key: cord-0513664-aong4ixd
authors: Kol, Barak
title: Natural dynamical reduction of the three-body problem
date: 2021-07-25
journal: nan
DOI: nan
sha: 4ca7bf366604ab2f8ec68d4f3ac78914c1b7d2a5
doc_id: 513664
cord_uid: aong4ixd

The three-body problem is a fundamental long-standing open problem, with applications in all branches of physics, including astrophysics, nuclear physics and particle physics. In general, conserved quantities allow to reduce the formulation of a mechanical problem to fewer degrees of freedom, a process known as dynamical reduction. However, extant reductions are either non-general, or hide the problem's symmetry or include unexplained definitions. This paper presents a dynamical reduction that avoids these issues, and hence is general and natural. Any three-body configuration defines a triangle, and its orientation in space. Accordingly, we decompose the dynamical variables into the geometry (shape + size) and orientation of the triangle. The geometry variables are shown to describe the motion of an abstract point in a curved 3d space, subject to a potential-derived force and a magnetic-like force with a monopole charge. The orientation variables are shown to obey a dynamics analogous to the Euler equations for a rotating rigid body, only here the moments of inertia depend on the geometry variables, rather than being constant. The reduction rests on a novel symmetric solution to the center of mass constraint inspired by Lagrange's solution to the cubic. The formulation of the orientation variables is novel and rests on a little known generalization of the Euler-Lagrange equations to non-coordinate velocities. Applications to special exact solutions and to the statistical solution are described or discussed. Moreover, a generalization to the four-body problem is presented.

In this paper, the reduction process is described in section 2. It starts by introducing a solution to the center of mass constraints, which transforms nicely under permutations, and hence is natural. It involves complex numbers in a novel way, and it is inspired by Lagrange's solution to the cubic. Next, the orientation of the plane defined by the three bodies is separated from motion within the plane simply by using a rotating body frame. We proceed to the separation of rotations within the plane from the geometry of the triangle formed by the three bodies. This requires to define certain spherical coordinates on C 2 and to handle correctly one of the angles, resulting in a famous quotient of it. Finally, we Legendre transform from angular velocities to angular momenta and obtain their Poisson brackets from a general theory for non-coordinate velocities and their conjugate momenta. This leads us to our final result, (2.49) . We end the section with an illustration of the resulting formulation and a discussion of it.

The formulation can be applied to study both special exact solutions and the statistical solution. In section 3, we provide a novel derivation of the uniformly rotating solutions, and sketch a novel derivation of the linear stability region for the equilateral solution. The section ends with an outlook on applications to the statistical solution and more.

This work has interesting implications for our-body problem, and a sketchy generalization appears in section 4.

Appendices provide background material on Lagrange's solution to the cubic, on bicomplex numbers and the isotropic oscillator and finally on the equations of motion in terms of non-coordinate velocities and the rotating rigid body.

Setup. Consider three point masses, m 1 , m 2 , m 3 . Let us take the bodies' position vectors, r a , a = 1, 2, 3, to be our initial dynamical variables. The kinetic energy is given by where G is Newton's gravitational constant, and r ab = | r a − r b |, are the inter-body distances. We note that the whole reduction procedure is independent of the choice of V as long as it depends on r ab only, namely V = V ({r ab }). Finally, the system can be defined by the Lagrangian L { r a ,˙ r a } 3 a=1 := T − V . 

The system is invariant under translations and hence the total linear momentum is conserved, the center of mass moves in uniform motion, and henceforth we work in the center of mass frame 0 = r CM := 1 M a m a r a (2.4) where M := m 1 + m 2 + m 3 is the total mass. The center of mass constraint reduces the configuration space, reducing the number of degrees of freedom from 9 to 6. This requires a choice coordinates on the reduced configuration space. We shall see that this choice creates a tension with the bodies' permutation symmetry, which will affect the whole reduction process, and hence we shall describe it in some detail.

The coordinates are chosen to be translation-invariant vectors: vectors, in order to preserve the transformation properties under rotations, and translation-invariant, in order to decouple them from the center of mass coordinates in the expression for the kinetic energy.

In the two-body problem, one chooses the relative coordinate to be

It is a translation-invariant vector, and under a 1 ↔ 2 exchange, it changes only by a sign.

In the three-body problem, there are three relative vectors r 12 := r 1 − r 2 , r 23 := r 2 − r 3 , r 31 := r 3 − r 1 (2.6)

They satisfy the constraint 0 = r 12 + r 23 + r 31 . Any pair of linear combinations of the relative vectors can serve as coordinates in the center of mass frame. There are two popular choices for that. The first set consists of the planetary coordinates r 13 , r 23 (2.7)

They are useful for the case where one of the masses, say m 3 , is much heavier than the other two: m 1 , m 2 m 3 . The second set consists of the lunar coordinates

These are useful for the hierarchical case where the magnitude of the first vector is much small than the second, namely bodies 1 and 2 are much closer to each other, relative to their distance from the third body. These coordinates are commonly called Jacobi coordinates, because they appeared in [7] , but they are a rather natural choice, and they appeared already in [6] . Both coordinate choices break the symmetry between the bodies. Hence, they are not optimal for the non-hierarchical case where all distances and all masses are comparable.

