/ Ju MATHEMATICAL METHODS IN PHYSICS. L i BY JAMES BYRNIE SHAW. Reprinted from the BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY 2d Series, Vol. XXI., No. 4, pp. 192-199 New York, January, 1915 7hwn-n:L remote storage £1 b II V- V* fn [Reprinted from Bull. Amer. Math. Society, Vol. 21, No. 4, Jan., 1915.] MATHEMATICAL METHODS IN PHYSICS. Sur quelques Progres recents de la Physique mathematique. Par Vito Volterra, Clark University Lectures of 1909, pub- lished by Clark University, 1912. 82 pp. Drei Vorlesungen iiher neuere Fortschritte der mathematischen Physik. Von Vito Volterra, mit Zusatzen und Ergan- zungen des Verfassers. Deutsch von Dr. Ernst Lanela. Sonderabdruck aus dem Archiv der Mathematik und Physik , III. Reihe, Band XXII, Heft 2/3. B. G. Teubner. Leipzig. Lemons sur V Integration des Equations differ entielles aux Derivees partielles. Par Vito Volterra. Professees a Stockholm. Nouveau tirage. Paris, Hermann. 1912. 3 + iv + 83 pp. The first of these books consists of three lectures delivered at Clark University, and afterwards printed by the Uni- versity.* They have since appeared in the second form in German in the Archiv der Mathematik und Physik, (3), 22 (1914), pages 97-182. In the latter form some of the details omitted in the original are supplied. The third book is a reprint of lectures delivered at Stockholm in 1906. There have been added some corrections, and some bibliographical notes. These lectures are striking examples of the intimate relationship between the advance of mathematics and that of physics. The fundamental notion of the Stockholm lectures is that the theories of the propagation of heat, of hydrodynamics, elasticity, Newtonian forces, and electromagnetism can all be treated from a single point of view — reducing indeed to differential equations of the same general form but of three types, the facts and the processes used following the types. A good supplementary paper to read along with the first part of the lectures on differential equations, containing examples and more detail, is to be found in the Annates de VEcole Normale, (3), 24 (1907), page 411. The most interesting part of the lectures is the introduction of the notion, due to Professor Volterra, of function of a line. In the fifth lecture this notion appears, and is indeed the guide to a generalization of the * The volume also contains lectures by Rutherford : ‘ ' History of the alpha-rays from radio-active substances”; Wood: “The optical properties of metallic vapors”; Barus: “Physical properties of the iron carbides.” MATHEMATICAL METHODS IN PHYSICS. 193 [Jan., analytic functions of a complex variable. The particular function of a line used in this lecture is the line integral V = Wo + f ( Xdx + Ydy +[Zdz), where u Q is a constant and the line integral extends from a point A to a point B. If we set X' = (dY/dz — dZ/dy), Y' = (dZ/dx — dX/dz ), Z' = ( dX/dy — dY/dx), then, n being the outward normal to the surface enclosed, the line integral in question around a loop will, by Stokes’s theorem, be the same as W =ff (. X' cos nx + Y' cos ny + Z' cos nz)dA. If we let the loop decrease and determine the limit of the ratio of W to the area enclosed, as the vanishing loop approaches a point, the limit in question is nothing else than the projec- tion, on the normal to the surface at the point, of the vector whose components are X', Y', Z', that is, of the curl of the vector X, Y, Z. This projection, if found for a point on the path of integration, Professor Volterra calls the derivative of the function of the line V with respect to the surface, and he represents the curl by the symbols X' = dV/d(yz) Y' = dV/d(zx ) Z' = dV/d(xy ). If now there is a function u whose gradient is (. X', Y', Z'), that is, if du/dx = X', du/dy = Y', du/dz = Z', then the convergence of the gradient of u is zero, and u is harmonic since X7 2 u = 0. When these relations are satisfied we have u and V so related that one may be called the con- jugate of the other. This is the generalization referred to of the theory of complex variables. An easy mathematical example is found by setting u = z 2 + y 2 - 2z 2 , which is harmonic, and whose gradient is the vector (2x, 2y, - 4s). It is conjugate to the function (integral around a loop) V = f (2 zydx — 2 zxdy), 1915.] MATHEMATICAL METHODS IN PHYSICS. 194 since the curl of (2 zy, — 2zx, 0) is (2x, 2 y, — 4s), the gradient of u. An easy physical example is the field of potential at a fixed origin of a single magnet pole in empty space, as the pole is moved into all possible positions, which gives the function u; and the field of a circuit carrying a unit electric current, as it is moved into all possible positions and shapes, which gives the potential V at the fixed origin. The two func- tions are conjugate, the first a function of a point, the reciprocal of its distance from the origin; the latter a function of a line, the solid angle it subtends at the origin. The function of a line of course need not be conjugate to a function u. In case X', Y', Z' is a vector whose convergence is zero, then it may be written as the curl of a vector X, Y, Z, the well-known relation of a vector to its vector potential; and the integral //<*' cos nx + Y' cos ny + Z' cos nz)dA = J (Xdx + Ydy + Zdz) gives a function of a line. It need be remarked that these definitions of function of a line and the differential of a func- tion of a line are generalized somewhat in the calculus of fonctionnelles. The notion of monogenicity is extended, under the name isogenicity, to space of three dimensions in the following form. Two functions of a complex variable 2 are monogenic if their differentials at any point have a ratio / which is a func- tion of the point but not of the direction of the differen- tials. Two functions of a line are isogenic if at each point of the line the vector derivative of each, that is, the curl of the vector which is to be integrated along the line, is parallel to the vector derivative of the other. In vector notation we would write the definition WB 1 =fWB 2 , where/ is independent of the derivative plane as defined above. The idea of function of a line may evidently be extended to functions of surfaces and hyperspaces, and leads into the calcul fonctionnel. These functions of lines and surfaces, and for the most general case, of hyperspaces, enable one to understand the applibation of the methods of Jacobi and Hamilton to the 195 MATHEMATICAL METHODS IN PHYSICS. [Jan., problems of the calculus