the statements in HORMANN'S Dissertation (GOttingen. 1887) it appears that WEIERSTRASS was in possession of essentially the same proof, but I have been unable to ascertain whether he has ever given it in his lectures. I reproduce in the text KNESER'S proof in a slightly simplified form. In ~40 of his Lehrbuch, KNESER gives another proof which, however, presupposes that Dt (t,, to) o 0 4 Compare ~16, p. 65, footnote 1. 5 Neither u nor v can be identically zero. For since, according to (20), 01 t) and 02 (t) are linearly independent and H1 = 0 in (t,,t), 01 (to) and 02 (t) are not both zero. and therefore uE0. v cannot be identically zero since U $0. "222 CALCULUS OF VARIATIONS [Chap. VI T(ti) = O, - (v)= U (35) respectively, and both vanish at to: u(to) -=0, (to)= 0. (36) Hence the determinant D(t, to) reduces, after an easy transformation, to D(t, to) = mv-nu, (37) where Ot at m = Uudt, n = Uvdt From (35) follows: vI ('u) - u/ (v) =- I H, (z ' - u'v) - - uU Integrating and remembering (36) we get H, (u '- -' ) = -. (38) Again, we obtain by differentiating (37) with respect to t: D = mv- V tnu and therefore'1 2 Du'g — D'Y H (39) From the preceding equation it follows that D has at to a zero2 of an odd order, except when (to) =0. After these preliminaries, we write the second variation in the form s2J= - e2k f w2dt + e2 fw [~ (w) +, U] dt 1If we denote by to' the root next greater than to of the equation u (t) = 0, the relation (39) shows that to y to'. For, since u has at to a zero only of the first order, the quotient D/u vanishes for to, and therefore D _ _ t m2dt u~ Jto Hlu2 which proves that D = 0 for to < t < to'. 2D cannot vanish identically; otherwise mt and therefore also u would vanish identically, which is incompatible with our assumptions. ~ 41]1 ISOPERIMETRIC PROBLEMS 223~ where le is an arbitrary positive constant and d d dw\ ` (w) - (H + k - H i dt \ dt Now let A~ and Z denote those particular integrals of the differential eqnations respectively, which satisfy the initial conditions: Tt-(to) u (to) - 0, i` (to) - u' (to) i(t0) - v (to) - 0, F'(t5) v' (to). then it follows from a general theorem' on differential eqnations containing a parameter that L (t'(t) - 'it(t)) O, L (;F(t) - V (t)) = 0 k=:o ~~~~~k=5 unifonrmly with r-esPect to the interval (toti) of t. Hence, if we put t t l, 3L=1 Zv l D~ (t, to) 4~n;Fi - j-H-1~ we have also L D (t, to) D (1, to), uniformly in (to, t1) A=O Now suppose that to <tl and that u, (t,) 0 Then D(t, to) changes sign at t', as has been shown above; we can therefore choose two quantities t, and t4 satisfying the inequalities to < t3 < to' < t4i < tl 1POINCARI, M~eanique cgleste, vol. i, p. 58; PICARD, TraitW dc'Analyse, vol. ITT, p. 157; and E. II A, p. 205. The assumption HI * 0 in (tot,) is essential for this conclusion. 224 CALCULUS OF VARIATIONS [Chap. VI and so near to to that D(t, to) has opposite signs at t3 and t4. Now select k so small that also D (t, to) has opposite signs at t3 and t4; then D(t, to) vanishes at least once at a point to between t3 and t4. But since D(to, to) is equal to zero, we can determine two constants cl, c2, not both zero, so that cl t (tu) + c2 (t)= 0, cli)(to ) + c2 (t)= 0 Now if we choose 'w = C1i t+ C2 v in (to to) wO 0 in (otl), and give the arbitrary constant,/ the value c2, then w satisfies the differential equation 'I (iw) + - 0, and the conditions (24) and (25). This function w makes 82J negative, viz.: 82J = -- E2k W2 cdt It remains to consider the exceptional case1 when it (to) -0. This can only happen when at the same time m (t1) 0 and (to) =0, as follows at once from (39) and (38), if we remember that H1 0 in (toti) and that it and it' cannot vanish simultaneously. In this case we can make 32J-O 0 by choosing ai 0 O and w, = t in (to to), w 0 in (t t) and by a slight modification of the method used by SCHWARZ2 for the proof of the necessity of Jacobi's condition in the unconditioned problem, it can be shown that 82J can be made negative by choosing 1 For this exceptional case, see BOLZA, Mathematische Annalen, Vol. LVII (1903), p. 44. 2 Compare ~16, p. 65, footnote 1. ~41] ~ ISOPERIMETRIC PROBLEMS 225 v - u! + ks in (tot', w - ls in (t I'tj) where,(to) 0 s (t,) O s (t'):# 0 I;tl s Udt 0() We thus reach in all eases the result that the third flecCssar-y condition for- a minihimumn is that D (t, to)# 0 for to <t<t, (III) orl to c) Knese-r's formn of the deter-minant D (t, to): Let 5 (t - tiw) be a point on the continuation of the extremal so beyond the point 0, taken sufficiently near to 0, or else the point 0 itself. Then it follows from our assumptions concerning the general solution (18) of the differential equation (I) that there exists1I a doubly infinite system 2 of extremals passing through the point 5: x (t, a, b), y t (t, ca, b), (40) and satisfying the following conditions: 1. The extremal Co is contained in the system Y, say for a ao, b - bo. 2. The functions and their first partial derivatives with respect to a and b are continuous in a domain 0 17-1 la -aol-dl b - bol (11 (41) where TO < t55 < to < ti < T~11 and di is a sufficiently small positive constant. 3. p2 -l #0 in the domain (41). 4. The value t t5, to vhich corresponds on the extremal (a, b) 1 if x = f(t, a, /, A), y = g(t, a, P, A) represents an extremal passing through the point 5 (say for t= t5), the quantities a, P, A, t5 must satisfy the two equations f (t5, a, /, A) -f(t55, a0, Po/, AO) O, g (t5, a, 3, )-g (t50, aOI f, A5O)= 0 Solving with respect to t5 and A and remembering (20), we obtain the results stated in the text. 226 CALCULUS OF VARIATIONS [Chap. VI the point 5, is a function of a and b, of class C' in the vicinity of aI0, b0. From the definition of t5, according to which, it follows by differentiation that (5k 5 t5 at5 t5 (5 [5 ~~~~~~~~~~(42) 5. X is a function of a, b of class C' in the vicinity of ao, bo, and the two derivatives XI(- x(ao, bo), A.= Xb (ao, bo) are not both zero, since 01 (t) and 02(t) are two linearly independent integrals of 'I'(u) = 0 (compare (33)). We shall denote by F (t, a, b), G (t, a, b), H(t, a, b) G~, (t, a, b),etc. the f unctions of t, a, b into which F, G, H, G_-, etc., change on substituting The integral K t aken along any extremal (a, b) of the system I from the point 5(t -t5) to an arbitrary point t, is a function of t, a, b, which we denote by x ([, a, b): X (t, a, b) =f (t, a, b)dt.(43) Finally we denote by A( t, a, b) the Jacobian of q5,if, x: Then Weierstrass's function D (t, t50) differs from the Jacobian A (t, ao, bo) only by a constant factor: D(t, t50) =CAX (t, ao, bo).(44) ~41.] ISOPERIMETRIC PROBLEMS 227 Proof: For the partial derivatives of x(t, a, b) we obtain the following values Xt G _- (,tG,, + ~t Gy Xa~ (G, 4ia + G,'/'~a + Gx ~ta + Gi'qrta) dt - G Da Applying the usual integration by parts and remembering that' -Gd G Y'U -G - -x'U dt ' dt ' we get /rt t t t x(I- IL U (qi Oa - Otq'a) dt + [Gx, O.+ GY, ]l G a, The terms outside of the sign of integration reduce to Gx, 0, + Gy a on account of (42). A similar transformation applies to xb. We substitute these values of Xt, X,, Xb in A (t, a, b) and then put a = ao, b b=, which makes t5 t,. Writing for brevity a=ao a= ao A -,t,- otV b, B l~t(b -ktlb M fUA dt NfUB dt ve obtain for the Jacobian the expression' A (t, ao, bo) -_ MB - NA (45) It is now easy to establish the relation (44); for if we substitute in one of the differential equations (8) for x, y the functions -P(t, a, b), q1(t, a, b), differentiate with respect to a and then put a m a,, b = bo, ve get T~ (A) Uth / = 0 similarly:,Y(B) + 2,U = I Compare equation (18) of ~25. 2 KNESER, Mat hemnatische Annalen, Vol. LV (1902), p. 95. 228 CALCULUS OF VARIATIONS [Chap. V1 Hence if we set u =- XA - XB, - - _ A2 B 2 A 2 + A2 1+ ^+ 1 2 2. and U satisfy the same differential equations as the functions u. v introduced under b). Moreover, ti and v vanish for t=t5o, since, on account of (42), A (t0) =0, B (to0) 0 Hence it follows that ut = C (t, fo), = (t, t50) + c'u (t, tpO where c and c' are constants. Taking now D(t, to) in the form corresponding to (37) we obtain immediately the relation (44). (1) Mayer's law of reciprocity for isoperimetric problems: The problem: To maximize or minimize the integral J while the integral K remains constant, and the "reciprocal problem": To maximize or minimize K while J remains constant, lead to the saime totality of extremals.l For, if we distinguish the quantities referring to the second problem by a stroke and make the substitution A ' =~ A~ ' ((46) we have - H which shows that the differential equations for the two problems become identical by the substitution X= 1/X. Now suppose that in both problems the given end-points are the same and that, moreover, the values prescribed in the two problems for the second integral are such that one and the same extremal o0, for which X)=t0, satisfies the 1This remark had already been made by EULER; see STACKEL, Abhandlungen auts der [ ariationsrechnung, I, p. 102. ~41] ISOPERIMETRIC PROBLEMS 229 initial conditions for both problems. Then the equivalence of the two problems still holds for the second variation. For since HI Hi=XA', (47) H1 has a permanent sign so long as H1 has, and vice versa. The sign is the same if X is positive, the opposite if X is negative. Further, the conjugate to the point 0 is the same in both problems: to= to. (48) For the system E of extremals through the point 0 is the same in both problems. Besides U= T; hence since the extremal @0 satisfies the differential equation T+ XO U =, we have, along @0: U = - oU, and therefore, according to (45), A(t, ao, bo) = -x0A (t, at, bo), (49) which proves our statement. This result is due to A. MAYER, and has been called by him the law of reciprocity for isoperimetric problems.' e) EXAMPLE XIII (see p. 210): From the expression (12) for H, it follows that X must be negative in case of a maximum. Equation (13) shows, then, that the vector from any point of the curve to the center must be to the left2 of the positive tangent. Of the two arcs which satisfy the differential equation and the initial condi1 Mathematische Annalen, Vol. XIII (1878), p. 60; compare also KNESER, Lehrbuch, pp. 131 and 136. 