Can we define natural, namely symmetric, coordinates in the center of mass frame? A solution is suggested by Lagrange's solution to the cubic equation. Lagrange realized that the key to a solution of a polynomial equation is to study expressions in terms of the roots that display a certain measure of symmetry among them. For more see appendix A. Inspired by this, we define w := r 1 + η r 2 +η r 3 (2.9) where η is the cubic root of unity, namely η = exp(2πj/3), and we denote here the imaginary unit by j, j 2 = −1, since i will have a different use below. The definition (2.9) is one of the central results of this paper. Let us discuss it. By construction w is a complex vector. Naturally, it is composed of 2 real vectors, and so it has the correct number of components to serve as coordinates in the center of mass frame. It is translation-invariant as a result of the identity 1 + η +η = 0. Under a cyclic rotation it is merely multiplied by a 2π/3 phase, while the exchange 2 ↔ 3 transforms it into its complex conjugate. These simple transformations under permutations render w a natural coordinate.

Transformed Lagrangian Given w, the relative positions can be expressed by

and similarly, replacing w → ηw in this expression gives r 12 , and w →ηw gives r 31 . The kinetic energy in the center of mass frame is given by

where the cyclic transformation denotes now w →ηw, in addition to 1 → 2 → 3 → 1.

One can naturally divide the system's coordinates into orientation coordinates and coordinates that describe the configuration up to rotations, namely, geometry coordinates. At any given moment, the positions of the three bodies define an instantaneous plane. We define a rotating body frame whose z axis is normal to the instantaneous plane, and 2d vectors ρ a specify the positions within the rotating (x, y) plane . As usual, inertial frame velocities are given by d dt inert ρ ab =˙ ρ ab + ω × ρ ab (2.12) where ω is the angular velocity vector that describes the rotation of the body frame, and ω is reserved for later use. Substituting, the kinetic energy (2.11) becomes

where I ij , i, j = 1, 2, 3 denotes the inertia tensor in the rotating (and center of mass) frame

(2.14)

L w denotes the the angular momentum due to the ρ a motion 15) where the wedge product between any 2d vectors u, v is defined by u ∧ v := ( u × v) ·ẑ; and finally, the kinetic energy due to the ρ a motion

We note several properties of I ij . Since the mass distribution is confined to the z = 0 plane one has 0 = I 13 = I 23 I := I 33 = I 11 + I 22 (2.17) In addition, since the mass distribution consists of only 3 point masses, one has

is twice the (signed) area of the triangle formed by the bodies. In other words, the 2 * 2 determinant of I αβ , α, β = 1, 2 factors into a part that depends only on positions, and a part which depends only on the masses.

The orientation-geometry decomposition is not done yet. To see that, let us count the number of generalized coordinates. We know that the 3d problem has 6 degrees of freedom (d.o.f) in the center of mass frame (CM). The orientation is specified by 3 angles, such as the Euler angles, and the ρ a at CM (or equivalently the associated 2d w) provide 4 other d.o.f., so altogether we have 7 generalized coordinates. This means that the these coordinates are redundant by a single degree of freedom. Rotations within the (x, y) plane are the origin of this redundancy: since a rotation of the body frame around the z-axis is equivalent to a rotation of the 2d vectors, the configuration depends on the two associated angles only through their sum.

Bi-complex w. In order to account for plane rotations, it is useful to complexify the plane, so that rotations would be represented by a phase multiplication. By complexification one means that any 2 vector ρ is mapped onto a complex number ρ → ρ := ρ · (x + iŷ). Similarly, the 2d j-complex vector w, defined in (2.9), is mapped onto a so-called bicomplex number

which is of the form w = a+b i+c j+d ij. The imaginary unit i represents a quarter rotation in the (x, y) plane. Hence, i, j are commuting imaginary units, namely i 2 = j 2 = −1 and i j = j i. For more on the algebraic structure of bi-complex numbers, and their application to the isotropic oscillator of mechanics, see appendix B.

Let us introduce a natural basis for the w space. Evaluating the w variable (2.20) for a right-handed and a left-handed equilateral triangles of unit sides motivates the definitions e R :=

where for concreteness, the equilateral triangles are chosen to be oriented such that the height from edge 2-3 to vertex 1 is pointed along +x within the (x, y) plane.

