Lobachevskii Journal of Mathematics

http://ljm.ksu.ru
ISSN 1818-9962

Vol. 25, 2007, 9–130 \begin{matrix}c \\ ◯ \end{matrix} Wu-Yi Hsiang and Eldar Straume

Wu-Yi Hsiang and Eldar Straume

KINEMATIC GEOMETRY OF TRIANGLES AND THE STUDY OF THE THREE-BODY PROBLEM

(submitted by V. V. Lychagin)

Contents

1. Introduction 10 1.1. The classical conservation laws 11 1.2. Least action principles 12 1.3. An alternative geometric approach 13 2. The basic setting and a presentation of the Main Theorems 15 2.1. Basic notions and terminology 15 2.2. Statement of the Main Theorems 20 3. Basic geometric and kinematic invariants of m-triangles 28 3.1. Ceva-type trigonometry 28 3.2. Analysis of angular velocities and kinetic energies 30 3.3. Linear motions of m-triangles 36 3.4. Eigenvalues and eigenframe of the inertia tensor 38 4. The spherical representation of shape space M^{∗} 41 4.1. Geometric interpretation of the polar distance r 41 4.2. Geometric interpretation of the longitude angle θ 43 4.3. Intrinsic form of the spherical representation 46 4.4. The reduced Newton’s equation in spherical coordinates 48 4.5. Ceva-type relations in the spherical representation 48 4.6. The vector algebra representation of the kinematic geometry 51 4.7. An integral formula for the distance function on M^{∗} 52 5. Motions of m-triangles with conserved angular momentum 54 5.1. Moving eigenframe and intrinsic decomposition of velocities 55 5.2. Final proof of the Main Theorems D, B, E1,E2 57 5.3. Geometric reduction of the least action principles 62 6. The Newtonian potential function 64 6.1. Vector algebra analysis of the Newtonian function 65 7. A geometric setting for the study of triple collisions 71 7.1. Geodesic rays and distance estimates 72 7.2. Existence of triple collision motions with minimal action 74 7.3. The uniqueness problem for triple collision motions with

minimal action 76

8. Case study of triple collision motions with zero energy 82 8.1. The basic setting and statement of Theorem G 82 8.2. Analysis of the potential function for equal masses 85 8.3. Reduction, regularity and singularity 87 8.4. Isosceles and collinear triple collision motions 92 8.5. Analytic uniqueness of triple collision motions 98 8.6. Global behavior of the shape of triple collision motions 106 8.7. Numerical solutions of triple collision motions 117 8.8. An outlook on the general case 121 References 130

1. Introduction

The classical three-body problem studies the motion of a system with three point masses under the action of the Newtonian gravitational potential. Let a_{i} = \overrightarrow{OP_{i}}, i = 1, 2, 3, be the position vectors of the points P_{i} with masses m_{i} > 0, with respect to a chosen inertial coordinate system for the Euclidean 3-space ℝ^{3}. Then a motion of the three point masses will be described as a curve in the (unrestricted) configuration space, namely the Euclidean space ℝ^{9} consisting of all triples (a_{1}, a_{2}, a_{3}).

Newton’s equation of motion is the following second order system of three vector differential equations

m_{i}ä_{i} = \frac{∂U}{∂a_{i}} = \frac{m_{i}m_{j}}{r_{ij}^{3}} (a_{j} − a_{i}) + \frac{m_{i}m_{k}}{r_{ik}^{3}} (a_{k} − a_{i}),  (1)

where {i, j, k} = {1, 2, 3}, and

U = \frac{m_{1}m_{2}}{r_{12}} + \frac{m_{2}m_{3}}{r_{23}} + \frac{m_{1}m_{3}}{r_{13}} , r_{ij} = ∥a_{i} − a_{j}∥ ,  (2)

is the Newtonian potential function. In brief, the central problem is to understand both the geometry and the analysis of the solutions of the above system (1), where each solution curve (or trajectory) is uniquely determined by the initial positions and velocities of the point masses.

1.1. The classical conservation laws. The equations (1) amount to solving a dynamical system in phase space of dimension 18. It is easily seen, however, that the classical 3-body problem (in fact, the n-body problem for all n) is invariant under Galilean transformations, namely the 10-dimensional Galilean group of 4-dimensional space-time ℝ^{3} × ℝ. Accordingly, there are 10 first integrals or conservation laws, and they actually reduce the integration problem to one of order 18 − 10 = 8. These are the conservation of linear momentum, angular momentum and total energy, and they are easily deduced as follows. First, by adding the three vector equations (1) we have

m_{1}ä_{1} + m_{2}ä_{2} + m_{3}ä_{3} = 0,

and hence the center of mass has the uniform motion

a_{CM} = \frac{1}{m̄}(m_{1}a_{1} + m_{2}a_{2} + m_{3}a_{3}) = p_{0} + tv_{0}, m̄ = ∑ m_{i},

where p_{0} and v_{0} are two constant vectors (and hence six scalar conservation laws) determined by the initial data. To make effective use of these we recall the Galilean principle of relativity of Newtonian mechanics, according to which one may choose an equivalent inertial frame of reference with origin at a_{CM} and axes in the same directions as before (or rotated). This justifies using a center of mass inertial reference frame, which effectively reduces the actual configuration space to the following 6-dimensional Euclidean space

M_{0} = ℝ^{6} ⊂ ℝ^{9} : ∑ m_{i}a_{i} = 0.  (3)

Next, the vector

Ω = m_{1}a_{1} × ȧ_{1} + m_{2}a_{2} × ȧ_{2} + m_{3}a_{3} × ȧ_{3}  (4)

is the total angular momentum of the system. It varies covariantly with rotations of 3-space, and it is also constant along a trajectory since by (1)

Ω̇ = ∑_{i} m_{i}a_{i} × ä_{i} = ∑_{i} m_{i}a_{i} × (∑_{j≠i} \frac{m_{j}}{r_{ij}^{3}} (a_{j} − a_{i})) = 0.

This is the law of conservation of angular momentum, whose geometric significance lies much deeper than that of conservation of linear momentum.

Finally, the total energy is defined to be

h = T − U,  (5)

where

T = \frac{1}{2}(m_{1} \left|{ \atop ȧ_{1}}\right|^{2} + m_{2} \left|{ \atop ȧ_{2}}\right|^{2} + m_{3} \left|{ \atop ȧ_{3}}\right|^{2})  (6)

is the kinetic energy and −U is the potential energy. Straightforward differentiation using Newton’s equation (1) gives

ḣ = Ṫ − U̇ = ∑ m_{i}ȧ_{i} · ä_{i} − ∑ \frac{∂U}{∂a_{i}} · ȧ_{i} = ∑ ȧ_{i} · (m_{i}ä_{i} − \frac{∂U}{∂a_{i}}) = 0.

Hence, h is constant along a solution curve of (1) and this is the law of conservation of energy.

1.2. Least action principles. Newton’s equation provides a characterization of the motion from the ”differential ” viewpoint, but it is also useful to characterize the motion as a boundary value problem, namely we ask: for a given pair {p, q} in the configuration space and time interval (t_{1}, t_{2}), what are those trajectories Γ(t), t_{1} ≤ t ≤ t_{2}, with Γ(t_{1}) = p, Γ(t_{2}) = q?

The idea of seeking solutions of the above problem as the extremals of a variational principle applied to virtual motions dates back to the 17th century, inspired by the success of Fermat’s principle of least time in geometric optics. Thus, a type of least action principle for classical mechanics was proposed already by Leibniz, Euler and Maupertuis. It was, however, Lagrange who finally provided a precise mathematical formulation: Lagrange’s least action principle. The solutions of the above boundary value problem are characterized by the variational principle of extremizing the action

J_{1} [Γ] = ∫_{Γ} T dt  (7)

among all virtual motions between a given pair of points and with the same constant total energy h.

It should be noted that time is allowed to vary in the above integral, that is, the limit of integration is not fixed. This awkwardness led Jacobi to reformulate the least action principle to the problem of determining the geodesics on a suitably defined Riemannian manifold (in modern terminology). In his famous lectures on mechanics [6], Jacobi essentially introduced the notion of kinematic metric on the configuration space M_{0}, namely the kinetic energy expression (6) defines a mass dependent Euclidean metric

ds^{2} = 2T dt^{2} = ∑ m_{i}(dx_{i}^{2} + dy_{i}^{2} + dz_{i}^{2}),  (8)

where a_{i} = (x_{i}, y_{i}, z_{i}) is the position vector of the point P_{i} with mass m_{i}.

On the other hand, T = U + h is also a function on M_{0} for a fixed energy level h, and this explains the following step:

Jacobi’s reformulation of Lagrange’s least action principle. The action integral

\sqrt{2}J_{1} [Γ] = \sqrt{2} ∫_{Γ} T dt = ∫_{Γ} \sqrt{T} ds = ∫_{Γ} \sqrt{U + h}ds = ∫_{Γ} ds_{h} (9)

is the arc-length of the virtual motion Γ in the metric space (M_{h}, ds_{h}^{2}), namely in the subspace

M_{h} = {p ∈ M_{0}; U(p) + h > 0} ⊂ M_{0}  (10)

with the squared arc-length element

ds_{h}^{2} = (U + h)ds^{2}.  (11)

Thus, according to Jacobi’s ”geometrization trick”, trajectories of Newton’s equation are precisely the geodesics in the above Riemannian space (M_{h}, ds_{h}^{2}). This also demonstrates why Jacobi in his study of mechanics, in fact, anticipated the general notion of a Riemannian metric. For more information on these issues we refer to Lützen[8].

On the other hand, in 1840 Hamilton formulated another least action principle, also inspired by the results of geometric optics. Hamilton’s principle of least action. The solutions of the above boundary value problem are characterized by the variational principle of extremizing the action integral

J_{2} [Γ] = ∫_{t_{1}}^{t_{2}} Ldt , L = T + U  (12)

among all virtual motions Γ(t) between a given pair of points, for a fixed time interval [t_{1}, t_{2}].

In general, the validity of the ”integral” viewpoint represented by any chosen variational principle is verified by calculating its infinitesimal limit, which must coincide with (or be equivalent to) Newton’s equation. In the case of (9) and (12) respectively, this amounts to the calculation of the geodesic equations of the metric ds_{h} and the associated Euler-Lagrange equations of the above Lagrangian function L, respectively. In both cases it is easily checked that these are equivalent to Newton’s equation.

1.3. An alternative geometric approach. Traditionally, the threebody problem is usually studied in the framework of Hamiltonian mechanics, canonical transformations and symplectic geometry, based on the least action principle of Hamilton and the Hamilton-Jacobi theory.

Moreover, the specific dynamics due to the Newtonian forces is usually assumed from the very beginning. Our present approach is, however, different from this, roughly for two major reasons:

• Firstly, we focus attention on the purely kinematic properties of virtual three-body motions in a Riemannian geometric setting and in the framework of equivariant differential geometry, and • secondly, the Newtonian dynamics is introduced as the final step, involving geometric reduction and conformal modification of the kinematic Riemannian structure, based on the least action principle of Lagrange and Jacobi (cf. Jacobi[6], Lecture 6).

Guided by the above program, the first author initiated studies in 1993 and was joined by the second author in 1994. The basic material, covering their work up to the winter of 1995, was presented in the two preprints [3], [4], and further studies of the n-body problem continued in the following years. However, the two basic preprints were never published and, unfortunately, they have had a rather limited circulation in the mathematical community.

On the other hand, in the recent years new and beautiful results on the three-body problem, and the more general n-body problem as well, have appeared in the literature, some of which are deeply related to the above geometric approach. This clearly suggests that new and unsolved problems along these lines are now becoming more feasible. To further stimulate this trend we propose hereby a review of the works [3], [4] from 1994-95, and Chapter 1 -7 of the present paper is, indeed, merely a faithful presentation of their actual contents.

The exposition has been updated by some changes in notation and terminology, together with a restructuring of ideas and proofs in order to unify and enhance the readability of the presentation. For the convenience of the reader, central results are now formulated as main theorems, such as Theorem A, B, …,F, G presented in Section 2.2.

In the final Chapter 8 we have included some additional and selected unpublished material from 1995, mostly concerning the moduli curves of triple collision motions in the special case of energy h = 0 (and, for technical reasons, the case of uniform mass distribution). The more general case is stated as an open problem in the last section. In this chapter we have done some explicit calculations which also serve as an illustration of how to apply the setting and the results up to Chapter 7.

The three-body problem has a vast literature with many excellent papers. In particular, the theoretical analysis of the problem and its influence during the last century is overwhelming. Our interest in the problem started with a study of Siegel’s monumental analysis of triple collisions involving clever applications of canonical transformations in the Hamiltonian setting. However, we were also astounded by the lack of basic geometric reasoning more directly linked to the kinematic geometry of three-body configurations, and apparently, this seemed to be typical in the more recent literature (such as Marchal’s book), of which we had only superficial knowledge. Perhaps it was, after all, worthwhile having a closer look at the underlying geometric structure of mass triangles and their motions in 3-space ?

With this ambition we started, hopefully with no prejudice due to the existing literature, and this lead us to the purely kinematic study described in the first five chapters of this paper, together with some preliminary investigations of the ensuing dynamics due to gravitational forces, most of which is presented in Chapter 7 and 8.

Thus, in our ”blindfolded” study during 1994-95 we deliberately avoided and did not consult any paper on the three-body problem (except Siegel’s work).This explains why the list of references were almost empty, and we must apologize for that. The present list purposely reflects this former state of affairs, but now there are at least some relevant titles which were available prior to 1995 and which may be useful for the reader. Some of these are also standard references of historical interest. In retrospect, various topics and results discussed in this paper are certainly more or less treated by the many authors who have contributed to the rich literature in classical or celestial mechanics. Therefore, we also apologize to those authors who may feel that we have failed to make the appropriate reference to their work appearing before 1995. On the other hand, in the present paper references newer than 1995 have not been considered.