2 If, as we always suppose, the positive y-axis lies to the left of the positive x-axis. __ __ _~_ ___ 230 CALCULUS OF VARIATIONS [Chap. VI tions only the one above the x-axis satisfies this condition. This arc may be represented in the form 0 = a- XO cos t to t < tl < to + 2 (50) y -= 0 - xO sint ( Hence we obtain 01 (t) =- X, cos t B A 02 (t) 0= - A sill t, 3 (t)- A0 4 Again,, U y - xy FIG. 45 U (= (1/x'2 q_ y'2)', which is equal to - 1/x along (o0, according to (13). This leads to the following expression for D(t, to): D (t, to) = 4X2 sin < (sin o- ocos w), (51) where t - to 2 Hence we easily infer that the parameter to of the conjugate point is: to = to + 2. (52) The arc (o satisfies, therefore, the condition to < to On the other hand, in the problem to maximize the integral T 1 (xy'- xy) -+ xo1/x+ y1 dt, without an isoperimetric condition, the conjugate point to' is determined by the equation ~(t, t,) - X- sin (t -t,) = whence1 to' = to 7r, The same result follows from the geometrical interpretation of Jacobi's criterion: The extremals through A are circles of radius Ao; their envelope is a circle about A of radius 2Ao, which is touched by each circle C through A at the point diametrically opposite to A on a. ISOPERIMETRIC PROBLEMS 231 so that, in accordance vith the general theory, to' > t"' f) EXAMPLE XIV (see p. 211): We have here (121 + X1) hence for a minimum it is necessary that y —> X>O Of the two solutions (14) of the differential equation (I) which satisfy the initial conditions, only the one in which the upper sign is taken in the expression for y+ \, fulfils this condition. For this solution ve obtain 0, (t),&go sinh t, 0,(t) go%(t sinh t - cosh t), 03(t) /3go U (XY_ U - I _ _12 + _ _ 1 (X 2+y2)/o cosh' t Hence follows fUO, dt-[~ cosh~ t11 t f UOdt [ tn-ht], t~~~~~~~~~~~t and the expression for D (t, to) reduces to 1 D(t, t0) -/30(2 cosh(t - to) -2 -(t- to) sinh (t-to)), (64) or, if we put t - t0 - 2o D (t, to) - 4P2 sinh w (sinh w - w cosh w). (Ma) The function sinh c is positive for every positive w, and the function 4 (w) -_sinh o - w cosh o is negative for every positive w, since 0 (co)= 0 and 0'(w) = - o sinh o lFirst given by A. MAYER, Mathenmatische Annalen, Vol. XIII (1878), p. 67. 232 CALCULUS OF VARIATIONS [Chap. VI Hence there exists no conjugate point, and the third necessary condition is always satisfied. The same result is even more easily obtained by using KNESER'S method:1 If we let the point 5 coincide2 with the point 0 and choose for the two parameters a, b the quantities a = t5 b = f, the system of extremals through the point 0 is represented by the equations x -xo b t - a), y-y = b (cosht-cosha), (55) Hence we obtain,t X(t, a, b)= /x'2+ y2 dt= b (sinht-sinha ), (56) and therefore A (t, a, b) = b2 [2 cosh (t - a) - 2 - (t - a) sinh (t - a)], which for a=ao(= to), b =bo(= o) reduces to the expression (54) for D(t, to). ~42. SUFFICIENT CONDITIONS The argumentation of ~28 applies, with slight modifications,3 to the present problem, and leads to a fourth necessary condition for a minimum: 1 Compare KNESER, Lehrbuch, p. 143. 2 Compare the introductory lines of ~ 41, c). 3 These modifications are: 1. The variations 5, -q must now satisfy the isoperimetric condition: 104 + Ki2 = K02, in addition to the conditions stated in ~28, a). To obtain sucn variations, let pi, qi(i=l,2, 3) be arbitrary functions of t of class C' satisfying the conditions: Pi (to) =, qi (to)=O, 1(t2) = 0, q (t2)=0, P2 (t) q3 (t2) -p3 (t2) q2(t2) 1 N i, N. having the same signification as in ~ 39, a). Then the functions t == 'Pl + e2P2 + e33 ' 7 =- el + q2- 2 q+ 2 q3 will satisfy all the required conditions if el,, e3 are determined by the equations () = 2, (t2) =( 2 04=, K 04-K - = - K, which is always possible under the above assumptions concerning pi, qi. 2. A J has to be replaced by A J+AoaK. ~42] ISOPERIMETRIC PROBLEMS 233 If we denote by E(x, y; p, q; j, 1X ) the function derived from H- F XG exactly in the same manner in which the E-function for the unconditioned problem is derived from the function F (see equation (48) of ~28), then the fourth, necessary condition for ca mininmum consists in the inequality' E (x, y; p, q; j5, q Ao) 0 (IV) which must be fulfilled along2 the arc @0 for every direction ), q. The question arises now whether the four conditions (I)-(IV) are sufficient for a minimum. a) Weierstrass's construction: Let C: x = + (s), y- = (s), so s s 1, (57) be any curve of class C', different from o0, joining the points 0 and 1, lying in the region 3 R and satisfying likewise the isoperimetric condition for s we take for simplicity the arc of the curve (. We propose to express the difference AJ = J Jo- Join terms of the E-function. For this purpose we take a point 5 on the continuation of the arc ~o beyond 0, but not on C, and consider with KNESER' the doubly infinite system E of extremals through the point 5: x: x= (t, a, b), y = (t, a, b) (58) introduced in ~41, c), the arc (0 being given by x = { (t, ao, bo), y = (t, ao, bo), to t t, 1 WEIERSTRASS, Lectures, 1879. 2 In the same sense as in ~28, a). 3 Compare ~ 24, b) and ~ 39. 4WEIERSTRASS considers instead the set of extremals through 0. Compare p. 240, footnote 1L 234 CALCULUS OF VARIATIONS [Chap. VI We., shall say that for the curve Q Weier-stross's construction is possible1 if the point 5 can be so chosen that the following conditions are fulfilled: A) Throngh every point 2 of the curve 6 there passes d uniquely defined extremal (2 of the system E: (t, a2, b y 4t, CZ 2 ~3 lying wholly in the region TR and such that the integral K taken along S2 from 5 to 2 has the same value as when taken from ) to 0 along C' and then from 0 to 2 along Q: 2 K52 K50+ K02; (60) FIG. and when 2 coincides with 0 or 1, the FIG. 46 extremal (2 coincides with (o. This means analytically: There exists a system of three single-valued functions t-t(s) a a aCGs) b 1 b (s) such that ~(t (s), a (s), b (s)) /.(s) q(t (s), a (s), b(s)) tli(s), (61) x(t (s), a (s), b(s)) = (s)-JK50 where X(t, a, b) has the same signification as in equation (43), and x( f G(sb(s), i(s), '(s), (s)) ds Moreover: t (so) =to a (so) a, b (so)= b, (62) t (s1) t1, a (s) a0, b (s1) b B) The three functions t (s), a(s), b(s) are of class C' in ICompare KNESER, Lehrbuch, p. 133. ~42] ISOPERIMETRIC PROBLEMS 235 C) If s2 be any value of s of the interval (sosl) and we denote: t2 = t (S2) a 2(), b = b (), t52- = t5 (a,, b2), then the functions and their first partial derivatives with respect to a and b are continuous in the domain t52 t c tv, I cc- c c | c i b-b2 d| c 2, d2 being a sufficiently small positive quantity, and moreover the function2 X(a, b) is continuous at (a2, b2). These conditions admit of the following geometrical interpretation: We adjoin to the two equations (58) the equation z=x(t, a, b). (58a) Interpreting then x, y, z as rectangular co-ordinates in space, the equations (58) and (58a) represent a curve in space, (', whose projection upon the x, y-plane is the extremal E, and whose z-co-ordinate indicates at every point! the value of the integral K taken along C from the point 5 to the point t. We thus obtain, corresponding to the system;, a doubly infinite system;' of curves in space, all passing through the point 5: x=x Y, = y, z =. The particular curve ~0 adjoined to the curve 0o passes, besides, through the two points 0' and 1': 0': x = o Y, yz = z0 = K0, I': x =, y= Y, z = z = K, 1 + In like manner we adjoin to the curve (s a curve in space, IFor the notation see ~41, c). 2 Compare ~41, c). SWEIERSTRASS, Lectures, 1879; compare also KNESER, Lehrbuch, p. 140. 236 CALCULUS OF VARIATIONS [Chap. VI (', by combining with the two equations (57) the third equation = (s) + K50. (57a) The curve G' passes likewise through the points 0' and 1'. The above assumptions A) and B) may then be couched in geometrical language as follows: Through every point 2' of the curve 6' there passes a uniquely defined curve of the set I'; it changes continuously as the point 2' describes the curve (' from 0' to 1' and coincides with C0' when 2' coincides with 0' or 1'. Under the assumption that Weierstrass's construction is possible for the curve (S, we consider as in ~20, b) and ~28, d) the integral J taken from 5 to an arbitrary point 2(s=s2) of ( along the uniquely defined extremal @2, and from 2 to 1 along (5, and denote its value regarded as a function of 82 by S(s2): S (s2) = J52 + J21 Then as in ~20, b) J=- - [S (s)-S (s)] The integral K taken along the same path has the constant value 1+ K50: K50 + = K52 + K21 since K0= K02 + K2K-I and K-52 02+K50. Hence it follows that we may write dS(s2) _ (dJ52 dK 52(d J21 d K21 (3) ds2 + ds A ds2 + ds2 + 2s) * (63) Proceeding now as in ~28, d) and remembering that the extremal (2 satisfies the differential equations Hx - dt = ~0 dHy dtH dt' where H =F+ 2 G, we obtain the result ~. 