of variations. The simple integral of the ordinary theory may be considered as a function of its limits and of the values of unknown functions at the limits. The extensions to multiple integrals are easily suggested by this view of the procedure, namely, we must consider the integrals to be functions of the lines, or surfaces, or hyper- spaces, that bound the space, and of the values of unknown functions on these contours. The detailed treatment of the partial differential equations is to be found in the last four lectures, and reference to them is necessary to have a clear notion of them. In the course at Clark University we find the dominant idea again to be the unifying principles of the application of mathe- matics to mechanics, elasticity, and mathematical physics. The first lecture is devoted to showing the reduction of physical problems to problems in the calculus of variations, the second to the advance in methods in elasticity, and the third to the problem of heredity, which leads to Volterra’s integro-differ- ential equations. These three lectures we will examine in some detail. In the first there is given a reduction of the problem of electrodynamics into terms of the variation of the definite integral P = fdtif '/s(a rs Z r X 8 + p rs Lrh)dA, in which the quantities a, (3 are quite arbitrary. This varia- tion in terms of purely arbitrary quantities shows us that we can devise an infinity of mechanical explanations or models of electrodynamic phenomena. But this analytic form of the problem enables us to introduce curvilinear coordinates, and thus consider curvilinear spaces. We also are enabled to find the integral invariants, and to apply Volterra’s reciprocity theorem which corresponds to Green’s theorem, and further to introduce the generalization of the Hamilton-Jacobi methods and the principle corresponding to stationary action and varying action. In order to accomplish this, the functions of lines, surfaces, and hyperspaces have to be used. We thus come into contact with the theory of the inversion of definite integrals, that is, the solution of linear integral equations, and with the study of functions of variables which run over assemblages of curves, of surfaces, etc. We are brought up to the functional calculus, or as it has been called, general 1915.] MATHEMATICAL METHODS IN PHYSICS. 196 analysis. The generalization of the Hamilton- Jacobi method replaces the canonical equations by partial derivatives, and the partial differential equation of Jacobi is replaced by a functional equation. There is an intimate connection between the partial dif- ferential equation, its characteristics, and the theory of waves.* It is simply necessary to consider the variable t as on the same footing as the variables x, y , z. The bearing of this is shown in the closing considerations of the first lecture. For example, in the very simple case of the equation of a vibrating mem- brane d 2 u/dt 2 — d 2 u/dx 2 — d 2 u/dy 2 = 0 we have the partial differential equation corresponding to the vanishing of the variation of the integral V = f f f [( du/dt ) 2 — ( du/dx ) 2 — (du/dy) 2 ]dxdydt. The surfaces of discontinuous derivatives and variation of V always equal to zero are then the envelopes of the character- istic cones of the partial differential equation. The general question of waves is therefore considered, and Minkowski’s universe noticed. This naturally leads to the Lorentz trans- formation, and to Poincare’s demonstration that under this transformation the integral whose variation was considered, and which may be called the action, remains invariant. In the second lecture the development of methods in the theory of elasticity is taken up, particularly those that are connected with the ideas already mentioned. The two great classes of methods of integrating the differential equations of elasticity may be called the method of Green with its exten- sions, and the method of simple solutions. Green’s method further has two divisions, in one of which the conception of Green alone is sufficient to solve the problem, in the other we must add consideration of the characteristics. Green’s method starts with Laplace’s equation, and depends upon a reci- procity theorem, by means of which from a fundamental solution one is enabled to determine a harmonic function inside a given region when its values on the contour are given. Betti carried the method of Green over into elasticity and extended the reciprocity theorem by the proposition: If two * Encyclopedic des Math., II 4 1 (II, 22, 8). 197 MATHEMATICAL METHODS IN PHYSICS. [Jan., systems of exterior forces determine two systems of displace- ments in an elastic body, the work done by either in producing the displacement due to the other is the same. The second division of methods along the line started by Green is found in Kirchoff’s work on the equation of retarded potential on four variables. He succeeded in solving it by means of a fundamental solution due to Euler, but in order to perceive the real difficulties in the way it is necessary to consider the problem on only three variables. The char- acteristics enter into the solution radically in this case. The lecturer shows the inherent difference between the cases of three dimensions and four dimensions. Again when the body is not simply connected and the functions can be polydromic the method of Green needs further extensions. The reci- procity theorem enters in as a theorem of the symmetry of a set of coefficients Eij = Eji, which occur in the linear equations that give the efforts in terms of the distortions. The method of simple solutions has been given great power by the development of the theory of integral equations, and the determination of methods of expansion in series of ortho- gonal functions. The third lecture introduces the new developments due to Professor Volterra himself and now well-known. These lead to the division of mechanics into the mechanics of no heredity, wherein the state of a system depends only upon the infini- tesimally near states preceding, and the mechanics of heredity, in which the state depends upon all the preceding states, thus introducing an action at a time-distance. A simple example is used to illustrate the new problem, the dependence of the angle of torsion of a wire upon the moment of torsion. Instead of Hooke’s law co = KM, we find it must be expressed by a more elaborate law dependent upon the time co = KM(t ) + r M(t)ip(t, t)cIt *Jt 0 +AC*-jC dT 2 M(Ti)M(T 2 )