2. The basic setting and a presentation of the Main Theorems

2.1. Basic notions and terminology. Let ℝ^{3} denote Euclidean 3space with the standard basis {i, j, k}. A three-body system consists of three labeled point masses (P_{i}, m_{i}) in 3-space, and its geometric model is the triangle with vertices P_{i} and the mass m_{i} > 0 attached to P_{i}, which we shall refer to as an m-triangle (or simply a triangle) or configuration. An m-triangle will also be identified with its triple (a_{1}, a_{2}, a_{3}) of position vectors a_{i} = \overrightarrow{OP}_{i}. These triples are usually denoted by boldface letters such as X, Y, but occasionally we also use the notation δ, δ_{1} etc. It is tacitly assumed that a fixed mass distribution (m_{1}, m_{2}, m_{3}) is given, and it is normalized so that m_{1} + m_{2} + m_{3} = 1.

Thus, the abundance of individual motions of three point masses jointly combine to a rich variety of virtual motions of m-triangles, through which the triangle changes its kinematic invariants such as size, shape and orientation (position) and velocity. This is our starting point for a systematic investigation of the kinematics of m-triangles, as a basis for a geometric approach to the three-body problem.

In addition to previously defined quantities, set

α_{j} = the central angle (at origin) opposite to the vertex P_{j}
ω_{j} = the (scalar) angular velocity of a_{j} for planary motion
I_{j} = m_{j} |a_{j}|^{2} , I = I_{1} + I_{2} + I_{3} = ρ^{2}  (13)
C_{1} = −m_{1}I_{1} + m_{2}I_{2} + m_{3}I_{3} etc. (cyclic permutation of indices)
T_{j} = \frac{1}{2}m_{j} \left|{ \atop ȧ_{j}}\right|^{2} , T = T_{1} + T_{2} + T_{3}, cf. (6)
Ω_{j} = m_{j}a_{j}×ȧ_{j}, Ω = Ω_{1} + Ω_{2} + Ω_{3}, cf. (4)
Δ = the area of Δ(P_{1}, P_{2}, P_{3}), Δ_{1} = the area of Δ(O, P_{2}, P_{3}) etc.

where I (respectively I_{j}) is the total (respectively individual) polar moment of inertia, and similarly T, Ω (respectively T_{j}, Ω_{j}) denote kinetic energy and angular momentum as in Section 1.1. See Figure 1.

Certain functions of the mass distribution appear frequently, so we introduce the notation

m_{j}^{∗} = \frac{1}{2}(1 − m_{j}): the dual mass distribution, with ∑ m_{j}^{∗} = 1 (14)
m̂_{1} = m_{2}m_{3} , m̂_{2} = m_{3}m_{1} , m̂_{3} = m_{1}m_{2}

and the basic elementary symmetric functions of the symbols m_{i} are

∑ m_{i} = 1, m̂ = ∑ m̂_{i}, m̄ = m_{1}m_{2}m_{3}.  (15)

2.1.1. Vector algebra and kinematics in the Euclidean space M_{0}. We will assume a center of mass reference frame as in Section 1.1, and hence an m-triangle X is a vector of the 6-dimensional Euclidean configuration space M_{0}, cf. (3). The zero vector X = 0 represents the one-point triangle (or the ”triple collision” configuration), and we say X is collinear or is an eclipse configuration (respectively is non-degenerate) if the subspace Π(X) ⊂ ℝ^{3} spanned by the position vectors a_{i} has dimension 1 (respectively 2). A virtual motion X(t) (or δ(t)) is a time parametrized curve in M_{0}, assumed to be (piecewise) differentiable so that its kinetic energy (6) is defined. The size of an m-triangle X is naturally measured by the Euclidean length

ρ = \sqrt{I} = |X|  (16)

in M_{0} with the kinematic metric (8), equivalently given by the following inner product of Jacobi type

X · Y = m_{1}a_{1} · b_{1} + m_{2}a_{2} · b_{2} + m_{3}a_{3} · b_{3},  (17)

where X = (a_{1}, a_{2}, a_{3}), Y = (b_{1}, b_{2}, b_{3}). For convenience, we define

ω × X = (ω × a_{1}, ω × a_{2}, ω × a_{3}), ω ∈ℝ^{3}  (18)
X × Y = ∑ m_{i}(a_{i} × b_{i})

and observe the general triple product identity

ω × X · Y = ω · X × Y, ω ∈ℝ^{3}.  (19)

The infinitesimal generators of the SO(3)-action on M_{0} are the rotational (or Killing) vector fields

X → ω × X, ω ∈ℝ^{3} ≃ so(3)

of fixed angular velocity ω. These vectors are tangential to the SO(3)- orbits. Thus, at each X the tangent space T_{X}M_{0} ≃ M_{0} has an orthogonal decomposition into vertical and horizontal vectors, where the vertical ones are the above Killing vectors ω × X and the horizontal vectors Y are characterized by X × Y = 0, due to (19).

For any virtual motion X(t) in M_{0}, the velocity vector at each time t has the above type of splitting, namely

Ẋ = \frac{d}{dt}X = Ẋ^{ω} + Ẋ^{h} = (ω × X) + Ẋ^{h},  (20)

where ω = ω(t) is commonly referred to as the (instantaneous) angular velocity of the motion. Correspondingly, kinetic energy splits as the sum

T = \frac{1}{2} |ω × X|^{2} + \frac{1}{2} |Ẋ^{h}|^{2} = T^{ω} + T^{h}  (21)

of purely rotational and horizontal kinetic energy, respectively. The motion is called horizontal if the velocity is always horizontal.

Using (19) we also deduce the following relationship between the angular momentum and angular velocity of a virtual motion, namely

Ω = X × Ẋ = X × Ẋ^{ω} = X × (ω × X).  (22)

Indeed, to each m-triangle X is associated the inertia operator

𝕀_{X} : ℝ^{3} → ℝ^{3}, ω → X × (ω × X)  (23)

relating the two vectors ω and Ω. This operator is invertible when X is nondegenerate, whereas Ω determines ω modulo a summand along the line Π(X) when X is an eclipse configuration. In any case, the rotational velocity component ω × X in (20) is uniquely determined by Ω and X. Consequently, the motion is horizontal if and only if Ω vanishes; in particular, such a motion must be planary (see Remark 23 ).

The above inertia operator corresponds uniquely to the associated inertia tensor

B_{X}(u, v) = (u × X) · (v × X)

= ∑ m_{j}(u × a_{j}) · (v × a_{j}), u, v ∈ ℝ^{3} (24)

which is a bilinear symmetric form on Euclidean 3-space. They are related by the identity

B_{X}(u, v) = 𝕀_{X}(u) · v.

For example, they provide an orthonormal eigenframe for each m-triangle, and hence a moving eigenframe for a motion of m-triangles, see Theorem D and Section 3.4.

2.1.2. Oriented m-triangles and their configuration space M. Since triangles in 3-space can be oriented we propose the following ”refinement” of the notion of an m-triangle. Define an oriented m-triangle to be a pair (X, n), where n ∈S^{2} ⊂ ℝ^{3} is a unit vector perpendicular to Π(X). In particular, a nondegenerate triangle can be oriented in two ways, namely we say the orientation is positive (respectively negative) if (a_{1}, a_{2}, n) is a right-handed (respectively left-handed) frame.

Clearly, for an m-triangle motion the orientation may be chosen so that the normals n(t) vary continuously along the motion, and then the orientation changes to the opposite one as the motion passes (transversely) through an eclipse configuration. In the study of planary motions we will (tacitly) assume the plane to be the xy-plane, with unit normals n = ±k.

Set M_{+} (respectively M_{−}) to be the set of positively (respectively negatively) oriented m-triangles, together with the set E of oriented eclipse configurations, where the latter includes the 2-sphere of orientations of the one-point triangle X = 0. Then the configuration space of oriented m-triangles is the union

M = M_{+} ∪ M_{−} ⊂ M_{0} × S^{2}, E = M_{+} ∩ M_{−}  (25)

and a virtual motion of oriented m-triangles is a parametrized curve on M

Γ(t) = (X(t), n(t))  (26)

As a submanifold of M_{0}×ℝ^{3} ≃ ℝ^{9}, M inherits a Riemannian structure and a natural isometric action of SO(3), also referred to as the congruence group. Let us have a closer look at this manifold and the two projection maps

M_{0} ⟵^{π_{1}} M →^{π_{2}} S^{2},

where π_{1} is a 2-fold covering over the non-degenerate m-triangles. On the other side, each (X, n) is SO(3)-equivalent to some (Y, k), where Y lies in the xy-plane ℝ^{2} and hence belongs to

ℝ^{4} ⊂ M_{0} : ∑ m_{i}a_{i} = 0 , a_{i} ∈ ℝ^{2}.  (27)

Then it is a useful observation (not mentioned in the 1994-95 preprints) that π_{2} is the SO(3)-equivariant projection of a homogeneous 4-plane bundle

ℝ^{4} → M ≃ SO(3) ×_{SO(2)} ℝ^{4} → SO(3)/SO(2) = S^{2},  (28)

where the 4-space in (27) is the fiber over the unit normal k of the xyplane, and SO(2) is the rotation group fixing the z-axis.

In (28) M is expressed as the SO(2)-orbit space of SO(3) × ℝ^{4}, where h ∈ SO(2) acts by (g, Y) → (gh^{−1}, hY), and the SO(2)-orbit of (g, Y) is identified with the oriented m-triangle (gY, gk). This is an example of a well known ”twisted product” construction.

2.1.3. Congruence moduli space and shape space. The orbit space

M̄ = M/SO(3) = M̄_{+} ∪ M̄_{−} , Ē = M̄_{+} ∩ M̄_{−}  (29)

is the (congruence) moduli space, where the union corresponds to the SO(3)-invariant splitting (25) of M, and the shape space is the subspace

M^{∗} = M_{+}^{∗} ∪ M_{−}^{∗} , E^{∗} = M_{+}^{∗} ∩ M_{−}^{∗}  (30)

corresponding to m-triangles of fixed size ρ = 1. Since an m-triangle (and its congruence classes) is scaled by the size function ρ ≥ 0 (16), M̄ = C(M^{∗}) has a natural structure of a cone over M^{∗}. So, we may regard M̄ as the union of two identical cones or ”half-spaces” M̄_{±} glued together along their common boundary Ē. This will be made more precise below.

Let us first investigate the topology of the above spaces from a trigonometric viewpoint, using the quadratic form (68) representing the squared area of m-triangles. The triple (I_{1}, I_{2}, I_{3}) is, indeed, a complete congruence invariant for (unoriented) m-triangles. There is only one ”halfspace” in the unoriented case and we may express it as the following cone in {I_{j}}-coordinate 3-space

M̄_{±} ≃ {(I_{1}, I_{2}, I_{3}) | I_{j} ≥ 0, Q(I_{1}, I_{2}, I_{3}) ≥ 0} ,  (31)

where the corresponding shape space M_{±}^{∗} is cut out by the plane I_{1} + I_{2} + I_{3} = 1. The eclipse variety Ē, defined by the condition Q = 0, is the cone over E^{∗}.

Now it is not difficult to see that M_{±}^{∗} is topologically a closed 2-disk with E^{∗} as boundary circle, and consequently the full shape space (30) is a 2-sphere M^{∗} ≈ S^{2} with a distinguished equator circle E^{∗} separating the two hemispheres M_{±}^{∗}. Note that the triple of central angles

(α_{1,}α_{2}, α_{3}), ∑ α_{j} = 2π

is a complete system of invariants for the shape of unoriented m-triangles and hence these angles also yield coordinates for each of the hemispheres.

It follows that the full moduli space M̄ is the cone over a 2-sphere and hence is homeomorphic to 3-space,

M̄ = C(M^{∗}) ≃ C(S^{2}) = ℝ^{3},  (32)

in such a way that Ē ≃ ℝ^{2} is the coordinate plane z = 0, M̄_{+} is the upper half space z ≥ 0 and M̄_{−} is the lower half-space z ≤ 0.

Finally, from the viewpoint of equivariant geometry, we observe that the pair M̄ ⊃ M^{∗} is the SO(3)-orbit space of the vector bundle (28) and its sphere bundle, namely

M^{∗} = (SO(3) ×_{SO(2)} S^{3})/SO(3) = S^{3}/SO(2) ≃ S^{2}  (33)

M̄ = (SO(3) ×_{SO(2)} ℝ^{4})/SO(3) = ℝ^{4}/SO(2) = C(S^{3}/SO(2)) ≈ ℝ^{3}.

For comparison reasons, if we only consider unoriented m-triangles, then the corresponding calculation of the moduli space as an orbit space will yield the closed half-space

M̄_{±} = M_{0}/SO(3) = M_{0}/O(3) = ℝ^{4}/O(2) ≈ ℝ_{±}^{3}.  (34)

2.2. Statement of the Main Theorems. In this summary we focus attention on six main topics, each of which is centered around one or two main theorems, labeled by A, B, C1, C2 … 2.2.1. Kinematic geometry of m-triangles and universal sphericality. For a virtual 3-body motion with vanishing angular momentum, that is, a horizontal motion Γ(t) in M, the kinetic energy T depends solely on the moduli curve Γ̄(t) in M̄, namely in terms of the local coordinates (I_{1}, I_{2}, I_{3}) it is the following ”differential” expression

T̄ = \frac{1}{8} \frac{İ}{I}^{2} + \frac{1}{2IQ} ( ∑_{i mod 3} m_{i}I_{i+1}I_{i+2}İ_{i}^{2} − C_{i}I_{i}İ_{i+1}İ_{i+2}) , (35)

where Q = 16m_{1}^{2}m_{2}^{2}m_{3}^{2}Δ^{2}. This is, indeed, a positive definite quadratic form on the tangent bundle of the moduli space M̄ and thus naturally defines a kinematic Riemannian metric

ds̄^{2} = 2T̄dt^{2}.  (36)

For a general virtual motion the same expression (35) is, in fact, obtained from T by removing the rotational kinetic energy. Therefore, by (21), the horizontal kinetic energy

T̄ = T^{h} = \frac{1}{2} |\frac{d}{dt}Γ̄(t)|^{2} = T − T^{ω}  (37)

may well be referred to as the kinetic energy in the moduli space.