Spherical coordinates for w. Considering w space as a 2d complex vector space over complex numbers involving i, namely C[i] 2 , one can expand a general state w in the e R , e L basis as follows

In this expression, r sets the overall triangle scale, ψ ∼ ψ + 2π is an overall rotation angle, θ ∈ [0, π] determines the relative magnitudes of the right and left components, and finally φ ∼ φ + 2π is the relative phase of the right and left components. Altogether r, θ, φ, ψ are spherical coordinates for w ∈ C[i] 2 . We define G to be the quotient space of planar three-body configurations, up to rotations, where G stands for geometry. In other words, G is the space of equivalence classes of congruent triangles. We can write

where the U (1) acts by overall phase rotations. We shall later see how to parameterize G in terms of quadratics in w. In coordinates, G is given by the w variable up to ψ-shifts, and it is parameterized by r, θ, φ. G includes information on both size and shape: the r coordinate determines the triangle size, while the θ, φ coordinates parameterize a sphere that describes triangles up to similarity, and is known as the shape sphere. Coordinates that are equivalent to r, θ, φ appeared in [11] in Eqs. (1, 2, 14) , and the term "shape sphere" appeared in [12] . Substituting (2.22) into (2.10), the relative position vector r 23 becomes ρ 23 = i r e iψ 0 e iφ/2 cos θ 2 − e −iφ/2 sin θ 2 .

(2.24)

In particular,

Similarly, r 31 , r 12 are given by the same expression after the substitutions φ → φ − 2π/3, and φ → φ + 2π/3, respectively. This means that expressions are invariant under a cyclic permutation combined with a third of a revolution in φ.

The definition of ψ 0 can be changed into ψ = ψ 0 + χ(θ, φ), which will be seen to be a gauge transformation. ψ 0 , defined in (2.22) , is symmetric between the R and L hemispheres, 0 ≤ θ ≤ π/2 and π/2 ≤ θ ≤ π, respectively. However, ψ 0 is singular for both R and L poles. A gauge that is regular at the R pole is given by

Kinetic energy. In spherical coordinates, and using the ψ + gauge (2.26), T w (2.16) becomes

where the largest principal moment of inertia (for rotations within the plane) (2.14), is given by

(2.28)

L G denotes the angular momentum in the geometry space G and it is given by

T G denotes the kinetic energy in G space, and is given by The derivation proceeds by projectingρ 23 along the radial and the tangential directions with respect to ρ 23 , performed in the r, θ, φ coordinates, and similarly for the other relative velocities. It will be detailed in a later version.

The kinetic energy (2.27) is in a form of a dimensional reduction (also known as Kaluza-Klein reduction) over the coordinate ψ. Therefore, the expression for L G depends on the choice of gauge for ψ, while the kinetic metric T G is gauge-independent, and represents the metric on the quotient space (2.23). The expression for T G appeared essentially in [12] .

Fixing the ψ gauge. Let us return to the issue of coordinate redundancy discussed at the beginning of this subsection. This redundancy can be removed by fixing a gauge for the ψ coordinate, and we choose to set

This can always be achieved through a choice of the x-y body axes, which in turn fixes the value ofω z . Clearly, the effect of this gauge is directly related to the choice of definition of ψ.

Relatedly, the kinetic energy depends onω z ,ψ only through their sum. Indeed, the relevant terms in T are (2.13,2.27)

where in passing to the second line, we have used (2.30), and in the last equality we have used ω defined by ω = ω +ψẑ .

Altogether, the expression for the kinetic energy in the center of mass frame becomes

where T G , I, L G , I αβ α, β = 1, 2 depend on the G-space variables and were defined in (2.31,2.28,2.29,2.14) respectively.

Let us study further the geometry of geometry space, namely three-body configurations up to rotations (2.23).

Invariants. So far, geometry space was described by the coordinates r, θ, φ. Alternatively, we can employ the invariants of the quotient. This approach will clarify the geometry at the origin of geometry space. We define the quadratic invariants

where w 1 , w 2 ∈ C[i] denote the j-real and imaginary parts of the bi-complex w, up to normalization, defined as follows

and the Pauli-like matrices τ r are given by

The quadratic invariants Q 0 , . . . , Q 4 are not all independent, but rather satisfy the relations

Geometrically, relation (2.40a) means that Q 1 , Q 2 , Q 3 describe a flat 2d space, and relation (2.40b) means that the quotient is a (light) cone in a 3+1 space. The cone singularity at the origin corresponds to a triple collision configuration. We note that the Q 0 , . . . Q 4 variables are closely related to the Stokes parameters S 0 , . . . , S 3 [20] , which are used to describe states of polarization of light and are reviewed in appendix B. More precisely, Q 0 , Q 4 are identical to the S 0 , S 3 while Q 1 , Q 2 , Q 3 up to the relation (2.40a) are analogous to S 1 , S 2 . However, the Q variables are distinguished by being compatible with a symmetry of order 3.