121 ~ 12] ISOPERIMETRIC PROBLEMS23 237 dSs2) _ E (X2, Y2; P2, q2; P2,~(2 2 the direction-cosines P2', 2 and P2, q2 referring to the curves @'~2 and 0, respectively. The resnit can again easily be extended to curves Cs having a finite nnmber of corners. Thus we finally reach the result' that whenever Weier-, strass's construction is Possible for the curve Q, Welerstrass''s theorem also holds: zAJ- F (X2 12; P2, q12; Ps, ~2X)s (12M)8 b) Hence we infer that A J 0 whenever E(X2, 112 P2, q2; P2 12 X2) 0 throughout (s~s1) If, moreover, the E-function vanishes only when -P2 -P-2, q i2 q and if besides A (t2, a2, b2) Z# 0 along G AJ cannot be zero, and therefore zAJ>O. Proof:.2If we differentiate equations Q61) with respect to s, we obtain dt c/a c/A - Otds c/0t s c/0, s cit ca db - ds' d bc/ dt c/a dbA c/s dcs c/s Now if P2 P2 i tq2 q2, we have at the point 2: and therefore, since' I wEIERSTRASs, Lectures, 1879; compare KNEsErz Lehi-buch, p. 134. 2IDue to KNESER, Lehe-buch, p. 131. 3 Compare ~ 41, c). 238 CALCULUS OF VARIATIONS [Chap. VI X' = G, Xt = G, also' X = kXt, on account of the homogeneity of G. Substituting these values in the above equations, we see that either a (t2, a2, b2) = 0 or else da db s =0, - =0 ds ' ds Hence if A (t2, b2, b,) 0 along2, C2 and b2 must be constant along (, and, on account of (62), their constant values must be a (s) = a, b (s) = 0, that is: ( is identical with the extremal,0, which is in contradiction to our assumption that G shall not coincide with 0o. Hence the statement is proved. c) In many examples the above theorem is sufficient to establish the existence of an extremum. EXAMPLE XIII (see p. 229): The system 2 is the totality of circles through the point 5: x - x5 b (cos t - cos a), y - y = b (sin t - sin a), the parameters being a = t5, b = - X. The ordinate z erected at the point t of the circle (a, b) is the length of the arc of this circle from the point 5 (t = a) to the point t: =Ib(t-a)1. (66) The system z' of curves in space is therefore a system of helices. Through every point (x, y, z) for which Z> 1/(X - 5)2+ (y _ y)2>, (67) 1 This means geometrically: If e2 touches A, then also e2' touches e'. 2The result remains true if A (t2, a2, b2) = at a finite number of points. ~42] ISOPERIMETRIC PROBLEMS 239 there passes one and but one curve of the system I' for which a <t < a + 2r, b > 0. (68) Moreover the inverse functions t, a, b of x, y, z thus defined are regular' in the vicinity of every point (x2, Y2, z,) satisfying the inequality (67), and take, at the points (xo, yo, Zo) and (xi, yl, Iz) the values to, ao, bo and ti, ao, bo respectively.2 Now we join the two points 0 and 1 by an ordinary curve (S, whose length has the given value I and which does not pass through 5. Then for every point 2 of (s the sum of the lengths of the arc 50 of the circle eo and of the are 02 of (S is greater than-never equal to —the distance between the two points 5 and 2, which in its turn is greater than zero, since (6 does not pass through 5, i. e., the condition 1 Proof: On setting t+a t-a 2 -' 2 the equations for the determination of t, a, b become x -- X5=- 2b sin y sin w, - 5= 2b cosy sin o, (69) z = 2bo. Hence if we put 1/(x-x - 5)- (y -,) = u, and suppose 0 < o <, we get u = 2b sin o, and therefore we obtain for the determination of o and y the equations: sin to s-n =, y -y-i (x - X) = uCe1 (70) where v = u/z. Since, according to equation (67), 0 < v < 1, the transcendental equation for o has one and but one solution in the interval: 0 < w < rr. Moreover if 0 <v2 < 1 be any particular value of v, this solution to is regular in the vicinity of v = v2, since the derivative of the function sin ow/o is: 0 for 0 < o < r. Similarly the equation for y has a unique solution in the interval 0 y<27r, which is a regular function of x, y in the vicinity of every point (x, y2) different from (X5, y5). The values of X and y being found, the quantities t, a, b are obtained immediately. They satisfy the inequalities (68) and are regular functions of x, y, z in the domain (67). 2 For, of the two arcs of circles of the system E which pass through the point (x, y) and have the given length z, the one is described in the positive sense (so that the center is to the left) if we start from the point 5, the other in the negative sense. For the former the inequalities (68) are fulfilled, for the latter, they are not. On the other hand the arcs 50 and 51 of %o are, according to ~41, e), described in the positive sense, and are therefore contained in the above system of uniquely defined solutions. 240 CALCULUS OF VARIATIONS [Chap. VI z2 > (xz - x)2 + (Y2 - y)2 > 0 is fulfilled.1 Hence it follows that Weierstrass's construction is possible for the curve (. Further we find easily that E(x2, Y2; P2, q2; p2, q2 12) = X( - cosa2), (71) where a2 is the angle between the positive tangents to the two curves (2 and (C at the point 2. X\ is negative in (SoSi) (since it is equal to - b2), and a2 cannot vanish identically in (soSi). For, according to (51), A (t2, a2, b2) = 4 Xj sin)2 (sin 2 - W2 cos 0)2) and therefore A(t2, a2, b2) = 0 in (sol), since 0 < &2 < r. Hence it follows that AJ<0, and thus we reach the result that the arc of circle &o furnishes a greater value for the area J than any other ordinary curve of the same length which can be drawn between the two points 0 and 1. The same reasoning, slightly modified,2 leads to the theorem If we had taken, instead of the system of extremals through 5, the system through 0, the above inequality would be true only with certain exceptions which would require a special discussion. Compare p. 233, footnote 4. 2The curve d is now closed; accordingly the points 0 and 1 coincide. If we let also the point 5 coincide with 0 and consider two points 3 and 4 of 6 for which so < S3 < S4 < s, we obtain by the same reasoning as above x S4 S (4) -S (3) = 2 (1 - cos a2) ds2. S3 Now let S3 and s1 approach so and sl respectively, then we get 51 Jo-IJ1 = 2 - 2 (1 -cos a2) d2 Jol being the area of a circle of the given perimeter 1. Hence Jol < J01 The previous method is not applicable when the curve 6 begins at the point 0 with a segment of a straight line, because then the inequality (67) is not satisfied for the point 3. In this case, take the point 3 beyond the end-point 6 of this rectilinear segment and let 3 approach 6. Then S(s3) approaches again Jol with the same result as before. ISOPERIMETRIC PROBLEMS 2-1 that among all closed curves of given length the circle includes the maximoun area. EXAMPLE XIV (see p. 231): Any admissible curve ( being given, we choose the point 5 so that for every point 2 of ( x2 > X5 Then through every point 2' of the space curve (' one and but one curve of the system1 z': X - X = b (t- a), 5 2 1 y - 5 = b (cosh t - cosh a), (72) z b(sinh sinhsinh a), ~ can be drawn for which FIG. 47 t>a, b>O. This follows from the determination of constants given in ~39, d). At the same time it is easily seen, in the same manner as in the preceding example, that all the conditions for Weierstrass's construction are fulfilled. Further we find E (x2, y2; P2, q2; P2, q2 1 X2) = (Y2 + X2) (1 - Cos a2), (73) where a2 has the same signification as in (71). But, according to ~41, f), Y2- +X2 = b2 cosh t2 > 0 since b2 > 0, and a2 cannot vanish identically along (5 since (t2, a2, b2) 0 along (. Hence we infer that Jo1 > J01, i. e., the catenary Co has its center of gravity lower than any other ordinary curve of equal length which can be drawn between the two points 0 and 1. d) "Field" about the arc ~0: Returning now to the general case, we meet with a peculiar difficulty which has 1Compare equations (55) and (56). 242 CALCULUS OF VARIATIONS [Chap. VI no analogue in the unconditioned problem. Suppose that for the arc 0o, which we assume to be free from multiple points, the conditions H > 0 (II' and t, < to (III' are fulfilled. Does it follow, then, that the are 0 can be surrounded by a neighborhood (p) such that for every admissible curve ( which lies wholly in this neighborhood, Weierstrass's construction is possible? In the unconditioned problem and under the analogous assumptions, this question could be answered in the affirmative;' for thle isoperimetric problem the questiotn /7as not yet been answelred. Only the following milder statement can be proved: If conditions (II') and (III') are fulfilled, a neighborhood2 (p') of the space curve ~0 adjoined to the are (0 can be assigned such that Weierstra ss's construction is possible for every admissible curve G whl ose corresponding space curve ( lies wholly in the neighborhood (p') of &o. The proof proceeds by the following steps: 1. If conditions (II') and (III') are fulfilled, we can take the point 5 so near to 0 that for the system of extremals through the point 5 not only the conditions enumerated in ~41, c) are satisfied, but, besides, the following:3 A (t, 0, bo) i 0 for t t- t. (74) 1 Compare ~28, d) and ~34. 2We understand by the neighborhood (p') of the arc 8%' the portion of space swept out by a sphere of radius p' whose center describes the arc %o'. 