Both the definition and the formula for the above metric (36) are dependent on the given mass distribution in a rather intricate manner. Therefore, it is a pleasant surprise that such a kinematically defined Riemannian structure (M̄, ds̄^{2}) turns out to be not only independent of the mass distribution, but it is, in fact, isometric to the Riemannian cone of the Euclidean sphere of radius 1/2, namely

Theorem A Let I = ρ^{2} be the moment of inertia and M^{∗} be the subspace of M̄ with I = 1, and set dσ^{2} = ds̄^{2}|_{M^{∗}} to be the restriction of the kinematic metric. Then

ds̄^{2} = dρ^{2} + ρ^{2}dσ^{2},  (38)
(M^{∗}, dσ^{2}) ≃ S^{3}(1)/U(1) = ℂP^{1} ≃ S^{2}(1/2),  (39)

where S^{3}(1) is the 3-sphere of radius 1 and S^{3}(1) → ℂP^{1} is the classical Hopf fibration.

The surprising emergence of spherical symmetry in the kinematic Riemannian space (M̄, ds̄^{2}) for arbitrary mass distribution naturally brings in the classical spherical geometry as a useful tool in the study of the three-body problem. We propose to call this fundamental fact the universal sphericality of the kinematic geometry of m-triangles.

The orientation reversing map (X, n) → (X, −n) of oriented m-triangles induces an isometric involution of (M^{∗}, dσ^{2}) with E^{∗} as its fixed point set, namely the distinguished equator which divides M^{∗} into two hemispheres M_{±}^{∗}. On this circle lie the three points 𝔭_{ij} representing the shape of the three types of binary collisions, cf. (97). Indeed, their relative positions on the circle determine the mass distribution uniquely, see Section 4.4 and (136).

2.2.2. Unique lifting property. Theorem B To a given curve Γ̄(t) in the moduli space M̄ ∖ {0}, together with a given constant vector Ω and initial configuration Γ(t_{0}), there exists a unique curve Γ(t) in M with Γ̄(t) as its moduli curve and with Ω as its conserved angular momentum. Moreover, the curve Γ(t) can be computed in terms of the C^{1}-data of Γ̄(t).

Consider the orbit map π : M → M̄, and observe that SO(3) acts freely outside the sphere π^{−1}(0) ≃ S^{2} and defines a principal bundle

π : M ∖ π^{−1}(0) → M̄ ∖ {0} .  (40)

In particular, above the half-spaces M̄_{±} ≃ ℝ_{±}^{3} there are locally trivializing diffeomorphisms

SO(3) × M̄_{±} → M_{±}.  (41)

Geometrically speaking, a motion of m-triangles can be represented by a time parametrized curve Γ(t) in M (or M_{±}) which (locally) consists of two components, namely a moduli curve Γ̄(t) that records the change of size and shape of the oriented m-triangles, and a position curve γ(t) in SO(3) that records the change of position. The latter curve is, of course, constrained by the fixed angular momentum.

In the special case of Ω = 0, namely the horizontal lifting of moduli curves, the proof of Theorem B follows directly from the standard theory of principal G-bundles with a connection (cf. e.g. [5]), applied to the above principal SO(3)-bundle. Therefore, we shall rather focus on the general case with Ω ≠ 0 and present two different proofs. The first proof involves the inertia operator (23), and the second proof is an application of Theorem D stated below. We refer to Section 5.2.2.

2.2.3. Angular velocities and kinematic Gauss-Bonnet formula.

Theorem C1 For a planary motion of oriented m-triangles with normal vector k and angular momentum Ω = Ωk, let ω_{i} = ϕ̇_{i} be the individual (scalar) angular velocity of the position vector a_{i} = \overrightarrow{OP}_{i}. Then

ω_{i} = ω_{i}^{0} + \frac{Ω}{I}, i = 1, 2, 3  (42)

where ω_{i}^{0} is a ”differential” expression purely at the moduli space level, namely

ω_{1}^{0} = \frac{1}{I}(I_{3}α̇_{2} − I_{2}α̇_{3}) etc. (cyclic permutation of indices)  (43)
= \frac{1}{8m_{1}m_{2}m_{3}ΔI} [(C_{3}I_{3} − C_{2}I_{2})\frac{İ_{1}}{I_{1}} − (C_{1} + 2m_{2}I_{3})İ_{2}  (44)
+ (C_{1} + 2m_{3}I_{2})İ_{3}] etc.

Remark 1. The proof of Theorem C1 holds for non-planary motions as well, that is, the plane Π(t) of the m-triangle is time dependent. Then ω_{i} stands for the (scalar) angular velocity of the velocity component of a_{i} in Π(t), and in formula (42) Ω must be replaced by the normal component Ω ·n(t) (cf. Theorem D). We refer to Section 3.2.1.

We introduce the following three kinematic 1-forms on the moduli space M̄ − {0} :

Θ_{1} = \frac{1}{I}(I_{3}dα_{2} − I_{2}dα_{3}) etc. (cyclic permutation of indices)
= \frac{1}{8m_{1}m_{2}m_{3}ΔI} [(C_{3}I_{3} − C_{2}I_{2})\frac{dI_{1}}{I_{1}} − (C_{1} + 2m_{2}I_{3})dI_{2}  (45)
+ (C_{1} + 2m_{3}I_{2})dI_{3}] etc.

In fact, they are invariant under scaling and may therefore be regarded as 1-forms on the shape space M^{∗} via the canonical retraction M̄ ∖ {0} → M^{∗}. They share the basic property

dΘ_{i} = 2dA, i = 1, 2, 3,  (46)

where dA is the area form of the 2-sphere M^{∗} ≃ S^{2}(1/2). Evidently, the 1-forms have singularities on the eclipse circle E^{∗}.

By suitably combining the kinematic 1-forms on appropriate regions on M^{∗} and applying Green’s theorem, the kinematic Gauss-Bonnet version as described by the next theorem follows immediately from Theorem C1 and (46).

Theorem C2 Let the shape curve of a piecewise differentiable motion of oriented m-triangles with Ω = 0 constitute the oriented boundary of a region D in M^{∗} ≃ S^{2}(1/2). Then the total change of position of the triangle is a rotation of angle equal to twice the oriented area of D, namely

Δϕ_{i} = ∫_{t_{0}}^{t_{1}} ω_{i}^{0}dt = ∫_{∂D} Θ_{i} = ∫∫_{D} 2dA.  (47)

The above type of integral (47) is an example of the geometric phase in the literature. Its value depends only on the shape curve and is independent of its parametrization. In the case of a planary motion with nonzero angular momentum, however, the total change of position in the above case (47) has an additional term called the dynamical phase, namely as a consequence of (42)

Δϕ_{i} = ∫∫_{D} 2dA + ∫_{t_{0}}^{t_{1}} \frac{Ω}{I}dt.  (48)

2.2.4. Moving eigenframe and Euler equations for m-triangles. Let (X, n) be a nondegenerate oriented m-triangle. Then we can choose eigenvectors of the inertia tensor (24) which constitute a positive orthonormal frame

(u_{1}, u_{2}, n) ∈SO(3),

where u_{1}, u_{2} lie in the plane Π(X) and u_{1} × u_{2} = n. By definition,

B_{X}(u_{1}, u_{1}) = λ_{1}, B_{X}(u_{2}, u_{2}) = λ_{2}, B_{X}(u_{1}, u_{2}) = 0  (49)

where the two eigenvalues λ_{i} may be expressed (cf. Section 3.4) as

\{λ_{1}, λ_{2}\} = \frac{1}{2}(I ± \sqrt{I^{2} − 16m_{1}m_{2}m_{3}Δ^{2}}) = \frac{I}{2}(1 ± sin φ), (50)

using spherical polar coordinates (φ, θ) on the 2-sphere M^{∗} centered at the north pole N, where φ is the colatitude with φ = 0 at the pole. The eigenvalue in the normal direction ±n is the largest eigenvalue

λ_{3} = λ_{1} + λ_{2} = I.

To a continuous motion of oriented m-triangles we may choose such an eigenframe

𝔉(t) = {u_{1}(t), u_{2}(t), n(t)}  (51)

varying continuously with the motion. In particular, t → 𝔉(t) is also a parametrized curve in SO(3).

Theorem D Let 𝔉(t) in (51) be a moving eigenframe attached to a differentiable motion Γ(t) of m-triangles, with Ω as the conserved angular momentum. Then the triple of inner products

g_{1} = Ω · u_{1}, g_{2} = Ω · u_{2}, g_{3} = Ω · n  (52)

satisfy the following system of ODE, namely

ġ_{1} = g_{2} [(\frac{1}{λ_{3}} − \frac{1}{λ_{2}} )g_{3} + \frac{1}{2}θ̇ cos φ]
ġ_{2} = g_{1} [(\frac{1}{λ_{1}} − \frac{1}{λ_{3}} )g_{3} − \frac{1}{2}θ̇ cos φ]  (53)
ġ_{3} = g_{1}g_{2}(\frac{1}{λ_{2}} − \frac{1}{λ_{1}} )

where the numbers λ_{i}(t) are the eigenvalues of the inertia tensor of Γ(t) and depend solely on the moduli curve Γ̄(t) = (ρ(t), φ(t), θ(t)).

Corollary 2. It follows from (53) that g_{1}(t_{0}) = g_{2}(t_{0}) = 0 at just one time t_{0} implies that g_{1}(t) = g_{2}(t) = 0 for all time. Thus, such a motion is planary if and only if the angular momentum vector is perpendicular to the m-triangle at just one time t_{0}.

Remark 3. The system (53) is the exact generalization of the classical Euler equations for a rigid body, see e.g. Arnold [1], p.143, where M_{i} and the fixed numbers I_{i} correspond to our g_{i} and λ_{i}, respectively. In (53) the additional terms are due to the change of shape, and the system is singular where the motion passes through an eclipse configuration, say, with λ_{1} = 0 and hence also g_{1} = 0. In particular, the eclipse takes place along a line perpendicular to Ω.

The triple (g_{1,}g_{2}, g_{3}) is the coordinate vector, with respect to the moving eigenframe, of the constant vector Ω = Ωk. It determines the position of the m-triangle, in particular its normal vector n, up to a rotation around the k-axis by a specific precession angle χ(t). This angle is calculated by quadrature from the formula

χ̇ = \frac{ṅ · k × n}{|k × n|^{2}} = \frac{Ω}{g_{1}^{2} + g_{2}^{2}} (\frac{g_{1}^{2}}{λ_{1}} + \frac{g_{2}^{2}}{λ_{2}} ) , cf. Section 5.2.1.  (54)

It follows that the m-triangle motion Γ(t) is largely described by two curves on the 2-sphere, namely the shape curve Γ^{∗}(t) = (φ(t), θ(t)) and the precession curve, that is, the curve traced out by the normal vector n(t). Thus, for the study of non-planar motions it is a basic problem to investigate the relationship between these two curves.

2.2.5. The reduced Newton’s equations. Now, let us focus attention on the dynamics of three-body motions, namely the Newtonian equation of motion (1). Such motions are, of course, a very special subclass of all the virtual three-body motions considered before.

The Ω-reduced Newton’s equation is a second order system of ODE for the moduli curves of three-body motions with a fixed angular momentum vector Ω, namely the three equations (with index i mod 3)

Ï_{i} = 4T_{i} − (\frac{m_{i} + 2m_{i+1}}{r_{i,i+1}^{3}} + \frac{m_{i} + 2m_{i+2}}{r_{i,i+2}^{3}} ) I_{i}  (55)

− (\frac{1}{r_{i,i+1}^{3}} − \frac{1}{r_{i,i+2}^{3}} ) (m_{i+1}I_{i+1} − m_{i+2}I_{i+2})

which are easily derived from the system (1). However, the individual kinetic energy terms T_{i} depend on Ω, of course, but are otherwise expressed solely at the level of the moduli space M̄.

The two cases of planary and non-planary motions differ substantially in complexity, so we will consider them separately. In the following two theorems we assume the initial position Γ(t_{0}) is not a collinear configuration (since otherwise the given initial data will be incomplete).

Theorem E1 A planary three-body motion Γ(t) is completely determined by its moduli curve Γ̄(t), initial position Γ(t_{0}) and angular momentum vector. The curve Γ̄(t) is a solution of the Ω-reduced Newton’s equation (55), with kinetic energy terms

T_{i} = \frac{İ_{i}^{2}}{8I_{i}} + \frac{1}{2}ω_{i}^{2}I_{i},  (56)

where ω_{i} is the i-th individual angular velocity (42).

Conversely, each solution curve of this Ω-reduced Newton’s equation can be realized as the moduli curve of a three-body motion in the xyplane, with a given initial position and normal vector Ωk as the conserved angular momentum.

Remark 4. The above Ω-reduced Newton’s equation may, of course, be expressed purely in terms of the coordinates {I_{j}} or the mutual distances {r_{ij}}, see (64). In fact, such an Ω-reduced Newton’s equation in terms of coordinates {r_{ij}} was derived by Lagrange [7]. We refer to Section 4.3.1 for another version in terms of spherical coordinates of M̄ ≃ ℝ^{3} as a cone over the 2-sphere.