In terms of the r, θ, φ coordinates, substitution of (2.22) shows that the invariants are given by

In this form the invariants are seen to be closely related to Cartesian coordinates associated with the spherical coordinates r 2 , θ, φ. Alternatively, one can define Cartesian coordinates on G-space, by considering r 2 , θ, φ to be spherical coordinates of a vector G, namely G 1 = r 2 sin θ cos φ, G 2 = r 2 sin θ sin φ, G 3 = r 2 cos θ. The three components of G are independent coordinates, which solve the relations on the Q variables (2.40). Similarly, we define the vector g to be the Cartesian coordinates associated with r, θ, φ (r instead of r 2 ).

Using (2.25) , the triangle edge lengths can now be expressed as

and similarly for r 2 31 , r 2 12 . Summing all three, and using (2.40a,2.41a) we find

This means that the geometric interpretation of the radial coordinate r is the root mean square of the side lengths. We note that one could take a wider perspective, and rather than study the planar configuration of the bi-complex w up to phase shifts, one could study the 3d complex position vector w (2.9) up to identification by 3d rotations, namely w/SO(3). The quadratic invariants Q 0 , . . . Q 3 can be expressed in terms of scalar products among w, w, and hence are invariants from the 3d perspective. On the other hand, Q 4 cannot be described in this way, and hence while it is an invariant of C 2 /U (1), only Q 2 4 is an invariant of w/SO(3). In addition, we note that Q 4 is proportional to the triangle area, more precisely

where ∆ is twice the (signed) triangle area, and was defined in (2.19) . This can be shown by following the definition of Q 4 and using translation invariance to set ρ 1 = 0.

Shape sphere. Geometry up to size is known as shape. For this reason, the unit sphere in geometry space, namely r = 1, is known as shape sphere [12] . For triangles, shape is the same as classification up to similarity, which is well known to be classified by the values of the three angles which are constrained to sum to π. This means that shape space should be a 2d surface. As we have seem, it turns out that this surface has the topology of sphere. More precisely, the shape sphere is the space of shapes of triangles with labeled vertices.

The shape sphere is shown in fig. 1 . The geometrical interpretation of various locations on the shape sphere is known [12] , and will be repeated here for convenience. The vertical Q 4 coordinate is proportional to the triangle area, as stated in (2.45). Hence, the Q 4 = 0 equator corresponds to collinear configurations. Within the collinear equator, the point Q 1 = 1 implies w 2 = 0 and hence it corresponds to the collision of the 2,3 vertices and it is denoted by C1. Similarly for C2, C3. Going away from the equator, the Q 4 = 1 pole is associated with w = 1 + i j, which corresponds to a right equilateral triangle, namely, such the 1,2,3 vertices are oriented in the positive mathematical direction (counter-clockwise). Similarly, the Q 4 = −1 pole corresponds to a left equilateral triangle.

Just as the Q invariants were noted above to be closely related to the Stokes parameters and the Pauli matrices (Pauli, 1927) [22] , the shape sphere is closely related to the Poincaré sphere, the sphere of polarizations that was introduced in Poincaré's lectures on optics [21] . These notions are also closely related to the Hopf fibration (Hopf, 1931) [23] and the Bloch sphere (Bloch 1946) [24] . The Pauli matrices are used to describe quantum operators on the states of a spin-half particle. The Hopf fibration represents the 3-sphere as a circle fibration over the 2-sphere. The Bloch sphere describes the states of a two-state quantum system, through an analogy with the spin-half system. Mathematically, the C 2 /U (1) quotient is at the root of all of the above-mentioned topics.

We record the form that several quantities, which appear in the motion potentials, obtain after transformation to the Q variables 

In order to incorporate into the reduction the conservation of angular momenta, we transform from angular velocity variables into the conjugate momenta, which are non other than the angular momenta

We have arrived at the final set of dynamical variables, namely

where g is a location in geometry space, which usually would be represented by the spherical coordinates r, θ, φ. Performing a Legendre transform over the Lagrangian (2.3) we obtain

L J is a function of the dynamical variables (2.48) (together with the generalized velocities associated with g). The expression for L J is organized into three parts L 0 , L 1 , L 2 according to powers of J, and their values are detailed below. The transform that defined L J was taken only with respect to part of the velocity variables, and therefore L J is a hybrid of Lagrangian and a Hamiltonian (sometimes known as Routhian). Later, we shall see that the equations of motion can be derived from L J . It is useful to have a term that refers to any kind of such function, whether it is a Lagrangian, a Hamiltonian or hybrid. In mechanics, the standard potential function encodes the forces through derivatives. In thermodynamics, one uses one of several thermodynamic potentials, such as the energy, the free-energy, the Gibbs free energy, all related among themselves by Legendre transforms. Similarly, we shall use the term motion potential to refer to any function from which the equations of motion can be derived. From this perspective, the standard potential can be distinguished by the term force potential.

The first term appeared already in (2.2,2.31) and is repeated here for convenience. It depends only on the geometry space variables

and

The first two terms of L 0 specify the kinetic energy on geometry space -in the radial and angular directions, respectively. The last term is minus the potential, and for concreteness we present the case of a Newtonian gravitational potential.