3For the proof remember (44), and notice that the condition for a permanent sign of 82J may also be written D(tl, t) =0 for to t<t, (compare ~41, a)). The statement follows then by a slight modification of the analogous proof given by C. JORDAN, Cours d'Analyse, Vol. Ili, No. 393. ~ 42j ~ 42j ILSOPERIMETRIC, PROBLEMS; 2. By an extension of the inethod of 384 we can now prove the existence of a "field" _19~ about the arc 0 If:43denotes the- domain and _Ap the image Of:4k in the xy, z-space defined by the transformation then the two positive qnantities k and E can be taken so small that the correspondence between:4k and O' is a oneto-one correspondence, and that at the same time in 34. The single-valned functions t, at, b of x, y, z thus defined are of class (7 in O', and a neighborhood (p') of the arc Ecan be inscribed in O. It follows now easily that for every admissible cnrve whlo~se adjoined space cw-rve lies whiolly in the "field" kj, We i e rstr a ss's construcition is possible. e) Sufficien t condcitions for at sent i-str-ong mniinnmn. Suppose now that in addition to the conditions (II') and (Ill') the inequality holds along the arc CS' for every direction P5, ecPt p) _p, q-(]. Then it follows fromn continuity considerations that we can take, k so small that along every admissible curve CS satisfying the above additioal onitinexcept at the points whereP2P2 q, at which E vanishes. 244 CALCULUS OF VARIATIONS [Chap. VI From Weierstrass's theorem and the inequality (75) it follows now that for every such curve ( AJ>. Hence, if we modify our original definition of a minimum and say: "The arc (0 furnishes a semi-strong minimum for the integral J if there exists a neighborhood (p') of the adjoined arc eo such that AJ 0 for every admissible curve G whose adjoined space curve 0' lies wholly in this neighborhood (p')," we can enunciate the Theorem:' The extremal %o (which we suppose free from?multiple points) furnishes a semi-strong minimum for the integral J with the isoperimetric condition K= 1, if the conditions (II'), (III'), (IV') are fulfilled. It must, however, be admitted that the restriction which we impose in the "semi-strong" minimum upon the variations of the arc eo, is rather artificial and alters completely the character of the original problem.2 1WEIERSTRASS, Lectures, 1882; compare KNESER, Lehrbuch, ~~36 and 38. Mayer's law of reciprocity extends to the sufficient conditions for a semi-strong extremum, since, in the notation of ~41, d), L =l1/AE. Compare KNESER, Lehrbuch, ~ 36. 2As a matter of fact the preceding theorem does not contain a solution of the isoperimetric problem originally proposed, but a solution of the following problem, which is usually (but unjustly) considered as equivalent to the isoperimetric problem, viz.: Among all curves in space which pass through the two points X = X, y=,z- =O and x= xl, y= y1, z= I and satisfy the differential equation dz c (X Y,) dz -t = GT (x, y, x, y'), to determine the one which maximizes or minimizes the integral J= F(x, y, x', y') dt. to CHAPTER VII HILBERT'S EXISTENCE THEOREM ~43. INTRODUCTORY REMARKS IF a function f(x) is defined for an interval (ab), it has in this interval a lower (upper) limit, finite or infinite, which may or may not be reached. If, however, the function is continuous in (ab), then the lower (upper) limit is always finite and is always reached at some point of the interval: the function has a minimum (maximum) in the interval. Similarly, if the integral tl J=- F(x, y, x', y')dt is defined for a certain manifoldness At of curves, we can, in general, not say a priori whether the values of the integral have a minimum or maximum. But the question arises whether it is not perhaps possible to impose such restrictions either upon the function F or upon the manifoldness ti (or upon both), that the existence of an extremum can be ascertained a priori. In a communication to the "Deutsche MathematikerVereinigung" (Jahresberichte, Vol. VIII (1899), p. 184), HILBERT has answered this question in the affirmative. He makes the following general statement: "Eine jede Aufgabe der Variationsrechnung besitzt eine Losung, sobald hinsichtlich der Natur der gegebenen Grenzbedingungen geeignete Annahmen erfi.llt sind und notigenfalls der Begriff der L6sung eine sinngemasse Erweiterung erfahrt," and illustrates the gist of his method by the example of the shortest line upon a surface and by Dirichlet's 245 246 CALOULUS OF VARIATIONS [Chap. VII problem. In a subsequent course of lectures (Gottingen, summer, 1900) he gave the details of his method for the shortest line on a surface, and some indications' concerning its extension to the problem of minimizing the integral 1i J-=j F(x, y, y'dx Xo We propose to apply, in this last chapter, Hilbert's method to the problem of minimizing the integral2 J t F(x, y, x', y') dt, with fixed end-points, under the following assumptions, where % denotes, as before, a region of the x, y-plane, and o0 a finite closed region contained in the interior of X: A) The function F(x,, x', y') is of class C"' and satisfies the honiogeneity condition FT(x, y, kx', ky') = kF (x, y, x', y'), ki> O throughout the domain GI: 11(x,y) in, x 2+y'2 O B) Tite finction F(x, y, cos y, sin y) is positive throughout the domain:,y) iln 0, 0 y 27r. C) Tlh fiunction F1 (x, y, cos y, sin y) is positive throughout the domain o0. 1 In his thesis, Eine neue Methode in der Variationsrechnung (Gottingen, 1901), ~~5-14, NOBLE has discussed the details of the proof for this case. But. his conclusions do not possess the degree of rigor which is indispensable in an investigation of this kind. In particular, the reasoning in ~~9, 10 and 13 is open to serious objections. 2 For the special case where F is of the form f(x, y)V/ x'2-+ '2, LEBESGUE has given a rigorous existence proof by an elegant modification of H ilbert's method in a recent paper, "Integrale, longueur, aire," Annali di Matematica (3), Vol. VII (1902), pp. 342-359. LEBESGUE applies H ilbert's method also to the more difficult case of a double integral of the form ff IIEG-_F2 dudv. ~44] HILBERT'S EXISTENCE THEOREM 247 D) The region i0 is convex (i. e., the straight line joining any two points of 0o lies entirely in the region T0) and contains the two given points which we denote' with HILBERT by A~ and A1. Under these assumptions we propose to prove 1. That for every rectifiable curve 2 in the region o0 the generalized integral Jf (according to WEIERSTRASS'S definition) has a determinate finite value. 2. That there always exists, in the region o0, at least one rectifiable curve o2, joining the two given points A~ and A1, which furnishes for the generalized integral Jo an absolute minimul m with respect to the totality of all rectifiable culrves uwhich can be drawn in S Ofrom A~ to A1. 3. That this minimizing curve QO is either a single arc of an extremal of class C', or else is made up of a finite number or of a numerablinfinitude of such arcs separated by points or segments of the boundary of the region LO. ~44. THEOREMS CONCERNING THE GENERALIZED INTEGRAL Ji In ~31 we have considered Weierstrass's extension of the meaning of the definite integral J- F(x, y, x', y') dt to curves having no tangent. Another definition of the generalized integral has been given by HILBERT2 in his lectures. This definition, while 1The advantage of this notation will appear in ~45. 2HILBERT'S own definition is asfollows (see NOBLE, loc. cit., p.18). Let u1 be a partition of the arc AB of a continuous curve into segments. Consider the totality of all analytic curves which can be drawn from A to B and which have at least one point in common with each of the segments. Let J1 denote the lower limit of the values of the integral J taken along these curves. Next, let II2 be a new partition derived from II1 by subdivision, J2 the corresponding lower limit, and so on. Then HILBERT defines the upper limit of the quantities: J1, J2, J31, ' ', J,, ' if it be finite, as the value of the definite integral J taken along the arc A B. 248 CALCULUS OF VARIATIONS [Chap. VII leading to the same value for the generalized integral as Weierstrass's definition, is better adapted to our present purpose, especially in the simplified form which has been given to it by OSGOOD.' a) Hilbert-Osgood's definition of the generalized integral: We shall use the following notation: P' and P" being any two points of the region o0, we denote by t(P'P") the totality of all ordinary curves which can be drawn in the region T0 from P' to P", and by i(P'P") the lower limit of the values which the integral J = Fl(x,, x', ')dt takes along the various curves of (P'P"). This lower limit is always positive. For, according to A) and B), the function F(x, y. cosy, sin 7) has a positive minimum value mn in the closed domain o0. Hence, if G be any curve of A (P'P"), we obtain, by taking the arc as independent variable on the curve 0, 0 <mP I'P'" I n J (PP"), (1) where I denotes the length of the curve ( and IP'P"I the distance between the two points P', P". Hence it follows that a0 < mP'P" (P'P"). (2) After these preliminaries, let x = x (t), y =, (t), to t E t, be a continuous curve lying wholly in the region o0. If the functions + (t), + (t) are not differentiable, the integral J taken along 2 has no meaning. In order to give it a meaning also in this case, we consider any partition II of the interval (tot1) 1 OSGOOD, Transactions of the American Mathematical Society, Vol. II (1901), p. 291, footnote. ~{44] HILBERT'S EXISTENCE THEOREM 249 nI: to < 1 < T2 * * < 1T,_1< tl, and denote by A. -- Po, PI, P2, "', P,,-1, B = P, the corresponding points of the curve i. Then we form the sum n-i Sn= - (PvPi) V=0 The upper limit of the values of SI for all possible partitions II we define as the value of the integral J taken along the curve 2 from A to B, and we denote it by J*'(AB), or simply Ji**. It is easily seen that SI, may also be defined' as the lower limit of the values of the integral J taken along all ordinary curves which can be drawn in Wo from A to B and which pass in succession through the points P1, P2, * * *, Pn-_ Hence it follows that it is always possible to select a sequence {v\} of ordinary curves joining A and B, lying in T0, and such that L JC = Jo*. V=oO The above definition of the generalized integral is a direct generalization of PEANO'S' definition of the length of a curve. For, in the particular case F = 1/x'2 + y'f, the sum SIS reduces to the length of the rectilinear polygon with the vertices A, P1 P2, * * *, Pn-1, B. We must next investigate under what conditions the generalized integral Je* is finite, and show that for ordinary 1This is the form which OSGOOD gives to Hilbert's definition; see the reference on p. 248, footnote 1. 