For the statement of the general (e.g. non-planary) version of the above theorem, let {u_{1}, u_{2}, n} denote a continuous eigenframe associated with the motion Γ(t), and let

(g_{1}, g_{2}, g_{3}), where g_{1}^{2} + g_{2}^{2} + g_{3}^{2} = Ω^{2},  (57)

be the coordinate vector of Ω relative to this frame, as in Theorem D. The individual kinetic energies depend on the components g_{k}, more precisely, they split into a tangential and normal component

T_{i} = T_{i}^{τ} + T_{i}^{η},  (58)

where the tangential term T_{i}^{τ} depends on the normal component g_{3} and T_{i}^{η} depends on g_{1} and g_{2}. We refer to Section 5.1 and 5.2.3 for a precise description of these quantities.

Theorem E2 A general three-body motion Γ(t) is completely determined by its moduli curve Γ̄(t), initial position Γ(t_{0}) and angular momentum vector Ω. The curve Γ̄(t) is characterized by the Ω-reduced Newton’s equations (55) with kinetic energy terms T_{i} (58) depending on the moving frame coordinates (57) of Ω and are thus coupled with the Euler equations, namely the first order ODE (53).

2.2.6. Reduction of the least action principles. Here we will only consider planary three-body motions and state the associated Ω-reduced least action principles, whose extremals are precisely the moduli curves of those planary three-body motions with a fixed angular momentum Ω = Ωk. In this case the total kinetic energy

T = T̄ + T^{ω} = T̄ + \frac{Ω^{2}}{2I}  (59)

and the Lagrange function L = T + U are, in fact, defined at the level of the moduli space M̄, and therefore the two action integrals

J̄_{1,Ω} = ∫_{Γ̄} T dt , J̄_{2,Ω} = ∫_{t_{1}}^{t_{2}} Ldt  (60)

apply to moduli curves Γ̄(t).

Theorem F The solution curves of the planary Ω-reduced Newton’s equation can be characterized as the extremal curves of J̄_{1,Ω} (respectively J̄_{2,Ω}) applied to curves Γ̄ in M̄ with fixed end points together with fixed energy h (respectively fixed time interval [t_{1}, t_{2}]).

2.2.7. Shape curves of triple collision trajectories. In Chapter 8 we initiate a general study of the geometry of moduli curves in the vicinity of a triple collision. According to a classical result of Sundman and Siegel, towards the collision these curves Γ̄(t) approach a ray solution, which in the generic case has the shape ±p̂_{0} of an (oriented) equilateral triangle. In the simplest case of equal masses, ±p̂_{0} are the poles of the 2-sphere M^{∗} ≃ S^{2}, and hence the corresponding shape curves approach one of the poles. Thus, it is natural to focus attention on the family of shape curves representing a triple collision with the limit shape of p̂_{0}. Here we state a theorem which is a simplified version of Theorem G_{1} stated in Section 8.1.

Theorem G Assume uniform mass distribution and zero total energy, and consider the family 𝔖 of arc-length parametrized shape curves Γ^{∗}(s), s ≥ 0, representing 3-body motions with a triple collision at s = 0, say Γ^{∗}(0) is the north pole of S^{2}. The family has the following properties:

(i) There is a unique curve Γ_{θ_{0}}^{∗} for each initial longitude direction θ_{0} at the pole, and Γ_{θ_{0}}^{∗} and Γ_{θ_{0}+π/3}^{∗} are congruent modulo a rotation of the sphere.

(ii) The six meridians representing isosceles triangles belong to the family 𝔖. They divide the sphere into six congruent sectors of angular width π/3, and each curve Γ_{θ_{0}}^{∗} stays within a sector, at least until the first eclipse (at the equator).

(iii) The curves Γ_{θ_{0}}^{∗}(s) are analytic in s, with no singularity before the first eclipse, and Γ_{θ_{0}+π}^{∗}(s) = Γ_{θ_{0}}^{∗}(−s).

(iv) The sign of the curvature of the above shape curves is the same inside a sector, and the sign is the opposite in neighboring sectors.

3. Basic geometric and kinematic invariants of m-triangles

3.1. Ceva-type trigonometry. In classical Greek geometry individual triangles - not their motions and kinematic relations - are the geometric objects of basic importance. A triangle Δ(P_{1}, P_{2}, P_{3}) is specified by its three vertices P_{i}, and its congruence properties involve the fundamental geometric concepts ”side”, ”angle” and ”area”, whose relationships are expressed by trigonometric identities and congruence theorems. In our study, however, we are rather concerned with m-triangles, that is, a positive mass m_{i} is attached to P_{i}. Thus it is natural and useful to reformulate the usual trigonometry into a kind of Ceva-trigonometry, depending on the given mass distribution.

Let us first establish the following three identities (cf. (13)):

Ceva-area law: Δ_{j} = m_{j}Δ,  (61)
Ceva-sine law: \frac{sin α_{i}}{m_{i} ∥a_{i}∥} = \frac{2Δ}{∥a_{1}∥ ∥a_{2}∥ ∥a_{3}∥} , i = 1, 2, 3,  (62)
Ceva-cosine law 2\sqrt{m_{i}m_{j}}\sqrt{I_{i}I_{j}} cos α_{k} = −C_{k}.  (63)

By calculating cross products such as

0 = a_{1} × ∑ m_{j}a_{j} = m_{2}a_{1} × a_{2} + m_{3}a_{1} × a_{3}
= (2m_{2}Δ_{3} − 2m_{3}Δ_{2})n,

the first law (61) follows directly, and then the sine law (62) follows:

2m_{3}Δ = 2Δ_{3} = |a_{1}| |a_{2}| sin α_{3} ⟹ \frac{sin α_{3}}{m_{3} |a_{3}|} =\frac{2Δ}{∥a_{1}∥ ∥a_{2}∥ ∥a_{3}∥}.

Furthermore, consider the ”small triangle” with side vectors {m_{i}a_{i}}, say, with one vertex at the center of mass O and an adjacent side along OP_{j} for some j. The triangle has outer angles α_{i}, and by applying the usual cosine law to it we deduce the cosine law (63).

Next, by combining the usual cosine law and its Ceva version, the relationship between the mutual distances s_{i} and the moments of inertia I_{i} is

s_{i}^{2} = r_{jk}^{2} = \frac{(1 − m_{i})I − I_{i}}{m_{j}m_{k}}, \{i, j, k\} = \{1, 2, 3\} ,  (64)

from which we also deduce

I = ∑_{i<j} m_{i}m_{j}r_{ij}^{2} = \frac{m_{1}^{∗}}{m_{2}m_{3}} C_{1} + \frac{m_{2}^{∗}}{m_{3}m_{1}} C_{2} + \frac{m_{3}^{∗}}{m_{1}m_{2}} C_{3},  (65)

where the first identity is known as Lagrange’s formula for the total moment of inertia with respect to the center of mass, and m_{i}^{∗} are the dual masses (14).

Finally, consider the ”Heron” quadratic form

H(a, b, c) = 2(ab + bc + ca) − (a^{2} + b^{2} + c^{2})  (66)

and recall the classical Heron’s formula for the area Δ

H(s_{1}^{2}, s_{2}^{2}, s_{3}^{2}) = 16Δ^{2}.  (67)

Set

Q(I_{1}, I_{2}, I_{3}) = H(m_{1}I_{1}, m_{2}I_{2}, m_{3}I_{3}) = 2 ∑_{i<j} m_{i}m_{j}I_{i}I_{j} − ∑_{j} m_{j}^{2}I_{j}^{2} (68)

= ∑_{i<j} C_{i}C_{j} = 4m_{1}m_{2}I_{1}I_{2} − C_{3}^{2} etc.

and consider again the ”small triangle” with side vectors m_{i}a_{i}. On the one hand, its area Δ̂ is related to Δ by

4Δ̂^{2} = m_{1}^{2}m_{2}^{2} |a_{1} × a_{2}|^{2} = 4m_{1}^{2}m_{2}^{2}Δ_{3}^{2} = 4m_{1}^{2}m_{2}^{2}m_{3}^{2}Δ^{2}

and on the other hand, 16Δ̂^{2} = Q(I_{1}, I_{2}, I_{3}) by (67) and (68). Consequently, we obtain the

Ceva-Heron formula: Q(I_{1}, I_{2}, I_{3}) = 16m_{1}^{2}m_{2}^{2}m_{3}^{2}Δ^{2}.  (69)

3.1.1. A simple torque formula. As a simple application of the Ceva-area law (61) we prove the following result concerning the individual torques due to gravitational forces acting at the vertices P_{i} of a nondegenerate m-triangle Δ(P_{1}, P_{2}, P_{3}).

Lemma 5. Let t_{i} be the torque of the Newtonian gravitational forces at P_{i}, i = 1, 2, 3,with respect to the center of mass O. Then

t_{i} = Ω̇_{i} = 2m_{1}m_{2}m_{3}Δ(\frac{1}{r_{i,i+1}^{3}} − \frac{1}{r_{i,i+2}^{3}} )n, i mod 3,

where n is the unit normal vector so that (a_{1}, a_{2}, n) is a right-handed frame.

Proof. Let F_{12} and F_{13} be the gravitational forces due to the mass points P_{2} and P_{3} acting on P_{1}, namely

F_{12} = \frac{m_{1}m_{2}}{r_{12}^{3}} (a_{2} − a_{1}), F_{13} = \frac{m_{1}m_{3}}{r_{13}^{3}} (a_{3} − a_{1}).

Then, by definition of torque and the area law (61)

t_{1} = a_{1} × (F_{12} + F_{13}) = \frac{m_{1}m_{2}}{r_{12}^{3}} (a_{1} × a_{2}) + \frac{m_{1}m_{3}}{r_{13}^{3}} (a_{1} × a_{3})

= (\frac{m_{1}m_{2}}{r_{12}^{3}} 2m_{3}Δ)n − (\frac{m_{1}m_{3}}{r_{13}^{3}} 2m_{2}Δ)n = 2m_{1}m_{2}m_{3}Δ(\frac{1}{r_{12}^{3}} − \frac{1}{r_{13}^{3}} )n

and similarly at the other two vertices.

Corollary 6. Corollary t_{1} = 0 if and only if r_{12} = r_{13}, and all t_{i} = 0 if and only if the triangle is regular (i.e. equilateral).

3.2. Analysis of angular velocities and kinetic energies. In the orthogonal splitting (20) of the velocity of a virtual motion X(t) = (a_{1}(t), a_{2}(t), a_{3}(t)), the horizontal component further splits into two summands

Ẋ^{h} = Ẋ^{ρ} + Ẋ^{σ} = \frac{ρ̇}{ρ}X + Ẋ^{σ}  (70)

representing the change of size and shape, respectively, and correspondingly the total kinetic energy splits as

T = T^{ω} + T^{h} = T^{ω} + (T^{ρ} + T^{σ}) = \frac{1}{2} |ω × X|^{2} + (\frac{1}{2}ρ̇^{2} + T^{σ}). (71)

In this chapter we will show that T^{h} actually equals the expression in (35), and in particular it depends only on the velocity of the image curve in M̄. This will justify our definition of T^{h} as the kinetic energy T̄ of the moduli curve, hence also our definition of the kinematic Riemannian metric on M̄

ds̄^{2} = 2T̄dt^{2} = dρ^{2} + 2T^{σ}dt^{2}.  (72)

Our first proof of Theorem A is by showing that the metric (72) actually transforms to the metric (138).

The differential expression (35), as a function on the tangent bundle of M̄, is calculated by eliminating from T its dependence on the angular momentum, namely the rotational energy. In fact, it suffices to consider a class of virtual motions whose term T^{ω} is easy to calculate and hence eliminate. For this single purpose we could as well assume T^{ω} = 0 from the outset and simply express T at the moduli space level. However, it is also illuminating to analyze the class of planary motions with a broader perspective.

3.2.1. Kinematics of planary motions and proof of Theorem C1 and C2. We assume the motion takes place in the xy-plane and write

Ω = Ωk, ω = ωk.

In this case (71) reads

T = T^{ω} + T^{ρ} + T^{σ} = \frac{1}{2} \frac{Ω^{2}}{I} + \frac{1}{2}ρ̇^{2} + T^{σ}.  (73)

On the other hand, from the orthogonal decomposition of each ȧ_{j} into its rotational and radial component

ȧ_{j} = ω_{j}(k × a_{j}) + \begin{matrix}ρ̇ & j \\ ρ & j \end{matrix} a_{j} , where ρ_{j}^{2} = I_{j},  (74)

the total kinetic energy also adds up to

T = \frac{1}{2} ∑ I_{i}ω_{i}^{2} + \frac{1}{8} ∑ \frac{İ_{j}^{2}}{I_{j}} .  (75)

Therefore, by combining (73) and (75) the ”intricate” energy term T^{σ}, responsible for the change of shape, is given by

T^{σ} = \frac{1}{2} ∑ I_{i}ω_{i}^{2} + (\frac{1}{8} ∑ \frac{İ_{j}^{2}}{I_{j}} − \frac{1}{2}ρ̇^{2}) − \frac{1}{2} \frac{Ω^{2}}{I} .  (76)

Now, start from the above expression to express T^{σ} purely in terms of the individual moments of inertia I_{j}.