The L 1 term couples geometry space and the rotating body, and it is given by

where L G is given by

cyc. denotes 1 → 2 → 3 → 1 together with φ → φ − 2π/3. The expression for L G is given in the ψ + gauge (2.26) and a gauge transformation with a gauge-function χ could shift it by ∆L 1 = −J 3χ . In geometry space, L 1 is analogous in form to a coupling of the motion in G-space to a vector potential, as noticed by [11] . This means that the motion in G-space experiences a velocity-dependent force, which is magnetic-like in form, and can be thought to originate from a Coriolis force. The magnetic-like two-form is given by

It can be seen that this 2-form does not involve the radial coordinate r. It carries non-zero magnetic monopole charge, proportional to J 3 and located at g = 0. The monopole charge is a consequence of the intrinsic charge associated with the Hopf fibration, which is used to reduce over ψ rotations. The author is not familiar with previous appearances of the expression for the magnetic field. On the side of the rotating body, L 1 will be understood to generate a precession of J around the z-axis.

The L 2 term also couples geometry space and the rotating body, only this term is quadratic in J and is given by

where ∆ is twice the triangle's area, which was defined around (2.18), and it can be expressed in G-variables through ∆ = 3 4 r 4 cos 2 θ , (2.57) and whereĪ αβ is the inverse of the I αβ matrix and is given bȳ

where ρ 23 is given within the ψ 0 gauge by

In geometry space, minus L 2 is interpreted as the centrifugal potential. It can be thought to generalize the familiar 2-body centrifugal potential V cent,2 body = L 2 2µ r 2 (2.60)

where L is the system's angular momentum, and µ is its reduced mass. Indeed, 2-body motion is necessarily planar and hence J 1 = J 2 = 0. Moreover, I = µr 2 , thereby the 3-body expression for the centrifugal potential reduces to that of the 2-body. As usual, a coupling in the motion potential, implies several terms in the equations of motion. In this case, in addition to a centrifugal force acting on the geometric variables g, we shall see that L 2 also implies the Euler equations for the rotating body.

Equations of motion. We performed several natural changes of variables in the kinetic energy in order to re-formulate the problem, and thereby re-phrase the equations of motion. One possibility to achieve the equations of motion would be to express ω in (2.36) in terms of a set of frame orientation angles, such as the Euler angles, which would be used as the fundamental dynamical variables. However, this procedure requires to make an arbitrary choice of the Euler-like angles, and obscures the rotational symmetry.

Alternatively, it would be nice to derive the equations of motion using ω as fundamental velocities variables. However, the standard Euler-Lagrange equations would not produce the correct equations of motion, as can be seen for the example of the rigid body in appendix C. Indeed, if we write ω i = β i jq j , where q i are generalized coordinates, then β i ≡ β i j dq j are 1-forms over G-space, such that dβ i = 0 and hence β i cannot be expressed as a differential of any generalized coordinates. In other words, β i define a non-coordinate basis of differential forms, also known as a non-holonomic basis.

In order to derive the equations of motion from a Lagrangian expressed in terms of noncoordinate velocities, we rediscovered the appropriate generalization of the Euler-Lagrange equations. This little-known generalization was originally found by Poincare in 1901 [19] and it is described in Appendix C. Here we only state the result for the case at hand.

In a Lagrangian formulation, one takes

where T ( ω, g) is given by (2.36) and V ( g) is an arbitrary potential. The above-mentioned differential 1-forms have the following non-zero exterior differentials

and hence the equations of motion read d dt

where q r = (r, θ, φ), the spherical coordinates in geometry space. The first line describes 3 generalized equations of motion which originate from variations with respect to ω, while the second line describes 3 standard Euler-Lagrange equations of motion that originate from variation with respect to the G-space variables.

In terms of the partial Legendre transform L J (2.49), the equations of motion are given by

where the Poisson brackets among the dynamic variables originate from the non-coordinate nature of the corresponding velocities according to the general rule (C.9), and in our case are given by

After a solution J = J(t), g = g(t) is found, one can further integrate to obtain the frame orientation angles.

Note that J 2 is conserved as a result of (2.49,2.64). Hence phase space is essentially 8 dimensional: 6 dimensions for g and its generalized velocities and 2 additional dimensions for J that lies on a sphere of fixed J 2 .

We comment that one could obtain a formulation where the vector potential is replaced by the gauge-invariant field-strength by performing a Legendre transform on the remaining velocity variables, and employing "covariant momenta" (in analogy with the covariant derivative).

The motion potential L J = L J ( J, g) given by (2.49) defines a reformulation of the threebody problem, and it is a central result of this paper. The dynamic variables J, g are compatible with the conserved charges in such a way as to reduce the number of effective degrees of freedom.