2PEANO, Applicazioni geometriche del Calcolo Infinitesimale, p. 161. 250 CALCULUS OF VARIATIONS [Chap. VII curves the generalized integral is identical with the ordinary definite integral. b) Conditions for the finiteness of the generalized integral: The function F(x, y, cos y, sin 7) has a finite maximum value M in the domain E0. Hence it follows that for every curve ( of AM(P'P") i (P'P"') (P'P") ll, (2a) 1 denoting again the length of the curve (. We may choose for the curve ( the straight line P'P", since, according to assumption D), the line P'P" lies wholly in the region o0. Then we obtain the further inequality i(P'P") - iP'PI". (3) From (2) and (3) follows at once n- 1 - 1 m I PvPv+i,II Sn I P, P+1 | (4) 1=(} V=0 But the upper limit of the sum P-Pv+1 V=O is, according to Peano's definition, the length of the curve i. Hence we obtain the Lemma: In order that the generalized integral JY' may be finite, it is necessary and sufficient that the curve 2 shall have a finite length (in Peano's sense). We confine ourselves, therefore, in the sequel to continuous curves 2 having a finite length ("rectifiable curves"' in JORDAN'S terminology).' From (4) it follows further that m AB | J** (AB) ML, (5) where L denotes the length of the curve S. J. I, No. 110. ~44] HILBERT'S EXISTENCE THEOREM c) Properties of the generalized integral: From the two characteristic properties of the lower limit it follows readily that for any three points P, P', P" of lo the inequality holds: i (PP') +(P'P")_ i(PP"). (6) Hence it follows that if II1 denotes a partition derived from II by subdivision of the intervals of II, then SI1 SI ~ Hence we easily infer that we get the same upper limit Ji* for the values of SII if we confine ourselves to those partitions II for which Tv1 - Tr < 8, (v = 0, 1, 2, * *, n - 1; s — to, = to ) = t), S being an arbitrary positive quantity. Following now step by step the same reasoning which JORDAN uses in his discussion of the length of a curve, we can easily establish the following properties of the generalized integral, always under the assumption that the curve 2 is rectifiable: 1. The generalized integral Jew (AB) is at the same time the limit which the sum Sin approaches as all the differences Tv+ - Tv approach zero. Combining this result with the inequality (4) we obtain the new inequality mL ~ Jf (AB). (7) 2. If P be a point on the curve 2 between A and B, dividing the arc 2 into the two arcs 21 and 22, then also the integrals J, (AP) and J* (PB) are finite, and2 J** (A B) = JQ* (AP) + Jf* (PB). (8) 3. The generalized integral J*' (AP) is a continuous3 1 Compare J. I, No. 107. 2 Compare J. I, No. 108. 3 Apply (8) and (5). 252 CALCULUS OF VARIATIONS [Chap. VII function of the parameter t of the point P and increases continually as P describes the arc AB from A to B. d) Comparison with Weierstrass's definition of the generalized integral: If P' and P" are two points of X0 whose distance from each other is less than the quantity po defined at the end of ~28, e), P' and P" can be joined by an extremal ( of class C' which furnishes for the integral J a smaller value than any other ordinary curve which can be drawn in the region Bo from P' to P". If the extremal e itself lies entirely in the region o0, the value which it furnishes for the integral J is equal to i(P'P"); if ( lies partly outside of i0, this value is equal to or less than i(P'P"). Now consider any partition II for which tV+1 - Tv < 8, (V = 0, 1, * * *, n - 1), 8 being chosen so small that IP'P"I < po for any two points P', P" of 2 whose parameters t', t" satisfy the inequality t'-t" < 8. Then we can inscribe in the curve 2 a polygon of minimizing extremals with the vertices A, P1, P2, *, Pn-1, B. As in ~31, d), let U. denote the value of the integral J taken along this polygon of extremals. If the curve 2 lies entirely in the interior of o0S, 8 can be taken so small that the polygon lies in the region o0, and therefore U1 = S. Hence J2* may in this case also be defined as the limit of U.. If 2 has points in common with the boundary of o0, U1 may be less than SI1. Nevertheless, also in this case the limit of UnI for LA=-O is J,*. In order to prove this statement we consider, along with ~45] HILBERT'S EXISTENCE THEOREM 253 the two sums Si and Un, the sum VT defined in ~31, c), i. e., the value of the integral J taken along the rectilinear polygon AP1 P2* * * Pn-_B. Since the region 0o is convex, this polygon lies entirely in To, and therefore we have the double inequality Un < S ~ V1. (9) From the first part of this inequality it follows that U,, has a finite upper limit Jf*. This upper limit is at the same time the limit which Uin approaches for L Ar 0, as can be inferred1 from the fact (proved in ~31, e)) that if II' be a partition derived from 1H by subdivision, then U1, U,. Hence it follows, according to ~31, c) and d), that Vn approaches the same limit as U.1; therefore we obtain, on account of (9), and remembering the equations (77) and (80) of ~31: Ja*= Jf, (10) i. e., we have the resultthat Hilbert-Osgood's definition leads for the generalized integral to the same value as Weierstrass's definition. Hence it follows, according to ~31, b), that for an "ordinary" curve the generalized integral coincides with the ordinary definite integral. ~45. HILBERT'S CONSTRUCTION We are now prepared to apply HILBERT'S method to the integral2 Jo. Accordingly we consider the totality of all rectifiable curves 2 which can be drawn in the region 0o from the point A~ to the point A1. The corresponding values of the integral J2 have a positive3 lower limit. We propose to 1 Compare J. I, No. 107. 2 On account of (10) we may use the symbol J* instead of J". 3 According to (5). 254 CALCULUS OF VARIATIONS [Chap. VII prove that under the assumptions A)-D) enumerated in ~43, there exists at least one rectifiable curve &Q drawn in io from A~ to At for which the integral J* actually reaches its lower limit. a) Construction of the point A-1: We consider the totality of ordinary curves fi (A~A1) which can be drawn in the region ]o from A~ to Al, and denote the lower limit i(A0~1) of the corresponding values of the integral J by K' i (AOA1). = K We can then select1 an infinite sequence of curves (1, (2, *''',, V.., belonging to AII(A~A1) such that the corresponding sequence of values of the integral J, which we denote by J1, J23 Jv1, ' approaches K as limit: L J, = K. V=oo On the curve,V there exists2 one and but one point A- such that J, (A~ A-)= J These points A' are infinite in number;3 they lie in the finite' 1Compare JORDAN'S definition of "point limite,' loc. cit., No. 20, and an analogous remark in E. II A, p. 14. 2 Since F is positive along qv the integral J taken along the curve Cv from A~ to a variable point P, in c r e a s e s continually as P describes the curve (,v from AO to A1; hence it passes through every value between 0 and Jv once and but once. 3They need not all be distinct; the conclusion holds even if there are only a finite number of distinct points among them. For in this case an infinitude of the points A- must coincide with at least one of the distinct points; this point has then the properties of the point A]. 4The existence of the accumulation point Al can also be proved without making use of the finiteness of o0. From (1) it follows that O 1 Jv Hence if we select G>J,(v = 1, 2, 3,- * *), which is always possible since L J, is finite, the points Al lie in the interior of the circle (A~, G/2mn), and therefore have an accumulation point. ~451 HILBERT'S EXISTENCE THEOREM 255 closed region o 0; hence there must exist at least one point A- in io such that every vicinity of Al contains an infinitude of the points Al. Moreover, we can select a subsequence {(S k of the sequence {Q(S such that LAl = Al kc=-o k b) Hilbert's lemma concerning the point Al: We consider next the totality of curves < (A~A). Then the fundamental lemma holds that the lower limit of the corresponding values of the integral J is -K: i (A~AI) = 1 i (A~AI ) = K. (11) Proof: We denote by /k the curve made up of the arc A~A' of the curve ek and of the straight line Ak A'; the latter lies entirely in S0 since io0 is convex. According to (2a) the integral J taken along the straight line A' A2 is at most equal to Mf A^,A-I. Therefore LJ, (AO~A) )={K k=- c k since L 1 v K and L IAA A-A1 =. kC-=oo k=oo c Hence it follows from the characteristic properties of the lower limit that i (A~A-) ~I. In the same way we prove that i (A"A1)). But, on the other hand, according to (6): i (A~A-) + i (A7A1) ) i (AA1). The three inequalities are compatible only if separately: i (A~A4) = K and i (AA1) = K. 256 CALCULUSz OF VARIATIONS [Chap. VI1 c) The jpoints A,1/2' Repeating the process of section a) with the points AO and Al we obtain a new point, A', lying in the region TWO and having the characteristic property that i(AOA]I) -i (A "A4) - i (.AOA1) - T K In like manner we derive from the two points A' and Al a point, A', satisfying the relation By an indefinite repetition of this process we. obtain an infinite set of points 1 =0, 1,2,.. all lying in property that More guenerallif the region 1?