Lemma 7. Let ω_{j} be the (scalar) angular velocity of a_{j}. Then

ω_{1} = \frac{1}{I}(I_{3}α̇_{2} − I_{2}α̇_{3}) + \frac{Ω}{I} etc. (cyclic permutation of indices) (77)

α̇_{1} = \frac{1}{8m_{1}m_{2}m_{3}Δ}(−2m_{1}İ_{1} + C_{3} \frac{İ_{2}}{I_{2}} + C_{2} \frac{İ_{3}}{I_{3}} ) etc.  (78)

Proof. Set θ_{j} to be the angle of a_{j} with respect to a chosen reference direction in the plane. Then ω_{j} = θ̇_{j} and

α_{i} = θ_{i+2} − θ_{i+1}, α̇_{i} = ω_{i+2} − ω_{i+1} i mod 3.

The total (scalar) angular momentum sums up to

Ω = ∑ m_{i}(a_{j}×ȧ_{j}) · k = ∑ Ω_{j} = ∑ I_{j}ω_{j}
= X × (ω × X) · k = Iω.

Consequently,

I_{1}ω_{1} = Ω − I_{2}ω_{2} − I_{3}ω_{3} = Ω − I_{2}α̇_{3} + I_{3}α̇_{2} − (I_{2}ω_{1} + I_{3}ω_{1})

and this proves formula (77).

Next, by differentiating the Ceva-cosine formula (63) for α_{1} with respect to t and use the expression for sin α_{1} from the Ceva-sine formula (62) for α_{1}, we obtain the formula (78).

Substitution of the expressions (78) into (77) also leads to the following formula involving only I_{j}’s, namely

ω_{1} = \frac{1}{8m_{1}m_{2}m_{3}ΔI} [(C_{3}I_{3} − C_{2}I_{2})\frac{İ_{1}}{I_{1}} − (C_{1} + 2m_{2}I_{3})İ_{2}  (79)

+ (C_{1} + 2m_{3}I_{2})İ_{3}] + \frac{Ω}{I} etc.

and this completes the proof of Theorem C1.

Furthermore, using either (77), (78) and the relation α̇_{1} + α̇_{2} + α̇_{3} = 0, or using (79) directly, we calculate

\frac{1}{2} ∑ I_{i}ω_{i}^{2} − \frac{Ω^{2}}{2I} = \frac{1}{2I} (I_{1}I_{2}α̇_{3}^{2} + I_{2}I_{3}α̇_{1}^{2} + I_{3}I_{1}α̇_{2}^{2})  (80)

= \frac{1}{8I} ( ∑_{i mod 3}(\frac{4I_{1}I_{2}I_{3}m_{i}}{Q} + I_{i} − I )\frac{İ_{i}^{2}}{I_{i}} + (2 − \frac{4C_{i}I_{i}}{Q})İ_{i+1}İ_{i+2})

and finally by insertion into (76) we deduce the formula

T^{σ} = \frac{1}{2IQ} ( ∑_{i mod 3} m_{i}I_{i+1}I_{i+2}İ_{i}^{2} − C_{i}I_{i}İ_{i+1}İ_{i+2}) .  (81)

Consequently, the metric (72) on M̄ may be written

ds̄^{2} = dρ^{2} + ρ^{2}dσ^{2},  (82)

where

dσ^{2} = \frac{1}{I^{2}Q} ( ∑_{i mod 3} m_{i}I_{i+1}I_{i+2}dI_{i}^{2} − C_{i}I_{i}dI_{i+1}dI_{i+2})

is the induced metric on the shape space M^{∗} = (I = 1). Indeed, the metric expression dσ^{2} is a tensor on M^{∗} since it is invariant under scaling in M̄.

On the other hand, on M^{∗} the relation I_{1} + I_{2} + I_{3} = 1 implies dI_{1} + dI_{2} + dI_{3} = 0, and therefore the above metric on M^{∗} can be restated as

dσ^{2} = \frac{1}{Q^{∗}} \{ \begin{matrix}[−I_{2}^{2} + (1 − m_{2})I_{2}] dI_{1}^{2} + [−I_{1}^{2} + (1 − m_{1})I_{1}] dI_{2}^{2} \\ − [2I_{1}I_{2} − (1 − m_{2})I_{1} − (1 − m_{1})I_{2} + m_{3}] dI_{1}dI_{2} \end{matrix} \} , (83)

where Q^{∗} denotes the restriction of Q to M^{∗} and we have used the mass normalization m_{1} + m_{2} + m_{3} = 1.

Lemma 8. The area form of (M^{∗}, dσ^{2}) is

dA = \frac{1}{2\sqrt{Q^{∗}}} dI_{1} ∧ dI_{2}.

Proof. As usual, the area form expresses as

dA = \sqrt{D}dI_{1} ∧ dI_{2},

where

D = \frac{1}{Q^{∗2}} \{ \begin{matrix}I_{1}I_{2}(1 − m_{1} − I_{1})(1 − m_{2} − I_{2}) \\ −\frac{1}{4} [2I_{1}I_{2} − (1 − m_{2})I_{1} − (1 − m_{1})I_{2} + m_{3}]^{2} \end{matrix} \} = \frac{1}{4Q^{∗}}

is the determinant of the metric (83).

Finally, we turn to the kinematic 1-forms (45) on M̄, whose definition is suggested by the expressions (79) for the individual angular velocities. Regarded as 1-forms on M^{∗} they are related to the area form by

dΘ_{1} = dI_{3} ∧ dα_{2} − dI_{2} ∧ dα_{3}

= \frac{1}{2\sqrt{Q^{∗}}} \{ \begin{matrix}dI_{3} ∧ (−2m_{2}dI_{2} + \frac{C_{1}}{I_{3}} dI_{3} + \frac{C_{3}}{I_{1}} dI_{1}) \\ −dI_{2} ∧ (−2m_{3}dI_{3} + \frac{C_{2}}{I_{1}} dI_{1} + \frac{C_{1}}{I_{2}} dI_{2}) \end{matrix} \}

= \frac{1}{2\sqrt{Q^{∗}}} (\frac{C_{2} + C_{3} + 2(m_{2} + m_{3})I_{1}}{I_{1}} )dI_{1} ∧ dI_{2}

= \frac{1}{\sqrt{Q^{∗}}} dI_{1} ∧ dI_{2} = 2dA.

This proves formula (46) and, as observed in Section 2.2.3, this also completes the proof of Theorem C2.

3.2.2. A purely kinematic proof of Theorem A. From the metric expression (82) it follows that M̄ is a Riemannian cone over the shape space (M^{∗}, dσ^{2}), expressed in (30) as the union of two isometric disks along their common boundary circle E^{∗}. Both disks are parametrized by the region Q^{∗} ≥ 0 in the (I_{1}, I_{2})-plane, where

Q^{∗}(I_{1}, I_{2}) = Q(I_{1}, I_{2}, 1 − I_{1} − I_{2}), 0 ≤ I_{i} ≤ 1  (84)

is the quadratic form (68) with I_{3} = 1 − I_{1} − I_{2}. In the following we will describe our original calculations in [4] leading to the discovery of the universal sphericality .

At first glance, the mass distribution {m_{i}} is intricately involved in the formula (83) of dσ^{2}, so we will focus attention on the mass dependent quadratic form Q^{∗}. The major step of the proof is, in fact, the algebraic approach of seeking better coordinates by transforming the metric tensor dσ^{2} into a simpler one. The geometric proof using the Hopf bundle (see Section 3.2.3 below) is, in fact, our second proof.

Intuitively, one expects that optimal simplicity and maximal symmetry is achieved by a suitable affine transformation of the (I_{1}, I_{2})-plane which transforms the region Q^{∗} ≥ 0 into the unit disk and makes the metric more ”transparent”. This simple idea was, indeed, the key leading to such a remarkable coordinate transformation.

As indicated in Figure 2, Q^{∗} = 0 defines an ellipse which is tangent to the triple of lines given by I_{1} = 0, I_{2} = 0 and I_{3} = 1 − I_{1} − I_{2} = 0. It is easy to see that its center of symmetry is the point (m_{1}^{∗}, m_{2}^{∗}), so we first set

Ĩ_{1} = I_{1} − m_{1}^{∗}, Ĩ_{2} = I_{2} − m_{2}^{∗}  (85)

and obtain

Q^{∗} = m_{1}m_{2}m_{3} − (1 − m_{2})^{2}Ĩ_{1}^{2} − (1 − m_{1})^{2}Ĩ_{2}^{2} − 2(m_{3} − m_{1}m_{2})Ĩ_{1}Ĩ_{2}.

This suggests a rotation through the angle

ψ_{0} = \frac{1}{2} tan^{−1} \frac{2(m_{1}m_{2} − m_{3})}{(m_{1} − m_{2})(1 + m_{3})}

and new coordinates x̃, ỹ defined by

Ĩ_{1} = x̃ cos ψ_{0} − ỹ sin ψ_{0}, Ĩ_{2} = x̃ sin ψ_{0} + ỹ cos ψ_{0}.

Then

Q^{∗} = m_{1}m_{2}m_{3} − μ_{1}x̃^{2} − μ_{2}ỹ^{2},  (86)

where

μ_{1} = \frac{1}{2}((1 − m_{1})^{2} + (1 − m_{2})^{2}) + \frac{1}{2}(m_{1} − m_{2})(1 + m_{3}) cos 2ψ_{0}
+ (m_{3} − m_{1}m_{2}) sin 2ψ_{0}
μ_{2} = \frac{1}{2}((1 − m_{1})^{2} + (1 − m_{2})^{2}) − \frac{1}{2}(m_{1} − m_{2})(1 + m_{3}) cos 2ψ_{0}
− (m_{3} − m_{1}m_{2}) sin 2ψ_{0}

and we notice the identity

(−(1 − m_{2})^{2} + (1 − m_{1})^{2}) sin 2ψ_{0} + 2(m_{3} − m_{1}m_{2}) cos 2ψ_{0} = 0.

Thus, by setting

x̃ = \sqrt{\frac{m_{1}m_{2}m_{3}}{μ_{1}}} x , ỹ = \sqrt{\frac{m_{1}m_{2}m_{3}}{μ_{2}}} y

the expression (86) transforms to

Q^{∗} = m_{1}m_{2}m_{3}(1 − x^{2} − y^{2}).  (87)

Therefore, the following combined transformation

I_{1} = cos ψ_{0}\sqrt{\frac{m_{1}m_{2}m_{3}}{μ_{1}}} x − sin ψ_{0}\sqrt{\frac{m_{1}m_{2}m_{3}}{μ_{2}}} y + m_{1}^{∗}  (88)
I_{2} = sin ψ_{0}\sqrt{\frac{m_{1}m_{2}m_{3}}{μ_{1}}} x + cos ψ_{0}\sqrt{\frac{m_{1}m_{2}m_{3}}{μ_{2}}} y + m_{2}^{∗}

will transform the formula (83) of dσ^{2} into

dσ^{2} = \frac{1}{4} \frac{(1 − y^{2})dx^{2} + (1 − x^{2})dy^{2} + 2xydxdy}{1 − x^{2} − y^{2}} .  (89)

From here, we simply set

x = sin φ cos θ , y = sin φ sin θ  (90)

which will transform (89) into the metric (107). This proves that

(M^{∗}, dσ^{2}) ≃ S^{2}(1/2).

3.2.3. The Hopf fibration and a geometric proof of Theorem A. The moduli space M̄ is, by definition, an SO(3)-orbit space with the induced differential structure, and according to (33) it is also the orbit space

M̄ = M/SO(3) ≃ ℝ^{4}/SO(2) ≈ ℝ^{3} = C(S^{2})  (91)

of the orthogonal transformation group (SO(2), ℝ^{4}). As a quotient of a Riemannian space by a compact group of isometries M̄ has the induced orbital distance metric which measures the distance between orbits in ℝ^{4} (or M).

Let S^{3} = S^{3}(1) ⊂ ℝ^{4} be the unit sphere and recall the well known classical Hopf fibration, which in the above metric setting reads

SO(2) → S^{3} → S^{3}/SO(2) = ℂP^{1} ≃ S^{2}(1/2),  (92)

where the projection is a Riemannian submersion and the quotient space is the round 2-sphere of radius 1/2. Combined with (91) we have an isometry

M̄ ≃ ℝ^{4}/SO(2) = C(S^{3}/SO(2)) = C(S^{2}(1/2))

of Riemannian cones over the 2-sphere M^{∗} ≃ S^{2}(1/2). The cone M̄ is homeomorphic to ℝ^{3}, but they are only diffeomorphic away from the cone vertex (or base point O) which corresponds to the origin 0 ∈ ℝ^{3}.

Finally, to complete the proof of Theorem A it remains to observe that the above orbital distance metric actually coincides with the kinematically defined one. The two metrics are, for example, determined by the kinetic energy they associate to ”motions” in M̄. These are the image curves of virtual m-triangle motions X(t), which can always be chosen with vanishing angular momentum, namely they are horizontal (cf. Section 2.1.1). These motions are planar, say X(t) is a curve in ℝ^{4} ⊂ M_{0}. Horizontal curves are those perpendicular to the SO(2)-orbits, and at a point X ≠ 0 the horizontal tangent vectors constitute the subspace H(X) ≃ ℝ^{3} consisting of all Y such that X × Y = 0.

Now, the orbital distance metric on M̄ is defined by demanding the projection π : ℝ^{4} → ℝ^{4}/SO(2) = M̄ to be a Riemannian submersion, that is, that the tangent map dπ takes H(X) isometrically to the tangent space of M̄ at π(X). Equivalently, the kinetic energy associated to a moduli curve is the same as the kinetic energy of a horizontal lifting. On the other hand, the kinematic metric (36) also associates to a moduli curve the kinetic energy T̄ of a lifting with vanishing angular momentum. Consequently, the two metrics on M̄ are identical.