The employed dynamical variables decompose into two sets. The first set consists of the J variables that describe the angular momentum in the body frame. They are the momenta canonically conjugate to ω, the angular velocity vector in the body frame. Hence, J describes the rotational motion of the configuration triangle. The second set of variables consists of g or in spherical coordinates (r, θ, φ). These describe the geometry of the triangle, namely its shape and size, see fig. 1 . The two sets are coupled as the geometry of the triangle determines its moment of inertia, which affects its rotation motion.

The corresponding decomposition of the mechanics into a rotational motion and a motion in 3d geometry space is illustrated in figure 2 .

Let us look for the origin of the chaotic nature of the system in terms of these two components. Already in the limit of planar motion, the rotational component of the motion becomes trivial, yet the system is chaotic. Indeed, the motion in geometry space is nonintegrable since it has 3 degrees of freedom and only a single conserved charge, namely the energy. Coupling to a rotating body cannot change this non-integrable nature. Moreover, given that the motion of a rotating body is integrable in the rigid body limit, geometry space can be considered to be the core chaotic component in the the three-body system.

Various ingredients of this formulation have already appeared in the literature. The r, θ, φ coordinates together with the vector potential appeared in [11] . The kinetic term in shape space, which is conformal to the round sphere, appeared in [12] together with the origin of geometry space from a quotient of a 4d space. In hindsight, these elements partially appeared also in the Helium atom context: the 3d geometry space and its conformal equivalence to the sphere in [16] , and the 4d predecessor in [17] .

The current paper includes two main novelties. First, the definition of the complex position vector w that solves the center of mass constraint (2.9). Secondly, the formulation in terms of J, the angular momentum components in the rotating system, see subsection 2.5. In comparing to previous work, [12] was limited to planar motion, while we address the full 3d problem. J has not been used as dynamical variables before, and in particular neither in [11] nor in [12] . While in hindsight, [18] can be recognized to have some relation to a formulation in terms of J within the Helium atom context, it is at most partial and special due to the special values of the masses and the distinct quantum context.

Finally, the definition of w naturally explains certain features that were previously observed yet deemed surprising. [11] introduces the spherical coordinates in geometry space without explanation. It starts with the equal mass case, where the correct definitions are Figure 2 . The natural dynamical reduction into orientation and geometry. Right -the three bodies define a triangle. Its dynamics is decomposed into its orientation and its geometry (shape + size). The dynamical variables for the orientation are taken to be the components of the total angular momentum within the rotating system. Left: Mechanics in geometry space. Geometry space is a 3-dimensional space in which each point describes the shape and size of a triangle formed by the three bodies. r, θ, φ are considered to be spherical coordinates in this space. The (yellow) surface shaped like a pipe joint is a surface of constant effective potential (shown for J = 0), so that motion is confined to be within it. The 3 solid (black) lines are sources of an attractive potential which originates with the Newtonian gravitational potential. Finally the (blue) arrows show the radial magnetic-field field, which originates from a Coriolis force. easier to guess, before generalizing to the general case of unequal masses. [12] arrives at these coordinates after a choice of coordinates on CP 1 and comments that "Remarkably, it turns out that if we put the binary collisions at the third roots of unity... then the equilateral points are automatically moved to the north and south poles." The definition of w provides the missing link and makes natural the transition to the geometry space coordinates and their properties.

This section describes an application of current formulation to planar three-body motion, and in particular to uniformly rotating configurations.

If the initial velocities are within the three-body plane then the motion remains planar throughout. For planar motion, J 1 , J 2 vanish identically, while J 3 is conserved. Hence, the rotating body motion is trivial, and L 2 specializes to

where L 0 , L 1 are defined in (2.50,2.53).

Equilibria. Equilibria of the reduced planar motion describe rigidly rotating solutions. The effective potential in geometry space V G ef f = V G ef f (r, θ, φ) is given by

where the second term is the centrifugal potential. If V is a power-law in the inter-body distance, one can solve the equilibrium potential with respect to r. In the remainder of this section, we assume the Newtonian gravitational potential. Solving the equation 0 = ∂V G ef f /∂r for r and substituting back into V G ef f one gets a potential that may be called the effective potential in shape space

Note that I V 2 is indeed r-independent, on account of the r-scaling properties of I and V .

To proceed towards the equilibria, it remains to differentiate V S ef f with respect to θ, φ. It turns out to be convenient to use the side-lengths, r 12 , r 23 , r 31 , rather than θ, φ. We require 0 = ∂ ∂r 23 log V S ef f = ∂ ∂r 23 (log I + 2 log(−V )) =

where we have used the expressions for V, I from (2.2, 2.51) . Solving for r 23 the factors of m 2 m 3 cancel out on account of the equality of gravitational and inertial masses, and one finds that the side lengths are all equal These are the equilateral solutions found by Lagrange [6] . Thus, we presented a derivation of them through the reduced Lagrangian in geometry space. Note that the angular velocity associated with an equilateral with sides a is given by

which is consistent with Kepler's third law and the fact that within the equilateral orbit each body moves as though attracted toward the center of mass (for this reason, these configurations are known as central configurations).