&0 and having the characteristic i ~ 17 4- 1K. (12) i (Ar AT") = (T"f- rf) K, (13) f q,? 1 q f T.~~~ where n', n" are integers, q', q" odd integers, and O =,r<Trl For, reducing Yr' and T" to the same denominator q,,~+ T - T r 2i we obtain, according to (6) and (12), i A"A211 ~ ~ ~ ~~~~.2 ~45] HILBERT'S EXISTENCE THEOREM 257 whence q q q +r. q+ i A~A) + i (AA 2 ) + i A A)- K. But on the other hand, we have, on account of (6), K i A~A Al) i (A~A) + i AA ) + i (A 2 A) The two inequalities are compatible only if in each of the above three formulae the equality sign holds, which proves (13). From (2) and (13) follows the important inequality I AT'AT" l < (T_ )K, (14) where IAT'A"'" denotes again the distance between the two points A', A'". Let us now denote by x (), (T) the rectangular co-ordinates of the point AT, T being one of the fractions q/2n considered above. Then lx(Tr)-X(T )1 AT'AT I, IY(T')-J(T")IIA.A7T, and therefore on account of (14) IY (T') - yr(") (T"-.Tr) ( d) The remaining points of Hilbert's curve: The meaning of the two functions x(t), y(t), which so far have been defined only for values of t of the set _ ( l q-= 0, 1, 2,..., 21 22 ' n=O, 1, 2,., can now be extended to all values of t in the interval O tl as follows: From the inequalities (15) we infer by means of the gen 258 CALCULUS OF VARIATIONS [Chap. VII eral criterion' for the existence of a limit, that if the independent variable t approaches in the set S any particular value t -a of the interval (01), then the functions x (t), y (t) approach determinate finite limits. In symbols, the limits2 L x (t) and L y (t) tls tls t=a t=a exist and are finite. Moreover, if a itself belongs to the set S, then Lx (t)= x (a), L y(t)= y (a) (16) tlS t S t=a t=a If a does not belong to the set S, we define, according to Hilbert, the functions x(t) and y(t) for tc a by the equations (16). The two functions x(t), y(t) thus defined for the whole interval (01) are continuous and "of limited variation.".3 For, the two inequalities (15), which have been proved for values T'<T" of the set S, can easily be shown to hold for any two values t' <t" of the interval (01), by considering two sequences {T'} and {Tr'} belonging to the set S and such that L TV t', L T= t. From the inequalities (15) thus extended, it follows at once that the two functions x(t), y(t) are continuous and "of limited variation." I Compare E. II A, p. 13. 2 The notation according to E. H. MOORE, Transactions of the American Mathematical Society, Vol. I (1900), p, 500. 3 Compare J. I, No. 67. Let f(t) be finite in the interval (totl), and let In: to < < T2. '. < Tn-1 < tl be a partition of this interval. If then the upper limit of the sum If l (r- +) -- f(i)I, (To= o, = tl) =0 if( ( ) to t for all possible partitions uI is finite, f(t) is said to be "of limited variation." ~46] HILBERT'S EXISTENCE THEOREM 259 Hence the curve 20 defined by the two equations So: x = (t), y=y(t), 0 t 1 (17) is continuous and has a finite length,l i. e., it is a rectifiable curve. As t increases from 0 to 1 the point (x, y) describes the curve 20 from the point A~ to the point A1. Moreover, the curve o2 lies entirely in the region o0, since o0 is closed. ~46. PROPERTIES OF HILBERT'S CURVE It remains now to prove that the curve 20 actually minimizes the integral J* and has the further properties stated in ~43. a) Minimizing property of Hilbert's curve: The fundamental equation (13) which has been proved for values T', T" of the set S only, can easily be extended2 to any two values t' < t" of the interval (01): i (At'At") (t" — t') K. (13a) But from (13a) it follows immediately that the generalized integral 1 Compare J. I, Nos 105, 110. 2For the proof, we introduce the same two sequences I r, T T' ' as above. Then we have, on account of (6), i (At'A -) + i (ATvArv) + i (A At" ) i (At'At" Passing to the limit v oo we obtain, on account of the continuity of the functions x(t), y(t), L At'A V=0=O, L IA^VAt"I =O, and therefore, on account of (3), L i(At'ATV)=0, L i(ATvAt")=0. V=Xi V=c Moreover L i(AvArTv ) = (t"-t') K V= -- OO on account of (13). Thus we obtain i (At 'At ). (t"-t')K. And by the method employed in proving (13) we finally show that the inequality sign is impossible. 260 CALCULUS OF VARIATIONS [Chap. VII Jt (AA'1) taken along HILBERT'S curve o2 is finite and that its value is equal to i(A~A1). For let II be any partition of the interval (01): II: To = 0 < T1 < T2 * *. < r,,_1 < =Tn Then we obtain, according to (13a), n — SI = i (ATATv+) =K =- i (A0A1) V=0 Hence also the upper limit of the values of Sn is equal to K, that is Jt (A0A1) = i(A~A). (18) From the definition of the symbol i(A0A1) as lower limit it follows now that if ( be any ordinary curve drawn in?0 from A~ to Al, then Jo$o (AOA') Jas (AOA-);Moreover, if S be any rectifiable curve drawn in i0 from A~ to Al, and e any preassigned positive quantity, we can always find, according to ~44, a), an ordinary curve 6 of I (A~A1) such that I J ((AOA1) - J (AOA1) I < E Hence it follows that J (AOA1) J (AO~A) (19) This proves the theorem enunciated at the beginning of this section: If the conditions A)-D) enumerated in ~43 are fulfilled, then there always exists at least one rectifiable curve joining the two points A~ and A1 and lying entirely in the region 1o, which furnishes for the integral J= F (x, y, x', y') dt, generalized, an absolute minimum with respect to the totality of rectifiable curves which can be drawn in o.from A~ to A1. ~461 HILBERT'S EXISTENCE THEOREM 261 b) Analytic character of Hilbert's curve: Let T' denote the totality of those values of t in the interval (01) which furnish points of the curve 20 in the interior of the region T0, T" the totality of those which furnish points of 0S on the boundary of o0. From the continuity of 20 it follows that every point1 t' of T' is an inner2 point of T'. Hence an interval (ad) contained in (01) and containing t' in its interior can be determined such that all points in the interior of (a/3) belong to T', whereas the end-points belong to T" except when they coincide with the points 0 or 1. The set T' consists, therefore, of a finite or infinite number of such intervals (a/3) which do not overlap. According to a theorem of CANTOR'S,3 the totality of these intervals is numerable, so that we may denote them by The curve 20 consists, therefore, either of a finite number or of a numerable infinitude of interior arcs separated by points of the boundary of 0o We are going to prove, according to HILBERT, that each interior arc of 0o is an arc of an extremal of class4 C". For let P(t) be a point of Hilbert's curve 20 in the interior of the region o0. Then according to ~28, e) a circle (P, o) can be constructed5 about P such that any two points P', P" in the interior of the circle can be joined by an extremal ( of class C" which lies entirely in the region io and which furnishes a smaller value for the integral J than any other ordinary curve which can be drawn in i0 from P' to P". 1Except the end-points of the interval (01) in case they should belong to T'. 2 Compare J. I, No. 22. 3 Mathematische Annalen, Vol. XX, p. 118. 4From our assumption C) it follows according to ~6, c) that every arc of an extremal of class C' which lies in c0, is ipsofacto also of class C". 5 Let d be a positive quantity, taken so small that the circle (P, e) lies in the interior of MO, and let po be defined for the region E0 as in ~28, e). Then choose for o the smaller of the two quantities d/3 and po/3. 262 CALCULUS OF VARIATIONS [Chap. VI1 On account of the continuity of the functions x(t), y(t) there exists a vicinity (t -, t + 8) of t such that the arc of the curve 20 corresponding to the interval1 (t-8, t -+ ) lies wholly in the interior of the circle (P, o). Let PI(ti) and P3(t3) be two points of this arc (t < t.3), and denote by @z the minimizing ex7/ — s, tremal joining P1 and P3. V P / )- We propose to prove that the ar'c P1P3 of HILBERT'S curve 20 is identiy.