3.3. Linear motions of m-triangles. According to Newton’s inertia law, in a center of mass reference frame and in the absence of forces, the trajectory of the three-body system in the Euclidean configuration space M_{0} will be a geodesic, namely a linear motion

δ(t) = (1 − t)δ_{1} + tδ_{2},  (93)

where δ_{1} = (a_{1}, a_{2}, a_{3}), δ_{2} = (b_{1}, b_{2}, b_{3}) are appropriate m-triangles. Such motions are also characterized by having constant velocity (δ_{2}−δ_{1}), and the motion (93) has constant angular momentum

Ω = δ_{1} × δ_{2}.  (94)

Moreover, twice the action integral (7) of the motion from δ_{1} to δ_{2} is the squared distance

2 ∫ T dt = |δ_{1} − δ_{2}|^{2} = ∑ m_{i} |a_{i}−b_{i}|^{2} .  (95)

On the other hand, it is clear that the two m-triangles δ_{i} lie in a common plane if the vector (94) is zero, namely the linear motion (93) has vanishing angular momentum. Then the motions (93) provide a useful tool in analyzing the kinematic geometry of m-triangles since their moduli curves are exactly the geodesics in the moduli space (M̄, ds̄^{2}). Their shape curves will be arcs along great circles (geodesics) on the round sphere M^{∗} = S^{2}.

Let us collect some simple facts, assuming δ_{1} and δ_{2} are m-triangles in the xy-plane (with normal vector k) and Ω = 0.

Example 9. If δ_{1} and δ_{2} have the same orientation, then the distance between their congruence classes δ̄_{i} in M̄ is

dist(δ̄_{1}, δ̄_{2}) = |δ_{1} − δ_{2}| .  (96)

Example 10. Consider two congruence classes δ̄_{1}, δ̄_{2} in M̄ whose shapes δ_{i}^{∗} are different and not antipodal points on S^{2}. It is not difficult to see that representative m-triangles δ_{i} can be chosen in exactly two ways, modulo a rotation of the xy-plane. The two choices are (δ_{1}, δ_{2}) and (δ_{1}, −δ_{2}) for suitable δ_{1} and δ_{2}.

The shape curves of the corresponding linear motions (93), for 0 ≤ t ≤ 1, are the two geodesic arcs Γ_{±}^{∗} between δ_{1}^{∗} and δ_{2}^{∗} whose union is a great circle. Each of the shape curves extends (as t → ±∞) to the whole circle, minus the limit point (δ_{1} ∓ δ_{2})^{∗} as |t| → ∞, which lies on the opposite arc Γ_{∓}^{∗}.

We also remark that the relative position of δ_{1} and δ_{2} can be calculated from the line integrals of the kinematic 1-forms Θ_{i} along the geodesic arc Γ_{±}^{∗}, see Theorem C1 and (45).

For easy reference, the shape of the three types of binary collisions are the following three points on E^{∗}

𝔟_{ij} : I_{i} = \frac{m_{i}m_{k}}{1 − m_{k}} , I_{j} = \frac{m_{j}m_{k}}{1 − m_{k}} , I_{k} = (1 − m_{k}),  (97)

where 𝔟_{ij} = 𝔟_{ji} represents an m-triangle (a_{1}, a_{2}, a_{3}) with a_{i} = a_{j} and I = 1, and {i, j, k} = {1, 2, 3}. There are two more points on the sphere M^{∗} which are of kinematic importance, namely the north pole and south pole {N,S}. Each pole is, of course, the geometric center of the corresponding pole M_{±}^{∗}, and we refer to Corollary 13 for an intrinsic geometric characterization of their shape.

Lemma 11. The north pole (respectively south pole) is the shape of the positively (respectively negatively) oriented m-triangle whose normalized individual moments of inertia equal the dual masses, namely

N (or S) : I_{j} = m_{j}^{∗} = \frac{1}{2}(1 − m_{j}), j = 1, 2, 3.  (98)

Proof. Let δ_{0} = (a_{1}, a_{2}, a_{3}) be an m-triangle with I_{j} as in (98). It suffices to show that the three points (97) on the equator E^{∗} have the same distance in M^{∗} to the point δ_{0}^{∗} ∈ M_{±}^{∗}. Let δ_{1} = (b_{1}, b_{2}, b_{3}) be the (unit size) m-triangle with b_{1} = \sqrt{2}a_{1} and b_{2} = b_{3} = −\frac{m_{1}}{1−m_{1}} b_{1}. It is easily checked that Ω = 0 in (94), that is, the linear motion between δ_{0} and δ_{1} has vanishing angular momentum.

The image of δ_{1} in M̄ is the point δ_{1}^{∗} = 𝔟_{23} on E^{∗}, and according to (96)

|δ_{0} − δ_{1}| = \sqrt{∑ m_{j} |a_{j}−b_{j}|^{2}} = \sqrt{2 − \sqrt{2}} = 2 sin \frac{π}{8}  (99)

equals the distance between δ_{0}^{∗} and δ_{1}^{∗} in M̄. Therefore, their (spherical) distance in M^{∗} is equal to π/4. On the other hand, it is clear from the above calculation (or by symmetry) that similar choices of δ_{1} with δ_{1}^{∗} = 𝔟_{12} or 𝔟_{31} lead to the same distance π/4.

3.4. Eigenvalues and eigenframe of the inertia tensor. The bilinear form B_{X} defined by (24) is identical to the well known inertia tensor in classical mechanics, for the special case of an m-triangle X viewed as a rigid body. This is useful in the kinematic study of non-planary motions of m-triangles. The geometric interpretation of the quadratic form is that it calculates the moment of inertia I_{ω} of the body with respect to the central axis through the vector ω, hence also the rotational kinetic energy due to the angular velocity ω, namely

2T^{ω} = B_{X}(ω, ω) = |ω × X|^{2} = |ω|^{2} I_{ω}.

By an eigenframe of X we mean an orthonormal basis in 3-space consisting of eigenvectors (i.e., along the principal axes) of B_{X}, whose eigenvalues are the associated moments of inertia. We will use the following notation for the eigenvalues and associated eigenframe,

λ_{1} ≤ λ_{2} ≤ λ_{3} ⟷ (u_{1}, u_{2}, n),  (100)

where u_{1} and u_{2} span the plane Π(X) if the triangle is nondegenerate, whereas in the collinear case λ_{1} = 0 and {u_{2}, n} can be any orthonormal basis of the normal plane Π(X)^{⊥}.

Lemma 12. The eigenvalues of B_{X} are related by

λ_{1} + λ_{2} = λ_{3} = I, λ_{1}λ_{2} = 4m_{1}m_{2}m_{3}Δ^{2},

and consequently

λ_{i} = \frac{1}{2}(I ± \sqrt{I^{2} − 16m_{1}m_{2}m_{3}Δ^{2}}), i = 1, 2.  (101)

Proof. We consider the case that X is nondegenerate, and then I is the moment of inertia with respect to the normal direction. Let {u_{1}, u_{2}} be an orthonormal frame of Π(X) consisting of eigenvectors of B_{X} , namely

B_{X}(u_{1}, u_{1}) = λ_{1}, B_{X}(u_{2}, u_{2}) = λ_{2}, B_{X}(u_{1}, u_{2}) = 0.

Then

λ_{1} + λ_{2} = |u_{1}×X|^{2} + |u_{2}×X|^{2} = ∑ m_{j}(2 |a_{j}|^{2} − (u_{1}·a_{j})^{2} − (u_{2}·a_{j})^{2})
= ∑ m_{j} |a_{j}|^{2} = I.

Set

a_{1} = a_{11}u_{1} + a_{12}u_{2}, a_{2} = a_{21}u_{1} + a_{22}u_{2}.  (102)

Then on the one hand

B_{X}(a_{1}, a_{1})B_{X}(a_{2}, a_{2}) − B_{X}(a_{1}, a_{2})^{2} =  (103)
|a_{1} × X|^{2} · |a_{2} × X|^{2} − [(a_{1} × X)·(a_{2} × X)]^{2} = | \begin{matrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix} |^{2} λ_{1}λ_{2},

where by the Ceva-area law (61)

| \begin{matrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix} |^{2} = |a_{1} × a_{2}|^{2} = 4m_{3}^{2}Δ^{2},

and on the other hand,

B_{X}(a_{1}, a_{1}) = m_{2} |a_{1} × a_{2}|^{2} + m_{3} |a_{1} × a_{3}|^{2} = 4m_{2}m_{3}(m_{2} + m_{3})Δ^{2}
B_{X}(a_{2}, a_{2}) = 4m_{1}m_{3}(m_{1} + m_{3})Δ^{2}  (104)
B_{X}(a_{1}, a_{2}) = −4m_{1}m_{2}m_{3}Δ^{2}.

When the expressions (104) are substituted into (103) we obtain

λ_{1}λ_{2} = 4m_{1}m_{2}m_{3}Δ^{2}

and then formula (101) follows.

Corollary 13. The poles (98) are the shapes uniquely characterized by any of the two equivalent conditions:

(i) λ_{1} = λ_{2} (i.e. ”umbilical” shape),

(ii) the m-triangle attains the maximal area

Δ_{max} = \frac{I}{4\sqrt{m_{1}m_{2}m_{3}}}  (105)

among all m-triangles with the same moment of inertia I.

Proof. Clearly, λ_{1} = λ_{2} if and only if the area Δ is given by the formula of (105). On the other hand, let us maximize the area function Q = C_{1}C_{2} + C_{2}C_{3} + C_{3}C_{1} (cf. (65), (68)), using Lagrange’s multiplier method subject to the constraint I = 1. It follows that

C_{i} = −m_{i}m_{i}^{∗} + m_{j}m_{j}^{∗} + m_{k}m_{k}^{∗}

or equivalently I_{j} = m_{j}^{∗}, by (13).

The following result will also be useful. Briefly, it says that a linear motion whose shape curve is a meridian arc from a pole to the equator, has a constant eigenframe.

Lemma 14. Let δ_{0} = (a_{1}, a_{2}, a_{3}) be an m-triangle with the shape of a pole (98), and let {u_{1}, u_{2}} be an orthonormal frame of the plane Π(δ_{0}). Moreover, let δ_{1} = (b_{1}, b_{2}, b_{3}) be a degenerate m-triangle satisfying δ_{0} × δ_{1} = 0 and u_{2} · b_{j} = 0 for all j. Then

B_{t}(u_{1}, u_{2}) = 0

holds along the linear motion δ_{t} = (1 − t)δ_{0} + tδ_{1}, where B_{t} is the inertia tensor of δ_{t}, cf. (24). Hence, {u_{1}, u_{2}} is an eigenframe for δ_{t} for each t.

Proof. From the above Corollary it follows that

B_{0}(u_{1}, u_{2}) = B_{1}(u_{1}, u_{2}) = 0.

Moreover, by the assumptions

(u_{2} × δ_{0}) · (u_{1} × δ_{1}) = ∑_{j} m_{j}(u_{2} × a_{j}) · (u_{1} × b_{j})

= ∑_{j} m_{j} [(u_{1} · u_{2})(a_{j} · b_{j}) − (u_{2} · b_{j})(u_{1} · a_{j})] = 0.

Therefore

B_{t}(u_{1}, u_{2}) = (u_{1} × [(1 − t)δ_{0} + tδ_{1}]) · (u_{2} × [(1 − t)δ_{0} + tδ_{1}])

= (1 − t)^{2}B_{0}(u_{1}, u_{2}) + t^{2}B_{1}(u_{1}, u_{2})

+ (1 − t)t[(u_{1} × δ_{0}) · (u_{2} × δ_{1}) ± (u_{2} × δ_{0}) · (u_{1} × δ_{1})] = (1 − t)t(u_{1} × u_{2}) · (δ_{0} × δ_{1}) = 0.

4. The spherical representation of shape space M^{∗}

By the spherical representation we refer to an identification of M^{∗} with a round 2-sphere, with a distinguished (northern) hemisphere M_{+}^{∗} whose natural orientation induces the positive orientation of the equator E^{∗} and hence the (eastward) direction of increasing longitude. We also assume the (cyclic) ordering 𝔟_{23}, 𝔟_{31}, 𝔟_{12} of the three binary collision points (lying on E^{∗}) is in the positive direction. Finally, the correspondence should represent the kinematic geometry and hence is an isometry

M^{∗} → S^{2}(1/2)  (106)

which identifies each shape δ^{∗} with a specific point on the sphere. We will develop methods enabling us to express the spherical coordinates in terms of intrinsic invariants of δ^{∗}, and conversely.

Let (r, θ) denote polar coordinates on S^{2}(1/2), where r is the polar distance which measures the spherical distance from δ^{∗} to the north pole N ∈ M_{+}^{∗} and θ is the longitude angle. For convenience, we also introduce spherical coordinates (φ, θ), where the angle φ = 2r is the colatitude with φ = 0 at the north pole. In these coordinates the Riemannian metric of the sphere M^{∗} expresses as

dσ^{2} = dr^{2} + \frac{1}{4} sin^{2}(2r)dθ^{2} = \frac{1}{4}(dφ^{2} + sin^{2} φ dθ^{2}),  (107)
0 ≤ r ≤ \frac{π}{2} , 0 ≤ φ ≤ π, 0 ≤ θ ≤ 2π.

Remark 15. The choice of the zero meridian θ = 0 is a matter of convenience, and until further notice our convention is that it passes through 𝔟_{23}. Indeed, only longitude differences (θ − θ′) is an intrinsic property of shapes of m-triangles, see Section 4.2 and 4.3.