We add some comments. Collinear solutions. V S ef f is invariant under a space reflection, which when translated to geometry space becomes a reflection through the equator. Hence, the gradient of V S ef f must lie within the equator. By restricting V S ef f to the equator one finds the three collinear solutions found by Euler [29] . These solutions were not found above while differentiating with respect to the r ab variables, since the transformation between them and the r, θ, φ variables is singular at the equator.

Type of equilibrium. By noting the behavior of V S ef f near the equator, one concludes that the extremum associated with equilateral motion is in fact a maximum.

Stability of Lagrange's equilateral solutions. The stability of the equilibria found in the previous subsection are closely related to the stability of the associated uniformly rotating solutions. In this subsection, we focus on the stability of Lagrange's equilateral solutions with uniform rotation, and we sketch the determination of the associated domain of linear stability.

Given m 1 , m 2 , m 3 and J 3 this solution is represented by the equilibrium point at the Right or Left pole on the shape sphere. Without loss of generality, let us consider the Right pole. Radial displacements can be shown to result in a stable normal mode, which is analogous to quasi-circular motion in the two-body problem.

For displacement in the plane tangent to the shape sphere, one has to take into account also the magnetic-like field (2.55). Qualitatively, one finds a negative (unstable) effective spring constant, yet the magnetic-like field can stabilize it. More specifically, the equation for the eigenfrequency ω is

where x := ω/Ω is the dimensionless frequency of oscillation, and Ω is the frequency of the equilateral solution (3.6) . This equation has real solutions, implying linear stability, whenever

This stability bound was found by Gascheau (1843) [25] , see also [26] since I was not able to locate the former reference. The current derivation is novel, reducing the dynamical stability analysis to a static one in geometry space.

Outlook. A future version will include the necessary details for the preceding discussion of stability.

In addition to the application of the formulation to the special exact solutions, we anticipate applications to the statistical solution. In a work in preparation, [5], the formulation would be shown to be essential to the determination of the system's regularized phase-volume. A future version will also apply it to a visualization of the system through allowed regions of motion (analogous to the Hill region).

In the three-body problem, we introduced symmetric vector coordinates in the center of mass frame, inspired by Lagrange's solution to the cubic (2.9). The quartic equation also has a general solution, which suggests a generalization to the four-body problem.

Denoting the masses by m a , a = 1, 2, 3, 4 and the positions by r a we define

These variables are translation-invariant vectors. They contain 9 degrees of freedom, which are necessary to cover the configuration space at the center of mass frame. As the labels 1, 2, 3, 4 are permuted, the s vectors transform nicely -they permute among themselves and/or change signs. In order to proceed and decompose the variables into orientation and geometrical variables, the invariants are the 6 scalar products Q rt = s r · s t , r, t = 1, 2, 3 and the gauge-field and associated field-strength become SO(3)-valued, and so, non-Abelian.

I thank Yogesh Dandekar, Lior Lederer and Subhajit Mazumdar for collaboration on a related project. This research, as well as [2] , was performed during the times of the COVID-19 pandemic, and benefitted from the associated isolation.

I dedicate this work to the memory of Ami Nathan, 15.6.1934-24.10 .2020, my father in law and a dedicated family man. is a fully symmetric function of x a , a = 1, 2, 3 and so is P := s 1 s 2 . Therefore, by the fundamental theorem of symmetric polynomials, both S, P can be expressed in terms of b/a, c/a, d/a. Now z = s 3 1 , s 3 2 are the roots of the quadratic

After solving this quadratic, one takes cubic roots to obtain s 1 , s 2 . Finally, (A.2) together with −b/a = s 0 := x 1 + x 2 + x 3 = is a system of linear equations which can be inverted to produce the three roots x a , a = 1, 2, 3. These transformations between the x and s variables are known as a discrete Fourier transform. In this way, Lagrange's method relies on the symmetry properties of expressions in the roots in order to solve the cubic. This is a crucial insight on the road to Galois theory.

The s 1 , s 2 variables are invariant under a shift in x and have nice transformation properties under permutations. In this way, they inspired the definition of z (2.9).

A bi-complex number w is an expression of form

where a, b, c, d are real numbers. Bi-complex numbers can be added and multiplied using the relations

These relations define an algebraic structure known as a commutative ring. Note that bi-complex numbers are distinct from Quaternions: while both are dimension 4 over the Reals, the latter is non-commutative, e.g. i j = −j i = k. Not all non-zero bi-complex numbers have an inverse. This is true of zero-divisors, namely non-zero elements u, v such that u · v = 0. The zero-divisors can be generated from the elementary zero divisors R := 1 + i j L := 1 − i j (B.