^, y r cal with the exthremal 23. FIG~ 48 Consider any point P2(t2) of the arc FIG. 48 P1P3 of 20 and denote by @3, @1 the m minimizing extremals joinng P1, P2 and P2, P3 respectively. Then it follows from the minimizing properties of the extremals @1, e2, @3 and from (13a) that J (P P2) = i(P P,) = (t2 - t,) K Jl (P2P) = i (P2P3) = (t - t) K, J(S (P P3) = i (PI P3) = (t, - t) K; hence, adding: J (P P3) = J( 3(PI P2) + J1( P2P3). The extremal @2 furnishes therefore the same value for the integral J as the curve made up of the two arcs @3 and @1. But this is in contradiction to the minimizing property of @2 unless the compound curve @3, il coincides with @2. Therefore the point P2 must be a point of @.; moreover ji% (PiP2) = i (P1P2) = (t - t,) K. Conversely, every point of the extremal @2 belongs at the same time to the arc PP?3 of 20. For, let P4 be any point of @2 between P1 and P3, and let 'U = JF (P1 P4) Then Or (0, 8), or (1 - 8, 1) in case P coincides with the point A0 or A1. ~ 46] HILBERT'S EXISTENCE THEOREM 263 0 < u < J9, (P P3) =(t3 -t,)K Hence if we define to by the relation u,-(t,-t t,)K, t4 lies between t1 and t3 and is therefore the parameter of some point P4 of 20 between P1 and P3. The point P4 belongs therefore also to E2 and we have J%2 (P1 Pi) = (t, - t,) K = J2 (P1 P4) Hence it follows that P4 must coincide with P4 since F is positive along.2 -Prom the relation between t4 and the quantity u (which may be taken as the parameter on @2), it follows, moreover, that the points are ordered on both arcs in the same manner, which completes the proof that the are P1P3 of 20 is identical with the extremal (2. Hence it follows that Hilbert's curve So is of class C" and satisfies Euler's differential equation in the vicinity of every interior point P, and therefore every interior arc of 20 is indeed an arc of an extremal of class C". From the assumption B) that F is always positive it follows finally that Hilbert's curve 20 can have no multiple points. ADDENDA P. 58, 1. 5: In order to justify the terms "next greater," "next smaller," it must be shown that an integral u of a homogeneous linear differential equation of the second order cdazP du dx have only a zro in an intrval (ab in can have only a finite number of zeros in an interval (ab) in which p and q are continuous. Proof: According to the existence theorem (compare footnote 1, p. 50), uz is of class C" in (a b). Suppose u had an infinitude of zeros in (a b); then there would exist in (a b) at least one accumulation point (compare footnote 1, p. 178) for these zeros. Now either u (c) 4 0; then a vicinity of c can be assigned in which u (x) = 0. Or else (c)= 0; then u'(c) z 0 (compare footnote 3, p. 58), and t u(c + -h) = h(t' (c) + (h)); hence a vicinity of c can be assigned in which c is the only zero of z(x). In both cases we reach therefore a contradiction with the assumption that c is an accumulation-point. The same lemma has to be used, p. 108,1. 6 up; p. 135,1. 13; p. 200,1. 4; p. 221,1. 1. P. 59, 1. 11. Simpler as follows: Choose X2 so that X1 < X2 < Xo and at the same time x, < X1 (the quantity introduced on p. 55). Then A(x, x2) and A(x, xo) are two linearly independent integrals of (9). Applying STURM'S theorem to these two functions we obtain the result that A(x, X2) $ 0 in (x,, x) P. 62,1. 6. Simpler proof: y.(x, 7o) and A(x, x0) are integrals of JACOBI'S differential equation; both vanish for x = Xo without being identically zero. Hence they can differ only by a constant factor. Compare footnote 2, p. 58, and footnote 1, p. 137. P. 81, 1. 18. From what has been proved in the first paragraph of p. 81, it follows that Hk is indeed a region in the specific sense of ~2, a). 265 2 6 6 CALCULUS OF VARIATIONS P. 83, 1. 13. Add: cd) The Field-Integral for the set of extremals through the point A. Let P(X2, 112) be any point in the field B. formed by the set of extremals through the point A (x5, 1y5), and let 7Y 2 (x2, Y2) be the value of -y for the nniqne extremal of the field which passes through the point P. Then the integral J taken along this extremal YI/-, (x, Y2) from the point A to the point P is a single-valued function of x2, 12 which we denote by J(x2, 112). Its valne is J(X2, 112) fF(m <(x, 72), Ox(', y2)) dx x5 where it is understood that Y2 is replaced by its expression V (x2, 112) in terms of x2 and 12. The partial derivatives of J(X2, 112) with respect to X2 and 2 have the following values: DJ (X21 Y2)_ aJDx2, 11~ -F(X2, Y2, P2) - P2FU5(X2, 12, P2) aDx, (1a) 9J(x2,12 )F (X, yP) vhere P2 denotes the slope of the extremal &2 at the point P. For DJ (X2, 112), 22 ( ax F(X2,,21 Y2)2aYf (F(Fqy + F. Y) dx a J (X~2 I Y2) ay2 X2a a11, 112 x) If we transform the integral as in 20, e), and make use of (12) we obtain (15a). In many respects it vould have been preferable first to prove the formulhe (15a) and to make use of them in the demonstration of WEIERSTRASS'5 theorem. Comparb the analogous formulhe (44) in ~37, and the still more general formulae (14) in ~ 34. P. 142,11. 4 and 5. After 0 insert: +2nmr where in is an integer. ADDENDA 267 P. 151, 1.14. Add: This result is due to ERDMANN; compare Journal fiir Mathematik, Vol. LXXXII (1877), p. 29. P. 152, 1. 8. WEIERSTRASS himself gives the condition in the following slightly different form: + Let 82 and 62 denote the numerical values of the angles which the directions p2, q2 and p2, q respectively make with the direction p2, q, so measured that 62 and 82 are 7r. Then _ + + + -_ sin 82 sin 2 = E (2, 22; P2, q2; p2, q2) ' E (x2, Y2; P2, q2; p2, q2). (64a) This form of the condition follows immediately from (64). For on account of (48) equation (64) may be written p21 ' (X, Y2,, 2 2) + q2fy' (X2,,P2, P2, ) P2Fx (X2, Y2, yP2 q2) + q2F(X' (xY2, 2,, But + _ + p2 =I [sin 8k22 + sin 82P2] q2 = 1 [sin 82 q2 + sin 82 q2], where 1 is a factor of proportionality. Substituting these values in the last equation, we obtain (64a). P. 169,1.7, and p. 175,1.15. Instead of "region" read "domain.' Compare ~ 2, a). P. 169,1. 8. Instead of: "of the set," read: "to the set." P. 172,1. 13. Add reference to KNESER, Lehrbuch, p. 48. P. 178, 1. 18. After "abgeschlossen" add the reference: E. I, p. 195. P. 180,1. 18. Add: Hence it follows that k, is a region in the specific sense of ~2, a). P. 182, 1. 7, and p. 185, 11. 4 and 6. The image of a region by a transformation of the kind here considered is again a region. Hence A, UTk, k are indeed regions. P. 200, 1. 7. Add: to is therefore identical with the quantity designated on p. 155 by to'. The use of the notation to in the present discussion is justified by the fact that in KNESER'S theory the conjugate point appears as a special case of the focal point corresponding to the case when the transversal $ degenerates into the point A. P. 246, 1. 1. HILBERT has published the details of his proof of 268 CALCULUS OF VARIATIONS Dirichlet's principle in the Festschrift zur Feier des 150-jahrigen Bestehens der konigl. Gesellschaft der Wissenschaften zu Gottingen 1901, and in the Mathematische Annalen, Vol. LIX (1904), p. 161. P. 246, 1. 2. I had at my disposal a set of notes of this course for which I am indebted to PROFESSOR J. I. HUTCHINSON. P. 247, 1. 17. After "numerable" add the reference: E. I, A, p. 186. P. 253,1.17. After "result" add: due to OSGOOD; see the referente on p. 248, footnote. INDEX [The numbers refer to the pages, the subscripts to the footnotes.] ABSOLUTE MAXIMUM, MINIMUM, 10. ACCUMULATION-POINT, of a set of points, 178,, 254,. ADMISSIBLE CURVES, 9, 11, 104, 121, 206. AMPLITUDE, of a vector, 9. BLISS'S CONDITION, for the case of two variable end-points, 113. BOUNDARY CONDITIONS: along segment of boundary, 43, 149; at points of transition, 42, 150, 267; when minimizing curve has one point in common with boundary, 152, 267. BOUNDARY, of set of points, 5. BRACHISTOCHRONE, 126,135,146; determination of constants, 1282; case of one variable end-point, 106,. CATENOID (see Surface of revolution of minimum area). CIRCLE, notation for, 9. CLASS C, C', C".... D', D..: functions of, 7; curves of, 8, 116; curves of class K, 161. CLOSED: region, 5; set of points, 178, 267. CONJUGATE POINTS, 60; for the case of parameter-representation, 135; for isoperimetric problems, 221; geometrical interpretation, 63, 137; case where the two end-points are conjugate, 65,, 204. CONNECTED SET OF POINTS, 5. CONTINUOUS FUNCTIONS: definitions and theorems on: existence of maximum and minimum, 134, 802; sign, 211; uniform continuity, 802; continuity of compound functions, 2ls; integrability, 125. CONTINUUM, 5. CONVEX REGION, 247. CO-ORDINATES: agreement concerning positive direction of axes, 8. CORNER: defined, 8, 117; corner-conditions, 38, 126, 210. CRITICAL POINT, 1091. CURVES: (a) representable in form y=f(x), 8; of class C, C',. D', 8; (b) in parameter-representation, 1152; of class C', C", 116; ordinary, 117; regular, 117; rectifiable, 1162; of class K, 161; Jordan curves, 180. CURVILINEAR CO-ORDINATES: in general, 181; Kneser's, 184. DEFINITE INTEGRALS: theorems on: integrable functions, 12,, 89,; first meanvalue theorem, 244; connection with indefinite integral, 892; integration by parts, 201; differentiation with respect to a parameter, 163. DERIVATIVES: notation, 6, 7; progressive and regressive, 7,; reversion of the order of differentiation in partial derivatives of higher order, 183. DIFFERENTIAL EQUATIONS: existence theorem, 284; dependence of the general integral upon the constants of integration, 543; upon parameters, 71,, 223,. DISCONTINUOUS SOLUTIONS, 36,125, 209. DISTANCE: between two points, notation, 9. DOMAIN, 5. END-POINTS, variable (see Variable endpoints). ENVELOPE: of a set of plane curves in general, 624, 1374; of a set of extremals, 62; theorem on the envelope of a set of geodesics, 166; extension of this theorem to extremals, 174; case when the envelope has cusps, 201; case when the envelope degenerates into a point, 204. EQUILIBRIUM, of cord suspended at its two extremities, 211, 231, 241. EQUIVALENT PROBLEMS, 183, 197, 228. ERDMANN'S CORNER CONDITION, 38. EULER'S (DIFFERENTIAL) EQUATION, 22; Du Bois-Reymond's proof of, 23; Hilbert's proof of, 24; Weierstrass's form of, 123; assumptions concerning its general integral, 54, 130; cases of reduction of