4.1. Geometric interpretation of the polar distance r. Let δ be a nonzero positively oriented m-triangle, that is, r ≤ π/4. We will investigate the relationship between the polar distance r and the geometric invariants of δ. Let δ_{1}^{∗} be the intersection point between the equator E^{∗} and the meridian passing through δ^{∗}. Choose unit size representatives

δ_{0} = (a_{1,}a_{2}, a_{3}), δ_{1} = (b_{1,}b_{2}, b_{3}),

of the pole N and δ_{1}^{∗} with δ_{0} × δ_{1} = 0, and observe that the above meridian is the shape curve of the linear motion δ_{t} = (1 − t)δ_{0} + tδ_{1}, 0 ≤ t ≤ 1. Henceforth, let t be the unique value such that δ_{t}^{∗} = δ^{∗}.

Let C be the cone surface (cf. also Definition 42) in M̄ spanned by the rays through points on the above meridian between N and δ_{1}^{∗}. C is isometric to a Euclidean sector of angular width π/4, see Figure 3. The cord distance between N and δ_{1}^{∗} is the number in (99). On the other hand, the cord distance between N and δ̄_{t} can be computed in two different ways, namely

dist(N , δ̄_{t}) = 2t sin \frac{π}{8} = sin \frac{π}{8} − cos \frac{π}{8} tan(\frac{π}{8} − r).

Hence,

t = \frac{1}{2}(1 − cot \frac{π}{8} tan(\frac{π}{8} − r))  (108)

and, moreover,

dist(O, δ̄_{t}) = \frac{cos \frac{π}{8}}{cos(\frac{π}{8} − r)}.  (109)

Next, let us compute the eigenvalues of B_{t} = B_{δ_{t}}, namely

λ′_{1} = B_{t}(u_{1}, u_{1}), λ′_{2} = B_{t}(u_{2}, u_{2}),

where by Lemma 14 we have chosen an orthonormal frame {u_{1}, u_{2}} of the plane Π(δ_{0}) with u_{2} · b_{j} = 0 for all j. It follows that

λ′_{1} = B_{t}(u_{1}, u_{1}) = |u_{1} × δ_{t}|^{2} = |u_{1} × (1 − t)δ_{0}|^{2}

= (1 − t)^{2}B_{0}(u_{1}, u_{1}) = \frac{1}{2}(1 − t)^{2}.

Therefore, since δ^{∗} and δ̄_{t} differ by the scaling factor (109), the eigenvalues of B_{δ}^{∗} are

λ_{1} = \frac{cos \frac{π}{8}}{cos(\frac{π}{8} − r)}λ′_{1} = \frac{1}{2}(1 − t)^{2} \frac{cos \frac{π}{8}}{cos(\frac{π}{8} − r)}  (110)

and λ_{2} = 1 − λ_{1}.

Finally, we can use (108), (110) and the identity

λ_{1}λ_{2} = 4m_{1}m_{2}m_{3}Δ^{2}

to solve for r as a function of the area Δ of δ^{∗}. We state the final result as follows:

Lemma 16. The polar distance r for an arbitrary given positively oriented m-triangle δ is given by the formula

cos(2r) = 4\sqrt{m_{1}m_{2}m_{3}} \frac{Δ}{I},  (111)

where ∑ m_{i} = 1 and Δ (respectively I) is the area (respectively moment of inertia) of δ.

Equivalently, by (101) there is the formula

sin 2r = \frac{1}{I} |λ_{1} − λ_{2}| .  (112)

4.2. Geometric interpretation of the longitude angle θ. Let δ_{0} be an oriented m-triangle whose shape δ_{0}^{∗} is the pole N or S (i.e. r = 0 or π/2). Recall from Lemma 14, it is possible to deform δ_{0}, through a linear motion with zero angular momentum, to the shape of any given degenerate m-triangle. Then the shape curve will be the meridian from the pole to a point δ_{1}^{∗} on the equator circle E^{∗}. Moreover, the line spanned by the final configuration δ_{1} is uniquely determined by δ_{0}, and there is a constant eigenframe throughout the deformation.

Now, let us consider two points δ_{1}^{∗}, δ_{2}^{∗} on E^{∗} and seek an interpretation of their spherical distance in M^{∗} = S^{2}(1/2).

Theorem 17. Let δ_{0} = (a_{1}, a_{2}, a_{3}) be an m-triangle, of maximal area for a fixed moment of inertia, and let δ_{1}, δ_{2} be degenerate (but nonzero) m-triangles satisfying the vanishing angular momentum condition

δ_{0} × δ_{1} = δ_{0} × δ_{2} = 0

for the linear motions from δ_{0} to δ_{i}, i = 1, 2. Then the angle ψ between the lines spanned by δ_{1} and δ_{2} is equal to the distance between the associated points δ_{1}^{∗} and δ_{2}^{∗} in the shape space M^{∗}, namely

ψ = \frac{1}{2} |θ_{2} − θ_{1}| .

Proof. We may assume all m-triangles are confined to the xy-plane, the shape δ_{0}^{∗} is the north pole and δ_{i}^{∗} has longitude angle θ_{i}, i = 1, 2. Consider the piecewise linear motion whose associated shape curve is the spherical triangle D in M^{∗} with vertices δ_{1}^{∗}, δ_{2}^{∗}, δ_{0}^{∗}. Starting from δ_{1}, the motion passes successively through the m-triangles δ_{1,}δ′_{2}, δ_{0}, δ′_{1}. Here δ′_{2} is congruent to δ_{2} and is situated in the same line as δ_{1}, whereas δ′_{1} is congruent to δ_{1} but is actually situated in a line making the angle ψ with the original line.

Next, let us apply the Gauss-Bonnet formula (47) to the region D, thus obtaining an equality between twice the area of D and the angle ψ.

Finally, we simply combine this with the fact that, as a geodesic triangle on the sphere of radius 1/2, the area of D equals one quarter of its angle at δ_{0}^{∗}, namely the angle |θ_{2} − θ_{1}|.

In general, it turns out that the longitude angle θ of an m-triangle is determined by the relative position and size of the eigenframe and normalized area Δ/I respectively. To make this relationship precise, let δ = (a_{1}, a_{2}, a_{3}) be a non-degenerate, positively oriented m-triangle in the xy-plane with normal vector

\frac{a_{1} × a_{2}}{|a_{1} × a_{2}|} = k

and assume the shape δ^{∗} is not the pole N.

Theorem 18. Let {u_{1}, u_{2}} be an eigenframe of δ, where u_{1} is the eigenvector of the inertia tensor B_{δ} associated with the smallest eigenvalue λ_{1}, and let ψ_{i} be the (oriented) angle from a_{1} to u_{i}. Then the following identity

tan \frac{θ}{2} = −\frac{1 + sin φ}{cos φ} tan ψ_{1} = \frac{1 + sin φ}{cos φ} cot ψ_{2}  (113)

relates the angle ψ_{i} to the spherical coordinates (φ, θ) of the shape δ^{∗} on the 2-sphere M^{∗}, where φ is the colatitude and θ is the longitude (eastward, with θ = 0 at 𝔟_{23}).

Remark 19. The above formula holds for any choice of eigenframe since ψ_{1} changes by ±π if u_{1} is replaced by −u_{1}.

Proof. The first step of the proof is to derive a formula which expresses ψ_{1} solely in terms of intrinsic invariants of the m-triangle, together with a simple recipe for calculating this angle. Then, by applying the Gauss-Bonnet formula (cf. Theorem C2) we shall deduce formula (113).

Since ψ_{2} = ψ_{1}±π/2 we need only prove the first identity in (113). First of all, in order to have the angle ψ_{1} uniquely defined we must specify the choice of eigenframe. Namely, let {u_{1}, u_{2}} be a positive frame and hence u_{1}×u_{2} = k, and moreover, we assume u_{1} chosen so that

−\frac{π}{2} ≤ ψ_{1} < \frac{π}{2} .  (114)

Let {e_{1}, e_{2}} be the orthonormal frame derived from {a_{1}, a_{2}} by the Gram-Schmidt algorithm, with e_{1} = a_{1}/ |a_{1}|. Using the expressions (104) it is not difficult to deduce

A = B_{δ}(e_{1}, e_{1}) = 4m_{1}m_{2}m_{3}(m_{2} + m_{3})\frac{Δ^{2}}{I_{1}}
B = B_{δ}(e_{1}, e_{2}) = ((m_{2} + m_{3})C_{3} − 2m_{1}m_{2}I_{1})\frac{Δ}{I_{1}}  (115)
C = B_{δ}(e_{2}, e_{2}) = I − A.

By writing

u_{1} = cos ψ_{1}e_{1} + sin ψ_{1}e_{2}  (116)
u_{2} = − sin ψ_{1}e_{1} + cos ψ_{1}e_{2}

and inserting these expressions into B_{δ}(u_{1}, u_{2}) = 0, we deduce the formula

tan 2ψ_{1} = \frac{2B}{A − C} = \frac{2B}{2A − I}  (117)

and similarly

λ_{1} − λ_{2} = B_{δ}(u_{1}, u_{1}) − B_{δ}(u_{2}, u_{2}) = (A − C) cos 2ψ_{1} + 2B sin 2ψ_{1}. (118)

How is ψ_{1} determined from (117) and (118)? Assume first B = 0. With the normalization ∑ m_{1} = 1, ∑ I_{i} = 1, this happens when

I_{1}(m_{3} − m_{1}m_{2}) + I_{2}(1 − m_{1})^{2} = (1 − m_{1})m_{3}.

For example, with uniform mass distribution this holds for the isosceles triangle with |a_{2}| = |a_{3}|. Note that {e_{1}, e_{2}} is also a positive eigenframe, and the eigenvalues λ_{i} equals A and C. Since λ_{1} < λ_{2}, by assumption, we deduce from (118) that A > C implies ψ_{1} = \frac{π}{2}, and A < C implies ψ_{1} = 0.

Next, assume B ≠ 0. By combining (117) and (118) we eliminate cos 2ψ_{1} and obtain the expression

λ_{1} − λ_{2} = \frac{(A − C)^{2} + 4B^{2}}{2B} sin 2ψ_{1}.  (119)

Consequently, sin 2ψ_{1} has the opposite sign of B, namely

0 < ψ_{1} < \frac{π}{2} if B < 0, − \frac{π}{2} < ψ_{1} < 0 if B > 0.  (120)

Finally, we turn to the proof of formula (113). On the sphere M^{∗}, let δ_{1}^{∗} be the intersection point of E^{∗} and the meridian passing through δ^{∗}, and consider the spherical triangle on M^{∗} with vertices 𝔟_{23}, δ_{1}^{∗}, δ^{∗}, whose area is denoted by Δ̄. The right angle at the vertex δ_{1}^{∗} has adjacent edges of length

s = \frac{|θ|}{2} ≤ \frac{π}{2} and \frac{π}{4} − r,

and by applying the spherical sine law and area formula to the magnified triangle on S^{2}(1) with area Δ̃ = 4Δ̄, we have

tan \frac{Δ̃}{2} = \frac{sin 2s sin(\frac{π}{2} − 2r)}{1 + cos 2s + cos(\frac{π}{2} − 2r) + cos 2s cos(\frac{π}{2} − 2r)}

= \frac{sin 2s}{1 + cos 2s} \frac{sin(\frac{π}{2} − 2r)}{1 + cos(\frac{π}{2} − 2r)} = tan s tan(\frac{π}{4} − r).

On the other hand, applying the Gauss-Bonnet formula (Theorem C2) to the triangle in Figure 4, it is not difficult to see that the total rotation of the vector a_{1} is through the angle ψ_{1}, hence

±ψ_{1} = 2Δ̄ = \frac{Δ̃}{2}

and consequently

tan ψ_{1} = ± tan \frac{θ}{2} tan(\frac{π}{4} − r).  (121)

We claim that the sign to be used in (121) is -1. This can be seen by considering the situation where δ_{1}^{∗} lies between 𝔟_{23} and 𝔟_{31}, by observing that ψ_{1} decreases as θ increases (i.e. when δ_{1}^{∗} approaches 𝔟_{31}). This completes the proof of formula (113).

4.3. Intrinsic form of the spherical representation. We will focus attention on the ”inverse” of the correspondence (106), namely

(φ, θ) → (I_{1}, I_{2}, I_{3}), ∑ I_{j} = 1.

The first step in this direction was, in fact, our kinematic proof of Theorem A (cf. Section 3.2.2). Namely, by substituting (90) into (88) and considering the special case of m_{3} = m_{1}m_{2}, it is easy to verify that the expressions (88) simplify to

I_{1} = m_{1}^{∗}(1 + sin φ cos θ), I_{2} = m_{2}^{∗}(1 + sin φ sin θ).  (122)

These formulas are, indeed, a special case of a general intrinsic description of the spherical representation, purely in terms of geometric concept.

The longitude distance between δ_{1}^{∗} = (φ_{1}, θ_{1}) and δ_{2}^{∗} = (φ_{2}, θ_{2}) is, by definition, the angle |θ_{1} − θ_{2}| mod 2π. Then it is easy to check that (122) can be stated as

I_{i} = m_{i}^{∗}(1 + sin φ cos θ̃_{i}), i = 1, 2, 3,  (123)

where θ̃_{1} (respectively θ̃_{2} or θ̃_{3}) is the longitude distance between δ^{∗} = (φ, θ) and the binary collision point 𝔟_{23} (respectively 𝔟_{31} or 𝔟_{12}).