3)

The notation R, L stands for right and left: for the isotropic oscillator these states correspond to right/left circular motion, while for the three-body problem they correspond to right/left equilateral triangles. Indeed, R · L = 0. Moreover, all other zero divisors can be generated from these through multiplication by a bi-complex number. R, L satisfy the relations

We note that after multiplication R, L obtain the following simple form:

The isotropic oscillator. The isotropic oscillator is the mechanical system consisting of a point mass m, free to move in the plane under the influence of a harmonic restoring force F = −mω 2 r. It can be defined through the Hamiltonian

One complexifies phase space through the definition of a complex vector

In terms of these variables, one has

where * denotes complex conjugation, and the Poisson brackets

-all other Poisson brackets among the complex variables vanish. It is natural to further complexify the plane. Since plane rotations commute with phase space rotations we use a different imaginary unit j which commutes with i and define a bi-complex number through a → a := a · (x + jŷ) (B.9)

Since time evolution is given by

the conserved charges can be expressed in terms of the Stokes parameters S 0 , S 1 , S 2 , S 3 defined by

where a bar denotes conjugation with respect to j, and a * continues to denote conjugation with respect to i. These expressions are equivalent to S 0 = a · a * , S i = σ i−1 αβ a * α a β , i = 1, 2, 3, α, β = 1, 2 where σ i , i = 1, 2, 3 are the Pauli matrices. If (a 1 , a 2 ) undergo a unitary transformation, S ≡ (S 1 , S 2 , S 3 ) undergoes an orthogonal transformation.

The Stokes parameters satisfy the relation

In addition, their Poisson brackets are given by

Consider an n-dimensional configuration space M, described by some generalized coordinates q i n i=1 , as well as non-coordinate velocities (also known as quasi velocities [28] ) v a = β a iq i , a = 1, . . . , n (C.1)

where β a i = β a i (q j ) defines a changes of basis from the coordinate velocitiesq i to v a . As an example, we shall see below that the angular velocity components of a rotating rigid body, are non-coordinate velocities.

Equivalently, the velocities define a basis of 1-forms over M

A basis of 1-forms is called a coordinate basis, or a holonomic basis, if the 1-forms can all be expressed as differentials of some functions. The non-coordinate nature of a basis is captured by the torsion coefficients C c ab = C c ab (q i ) defined by

The dual basis of vector fields e a is defined by β a · e b = δ a b . In components e b = e i b ∂ i and e i b is the inverse matrix for β a i . In terms of e a , the structure constants can be defined equivalently through the Lie brackets [e a , e b ] = C c ab e c . Suppose the Lagrangian is given in terms of the generalized coordinates and the velocities v a L = L q i , v a . where ∂ a ≡ e i a ∂ i is a derivative with respect to the vector field e a . This generalization was found by Poincaré (1901) [19, 28] .

For coordinate velocities C c ab = 0 and hence the term C c ab v b ∂L ∂v c vanishes and the equations reduce to the Euler-Lagrange equations.

Derivation. The derivation of the Euler-Lagrange equations relies on the commutation of the variation and the time derivative. For non-coordinate velocities, this no longer holds, and a correction term arises. Indeed and similarly 2 other equation by cyclic permutations. ω is the angular velocity vector in a body frame whose axes are the body's principal axes, I i , i = 1, 2, 3 are the principal moments of inertia, and d dt inert denotes a time derivative in the inertial frame. The kinetic energy of the body is given by T = 1 2 I ij ω i ω j . A free body has no potential energy and hence the Lagrangian is defined to be L := T = 1 2 I ij ω i ω j . (C.12)

If one applies the standard Euler-Lagrange equations carelessly, one finds (d/dt) ∂L/∂ω i = 0. This is different from the Euler equations of motion (C.11). However, we notice that ω i are non-coordinate velocities with torsion coefficients

where ijk is the Levi-Civita tensor. This can be shown by expressing ω i in terms of Euler angles, 1 and we note that the coefficients are constant over M and coincide with the structure constants of SO(3). The modified equations of motion (C.5) read

which now fully agree with Euler's equations.

We can now proceed to a Hamiltonian formulation. The conjugate momenta S i = ∂L ∂ω i = I ij ω j (C. 16) are the coordinates of the spin (angular momentum) vector in body coordinates. By (C.9, C.13) their Poisson brackets are given by

The Hamiltonian is given by

where I −1 is in the inverse matrix for the moment of inertia tensor. Finally, the equations of motion are given by

which reproduce the Euler equations.

1 E.g. ω x = − sin θn cos ψnφn + sin ψnθn ω y = sin θn sin ψnφn + cos ψnθn ω z = cos θnφn +ψn (C.14)

where θn, φn, ψn are Euler-like angles. These expressions are of the form (C.1).

Philosophiae Naturalis Principia Mathematica

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Sur l'élimination des noeuds dans le problème des trois corps

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A symmetric reduction of the planar three-body problem

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Théorie mathématique de la lumière II

Zur Quantenmechanik des magnetischen Elektrons

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