order, 26,, 29. EULER'S ISOPERIMETRIC RULE, 210. EVOLUTE, of plane curve, 1743. EXISTENCE THEOREM: for a minimum " im Kleinen," 146; for a minimum " in Grossen," 245; for differential equations, 28; in particular for linear differential equations, 50. EXTRAORDINARY VANISHING OF THE EFUNCTION, 142. EXTREMAL: defined, 27, 123, 209; construction of extremal through given point in given direction, 28, 124; set of extremals through given point, 60; set of extremrals cut transversely by a given curve, 111; construction of extremal through two points, sufficiently near to each other, 146; problems with given extremals, 30. EXTREMUM: defined 10 (compare Minimum, Maximum). FIELD: defined, 79; theorem concerning existence of, 79; applied to set of extremals thro 4ll A, 82; improper, 832; for case of i-qrameter-representation, 144, 176; for isoperimetric problems, 241; field-integral, 266. FIRST NECESSARY CONDITION (see Euler s differential equation). FIRST VARIATION: defined, 17; vanishing of the, 18; transformation by integration by parts, 20, 22; for case of variable end-points, 102,; for case of 269 270 CALCULUS OF VARIATIONS parameter-representation, 122, 123; for isoperimetric problems, 209. FOCAL POINT: of a transverse curve on an extremal: defined, 109; equation for its determination, according to Bliss, 108, 155; according to Kneser, 200; geometrical interpretation, 111, 156; case where end-point B coincides with focal point, 204. FOURTH NECESSARY CONDITION (see under Weierstrass). FREE VARIATION, points of, 41. FUNCTION E (x, y; p, j)): defined 34, 75; relation between E (x, y; p, p ) and Fy'y', 76; geometrical interpretation of this relation, 77. FUNCTION E, (, y; p, J), 76. FUNCTION E (x, y; p, q; p, T): defined, 138; homogeneity properties, 140; relation between E-function and F,, 141; ordinary and extraordinary vanishing, 142; Kneser's geometrical interpretation, 195. FUNCTION E1 (x, y; p, q; pr, q), 145. FUNCTION 1,, 121. FUNCTION F2, 132. FUNDAMENTAL LEMMA, of the Calculus of Variations, 20. GENERALIZED INTEGRAL, 157, 248 (compare Integral taken along a curve). GEODESIC CURVATURE, 129. GEODESIC DISTANCE, 176. GEODESIC PARALLEL CO-ORDINATES, 164. GEODESICS, 128, 146, 155; Gauss's theorems on, 164,165; theorem on the envelope of a set of, 166. HILBERT'S: construction, 253; existence theorem, 245; invariant integral, 92, 195. HOMOGENEITY CONDITION, 119; consequences of, 120. IMPLICIT FUNCTIONS, theorem on, 352. IMPROPER: field, 832; maximum, minimum, 11, IN A DOMAIN, use of the word explained, 5,6. INFINITESIMAL, 6. INNER POINT, 5. INTEGRABILITY CONDITION, 29. INTEGRABLE FUNCTIONS, theorems on, 125, 892. INTEGRAL taken along a curve, definition and notation, 8; for case of parameter-representation, 117; condition for invariance under parameter-representation, 119; extension to curves without a tangent, (a) Weierstrass's, 157, (b) Hilbert-Osgood's, 248. INTEGRATION, by parts, 20, 20,. INTERVAL, defined, 5. INVARIANCE, of E and F,, 183. ISOPERIMETRIC CONSTANT, 209; Mayer's theorem for case of discontinuous solutions, 209,. ISOPERIMETRIC PROBLEMS: in general. 206-44; special, 4, 210, 229, 238; with variable end-points, 1132. JACOBIAN, 572. JACOBI'S CONDITION, 67; proofs of its necessity, 65,, 66; Weierstrass's form of, 135; Kneser's form of, 136; for case of one variable end-point, 109, 155, 200; for isoperimetric problems, 225, 226. JACOBI'S: criterion, 60, 135; differential equation, 49, 133; theorem concerning the integration of Jacobi's differential equation, 54, 135; transformation of the second variation, 51. JORDAN CURVE, 180. KNESER'S: theory, 164-205; curvilinear co-ordinates, 184; sufficient conditions, 187; theorem on transversals, 172. LA GRANGE'S DIFFERENTIAL EQUATION, 223. LEGENDRE'S CONDITION, 47; Weierstrass's form of, 133; for isoperimetric problems, 217; Legendre's differential equation, 46. LENGTH OF A CURVE: Jordan's definition, 157,; Peano's definition, 2492. LIMIT: definition and notation, 72; uniform convergence to a, 19,; criterion for the existence of, 258,. LIMITED VARIATION, functions of, 2583. LIMIT: lower and upper, 33,102; attained by continuous function, 134, 802. LIMIT-POINT (see Accumulation-point), LINDELOF'S CONSTRUCTION, 64. LINEAR DIFFERENTIAL EQUATIONS OF THE SECOND ORDER: existence theorem, 50,; Abel's theorem, 582; Sturm's theorem, 582. LOWER LIMIT, 33, 102. MAXIMUM (see Minimum). MAYER'S LAW of reciprocity for isoperimetric problems, 229, 244,. MEAN-VALUE THEOREM, first, for definite integrals, 244. MINIMUM: of a continuous function, 134, 802; of a definite integral, absolute and relative, 10; proper and improper, 11; weak and strong, 69, 70; for case of parameter-representation, 121; semistrong in case of isoperimetric problems, 244; existence of a minimum " im Kleinen," 146; Hilbert's a-priori existence proof of a minimum "im Grossen," 245-63. NEIGHBORHOOD OF A CURVE, 10; neighborhood(p) of a curve, 13, 121. NEIGHBORING CURVE, 141. NUMERABLE SET OF POINTS, 261, 268. ONE-SIDED VARIATIONS (see also Boundary conditions): analytic expression for, 42, 148; necessary conditions for a minimum with respect to, 42, 149; sufficient conditions, 42. OPEN REGION, 5. ORDINARY CURVES, defined, 117. INDEX 271 ORDINARY VANISHING OF THE E-FUNCTION, 142, 266. OSGOOD'S THEOREM concerning a characteristic property of a strong minimum, 190. PARAMETER REPRESENTATION, curves in, 115. PARAMETER-TRANSFORMATION, 116. PARTIAL DERIVATIVES (see Derivatives). PARTIAL VARIATION, of a curve, 37. POINT-BY-POINT VARIATION, of a curve, 41. POINT OF A SET, 124. POSITIVELY HOMOGENEOUS, 119. PROGRESSIVE DERIVATIVE, 7,. PROPER MINIMUM, 11. RECTIFIABLE CURVES, 116.,, 250,, 251,, 2512, 2513 (compare Length). REGION: defined, 5; open, 5; closed, 5. REGRESSIVE DERIVATIVE, 71. REGULAR CURVES, 117; functions, 212; problems, 29, 40, 97, 125. RELATIVE MAXIMUM OR MINIMUM, 10,104. SECOND NECESSARY CONDITION (see Legendre's condition). SECOND VARIATION, 44-67; Legendre's transformation of, 46; Jacobi's transformation of, 51; for case of variable end-points, 1022; Weierstrass's transformation of, for case of parameterrepresentation, 131; for case of variable end-points in parameter-representation, 102, 155; for isoperimetric problems, 216-25. SEMI-STRONG EXTREMUM, 244; sufficient conditions for, 244. SET OF POINTS: definition, 10,; inner point of, 5; boundary point of, 5; accumulation points of, 178t; closed, 178, 267; numerable, 261, 268; upper and lower limits of one-dimensional set, 33, 102; connected, 5; continuum, 5. SIGN OF SQUARE ROOTS, agreement concerning, 21. SLOPE RESTRICTIONS, 1011. SOLID OF REVOLUTION, of minimum resistance, 73,, 1423. STRONG EXTREMUM: defined. 70; sufficient conditions for (see Sufficient conditions). STRONG VARIATION, 72. STURM'S THEOREM, on homogeneous linear differential equations of the second order, 582. SUBSTITUTION SYMBOL, 5, 6. SUFFICIENCY PROOF, for geodesics, 165. SUFFICIENT CONDITIONS FOR WEAK MINIMUM, 70. SUFFICIENT CONDITIONS FOR STRONG MINIMUM: when x independent variable, in terms of E-function, 95; in terms of Fyy', 96; for one-sided variations, 421; in case of one movable endpoint, 109; in case of two movable end-points, 1132. SUFFICIENT CONDITIONS FOR STRONG MINIMUM: for case of parameter-representation, Weierstrass's, 143-46; extension to curves without a tangent, Weierstrass's proof, 161, Osgood's proof, 192; Kneser's sufficient conditions for case of one movable endpoint, 187; for isoperimetric problems, Weierstrass's, 237, 243. SURFACE OF REVOLUTION OF MINIMUM AREA, 1, 27, 48, 64, 97, 153. TAYLOR'S THEOREM, 142. THIRD NECESSARY CONDITION (see Jacobi's condition). THIRD VARIATION, 59,. TOTAL DIFFERENTIAL, 253. TOTAL VARIATION, 14. TRANSVERSE: curve transverse to an extremal, 106; condition of transversality, 36, 106; in parameter-representation, 155; for isoperimetric problems, 210. TRANSVERSAL: to set of extremals, 168; degenerate, 169; Kneser's theorem on transversals, 172, UNFREE VARIATION, points of, 41. UNIFORM CONTINUITY, 802. UNIFORM CONVERGENCE, to a limit, 191. UPPER LIMIT, 33, 10,. VARIABLE END-POINTS: general expression of first variation for case of, 1023; of second variation, 1022; one end-point fixed, the other movable on given curve, treated (a) by the method of differential calculus, 102-113, (b) by Kneser's method, 164-205 (for details see Transversality, Focal point, Sufficient conditions); case when both end-points movable on given curves, 113. VARIATION: of a curve, 141; total, 14; definition for first, second, etc., 16; special variation of type er, 15; of type w (x, e), 18; for case of parameter-representation, 122,1221; weak and strong, 72. VARIED CURVE, 141. VICINITY (3) OF A POINT, 5. WEAK EXTREMUM: defined, 69; sufficient condition for, 70. WEAK VARIATIONS, 72. WEIERSTRASS'S: construction, 84, 144, 234; corner-condition, 126; E-function, 35,138; form of Euler's equation, 123, of Legendre's condition, 133, of Jacobi's criterion, 135; fourth necessary condition, 75, 138, 233; lemma on a special class of variations, 33,139; transformation of second variation, 131. WEIERSTRASS'S SUFFICIENT CONDITIONS, 95, 96, 143; extension to curves without a tangent, 161; for isoperimetric problems, 237, 243. WEIERSTRASS'S THEOREM (expression of AJ in terms of the E-function), 89, 144; Hilbert's proof of, 91; for case of variable end-points 189, 194, 195; for isoperimetric problems, 237. WRONSKIAN DETERMINANT, 571. ZERMELO'S THEOREM, on the envelope of a set of extremals, 174.