On the other hand, consider the three distance functions on M^{∗} = S^{2}(1/2)

σ_{i} = σ_{i}(δ^{∗}) = dist(δ^{∗}, 𝔟_{i+1,i+2}) (i mod 3)  (124)

which measure the (spherical) distances to the points 𝔟_{ij}. By the spherical cosine law applied to the triangle with vertices N , 𝔟_{ij}, δ^{∗}, it follows that

cos 2σ_{i} = sin φ cos θ̃_{i}, i = 1, 2, 3

and consequently (123) has the invariant form

I_{i} = m_{i}^{∗}(1 + cos 2σ_{i}), i = 1, 2, 3.  (125)

This may be stated as

cos 2σ_{i} = \frac{I_{i}}{m_{i}^{∗}} − 1 = \frac{1}{m_{i}^{∗}} Ĩ_{i}, cf. (85),  (126)

or equivalently

σ_{i} = arccos \sqrt{\frac{I_{i}}{1 − m_{i}}} (∑ I_{i} = 1).  (127)

Thus, in order to establish (125) or (123) as a general formula it suffices to verify formula (127) in general. Again, the basic idea we use is to construct a suitable linear motion in M_{0}, as we did in Chapter 3, namely we consider the linear motion whose shape curve is the (shortest) geodesic from δ^{∗} to 𝔟_{ij}. We shall calculate the length of this curve in the following way.

Let δ = (a_{1}, a_{2}, a_{3}) be a given m-triangle, normalized with I = 1, and consider the linear motion of vanishing angular momentum

δ(t) = (1 − t)δ + t(\sqrt{\frac{1 − m_{1}}{I_{1}}} a_{1}, −\frac{m_{1}}{\sqrt{(1 − m_{1})I_{1}}} a_{1}, −\frac{m_{1}}{\sqrt{(1 − m_{1})I_{1}}} a_{1})

between δ = δ(0) and the normalized m-triangle δ_{1} = δ(1) with the shape δ_{1}^{∗} = 𝔟_{23}. Its shape curve is the desired geodesic.

The length |δ − δ_{1}| of the segment in M_{0} from δ to δ_{1} is also the length σ̄_{1} of the chord in M̄ between δ^{∗} and δ_{1}^{∗}, see (96) and Figure 4. By applying the Ceva-cosine law (63) to the calculation of inner products a_{i} · a_{j} we arrive at the expression

σ̄_{1}^{2} = |δ − δ_{1}|^{2} = 2(1 − \sqrt{\frac{I_{1}}{1 − m_{1}}} ),

and consequently

σ_{1} = 2 arcsin \frac{σ̄_{1}}{2} = arccos \sqrt{\frac{I_{1}}{1 − m_{1}}} .

4.4. The reduced Newton’s equation in spherical coordinates. Let us utilize the structure of M̄ as a cone over the 2-sphere to express the reduced Newton’s equation of Theorem E1 in terms of the spherical coordinate system (ρ, φ, θ), where ρ = \sqrt{I} measures the distance from the base point O (cone vertex). The relationship between coordinate functions {I_{j}} , {r_{ij}} and {ρ, φ, θ} is expressed by (64) and (123), thus enabling us to transform the equation to a system purely in terms of ρ, φ, θ. This change of variable is, however, rather messy, but an equivalent system can be worked out in several ways. For example, we obtain the following system of three second-order equations

(i) 0 = ρ̈ + \frac{ρ̇^{2}}{ρ} − \frac{1}{ρ}(U + 2h)
(ii) 0 = φ̈ + 2\frac{ρ̇}{ρ}φ̇ − \frac{1}{2} sin(2φ)θ̇^{2} − \frac{4}{ρ^{2}} \frac{∂U}{∂φ}  (128)
(iii) 0 = θ̈ + 2\frac{ρ̇}{ρ}θ̇ + 2 cot(φ)φ̇θ̇ − \frac{4}{ρ^{2}} \frac{1}{sin^{2} φ} \frac{∂U}{∂θ}

valid for planary three-body motions with a fixed energy level h. The angular momentum Ω constant is not implicit in these equations since it is an integration constant defined by the initial value problem. In fact, we have also the equations

T − U = h Ï = 2T + 2h,

namely the energy equation and the Lagrange-Jacobi equation (cf. (217)). The latter is precisely equation (i) in (128), and the energy integral

U + h − \frac{Ω^{2}}{2ρ^{2}} = \frac{1}{2}ρ̇^{2} + \frac{ρ^{2}}{8} (φ̇^{2} + (sin φ)^{2} θ̇^{2})

makes any of the two equations (ii) or (iii) superfluous.

4.5. Ceva-type relations in the spherical representation. The classical Ceva theorem tells us that the lines from the vertices to the center of mass of an m-triangle δ divide the triangle into subtriangles whose areas are in the proportion

Δ_{1} : Δ_{2} : Δ_{3} = m_{1} : m_{2} : m_{3}.

On the other hand, a point δ^{∗} on a hemisphere M_{±}^{∗} divides it into three spherical triangles with areas A_{i}, with the common vertex δ^{∗} and the binary collision points 𝔟_{12}, 𝔟_{23}, 𝔟_{31} as the other vertices, cf. Figure 5. In this way, various (normalized) geometric invariants of δ such as sides, areas, angles (cf. Figure 1) have their spherical ”dual” counterparts, although the dual quantity may be of a different type. There are, for example, the three central angles α_{i} (respectively π_{i}) of δ (respectively at the shape point δ^{∗}) with ∑ α_{i} = ∑ π_{i} = 2π. According to the following lemma, the areas A_{i} are dual to the angles α_{i}, and later we also show the areas Δ_{i} are dual to certain sides of the spherical triangles.

Lemma 20. Let δ = (a_{1}, a_{2}, a_{3}) be an m-triangle with central angle α_{i} opposite to the vector a_{i}. Then the area of the spherical triangle in M^{∗} with vertices 𝔟_{12}, δ^{∗}, 𝔟_{31} equals

A_{1} = \frac{1}{2}(π − α_{1}).

Proof. Let L_{2}, L_{3} be the lines spanned by the vectors a_{2} and a_{3} respectively. There is an obvious piecewise linear motion with Ω = 0 which starts at δ and collapses the triangle to a degenerate configuration of shape 𝔟_{31} along L_{2}, and continues along L_{2} until the shape of 𝔟_{12} is reached. This motion keeps the direction of a_{2} unaltered.

On the other hand, the linear motion with Ω = 0 which collapses δ to a configuration of shape 𝔟_{12} along L_{3}, will rotate a_{2} to a vector along L_{3} which lies opposite to a_{3}. Hence, the total change of position when δ is deformed according to the above piecewise linear motion whose shape curve encloses the spherical triangle, is equal to the angle π− α_{1}. Finally, by the Gauss-Bonnet formula, this is twice the area A_{1} of the triangle.

Next, we turn to the mutual distances, that is, the sides of the mtriangle δ

s_{1} = r_{23} = |a_{2} − a_{3}| etc.

and ask for their spherical counterpart, namely the spherical distances σ_{i} from δ^{∗} to the binary collision points. The quantities s_{i} have, indeed, a nice geometric interpretation in the vector algebra representation described below, see (141). But first, by combining (64) and (126), the identity

s_{i}^{2} = \frac{1 − m_{i} − I_{i}}{m̂_{i}} = \frac{1 − m_{i}}{m̂_{i}} (1 − \frac{I_{i}}{1 − m_{i}}) = \frac{1 − m_{i}}{m̂_{i}} sin^{2} σ_{i}  (129)

holds, where we have assumed normalization I = 1 and (as usual) ∑ m_{i} = 1, cf. also (14) for notation. By summation over i the condition I = 1 reads

1 = ∑ m̂_{i}s_{i}^{2} = ∑(1 − m_{i}) sin^{2} σ_{i} or ∑ m_{i}^{∗} cos 2σ_{i} = 0, (130)

where the first identity is just the normalized version of Lagrange’s formula (65).

Now, let δ_{0} be an m-triangle whose shape is a pole δ_{0}^{∗} = N or S, and let

{α_{1}, α_{2}, α_{3}}, {β_{1}, β_{2}, β_{3}}  (131)

denote the central angles α_{i} of δ_{0} and the central angles β_{i} = π_{i} at δ_{0}^{∗}, respectively, see Figure 5. In particular, β_{1} is the longitude distance between 𝔟_{31} and 𝔟_{12}, or equivalently, β_{1}/2 is their distance in M^{∗}. The relationship between the two triples in (131) follows from the above lemma, namely

β_{i} = 4A_{i} = 2π − 2α_{i}.  (132)

On the other hand, the triple of angles and the (normalized) mass distribution uniquely determine each other. In one direction, we apply the Ceva-cosine law to δ_{0} and obtain

cos α_{1} = \frac{−\sqrt{m_{2}m_{3}}}{\sqrt{1 − m_{2}}\sqrt{1 − m_{3}}} , cos β_{1} = \frac{m_{2}m_{3} − m_{1}}{(1 − m_{2})(1 − m_{3})}, etc. (133)

In particular, the angles are in the range

\frac{π}{2} < α_{i} < π 0 < β_{i} < π.  (134)

We also note that cos β_{i} can be calculated by applying a distance function σ_{j}, j ≠ i, see (124). For example, cos β_{1} = cos(2σ_{3}(𝔟_{31})), and by (126) and the fact that 𝔟_{31} has I_{3} = \frac{m_{2}m_{3}}{1−m_{2}} , we deduce again the above expression for cos β_{1}.

In the other direction, we would like to know which angles in the range (134) are actually realizable for some mass distribution. The condition on the angles is

2 sin β_{i} < ∑ sin β_{j}, for each i,  (135)

and the corresponding (normalized) mass distribution is defined by

m_{1} = 1 − \frac{2 sin β_{1}}{∑ sin β_{j}} = \frac{1 + ∑ cos β_{j}}{−1 + ∑ cos β_{j} − 2 cos β_{1}} , etc.  (136)

We omit the simple proof of this, remarking that the realizability condition (135) also has a nice geometric interpretation. Namely, consider the three binary collision points 𝔟_{12}, 𝔟_{23}, 𝔟_{31} as the vertices of a triangle in the Euclidean disk (of radius 1/2) with the equator circle E^{∗} as boundary. In general, let us call a triangle central if its circumcenter O lies in its interior. Then the realizability condition simply says that the above triangle must be central. For another property of this triangle we also refer to Lemma 21.

4.6. The vector algebra representation of the kinematic geometry. Since the kinematic study of m-triangles and their motions essentially involves spherical geometry, we are naturally led to the vector algebra in the Euclidean 3-space ℝ^{3}, where inner products and determinants are the basic invariants. Therefore, it is sometimes convenient to represent M^{∗} by the sphere S^{2}(1) of unit vectors in ℝ^{3} and hence its cone M̄ becomes the whole Euclidean space 3-space. Thus we introduce the vector algebra representation of the moduli space M̄ by constructing the following transformation between Riemannian cones

M̄ = C(S^{2}(1/2)) →^{Ψ} C(S^{2}(1)) = ℝ^{3},  (137)
Ψ : (ρ, r, θ) → (ρ^{2}, 2r, θ) = (I, φ, θ),

which magnifies M^{∗} to a sphere of radius 1 and squares the distance to the origin. It is a diffeomorphism away from the base point (or origin) O, namely the class of the triple collision ρ = 0.

In (137), (I, φ, θ) are the usual spherical coordinates in 3-space associated with the Euclidean coordinates (x, y, z), that is,

x = I sin φ cos θ, y = I sin φ sin θ, z = I cos φ,

where 0 ≤ φ ≤ π, 0 ≤ θ ≤ 2π. The kinematic metric (38) on M̄, expressed as a metric on ℝ^{3}, now becomes the following conformal modification of the Euclidean metric, namely

ds̄^{2} = dρ^{2} + ρ^{2}dσ^{2} = \frac{dx^{2} + dy^{2} + dz^{2}}{4\sqrt{x^{2} + y^{2} + z^{2}}} .  (138)

A variable point on M^{∗} will be represented by a unit vector

δ^{∗} → p = (x, y, z) ∈ S^{2}(1) ⊂ ℝ^{3}

and we fix the following notation and location of binary collision points (cf. (133) and Figure 6)

𝔟_{23} → b̂_{1} = (1, 0, 0),
𝔟_{31} → b̂_{2} = (cos β_{3}, sin β_{3}, 0),  (139)
𝔟_{12} → b̂_{3} = (cos β_{2}, − sin β_{2}, 0).

Moreover, (126) now reads

p · b̂_{i} = cos 2σ_{i} = \frac{1}{m_{i}^{∗}} Ĩ_{i} = \{ \begin{matrix}x & i = 1 \\ x cos β_{3} + y sin β_{3} & i = 2 \\ x cos β_{2} − y sin β_{2} & i = 3 \end{matrix}  (140)

Recall that 2σ_{i} is the spherical distance between p and b̂_{i}; hence, by (129) there is the simple formula

s_{i} = \frac{1}{2}\sqrt{\frac{1 − m_{i}}{m̂_{i}}} |p − b̂_{i}|  (141)

which expresses the three mutual distances s_{i} for the normalized mtriangle δ as the Euclidean distances (modulo a fixed factor) from p ∈S^{2} to the three fixed points b̂_{i} lying on the circle x^{2} + y^{2} = 1, z = 0.

Lemma 21. There is a unique mass distribution (m′_{1}, m′_{2}, m′_{3}) such that (b̂_{1}, b̂_{2}, b̂_{3}) becomes an m-triangle with center of mass at origin, that is,

m′_{1}b̂_{1} + m′_{2}b̂_{2} + m′_{3}b̂_{3} = 0,

namely the dual masses m′_{i} = m_{i}^{∗}, cf. (14).

Proof. Since the triangle is central the origin lies in its interior, so there are barycentric coordinates q_{i} > 0, unique up to a common multiple, such that ∑ q_{i}b̂_{i} = 0. On the other hand, by combining (130) and (140), ∑ m_{i}^{∗}b̂_{i} · p = 0 holds for all p and consequently ∑ m_{i}^{∗}b̂_{i} = 0.

4.7. An integral formula for the distance function on M^{∗}. The kinematic Riemannian metric dσ^{2}, expressed in terms of coordinates I_{1}, I_{2} as in (83), may be viewed as the infinitesimal version of an integral formula for the distance function σ(p, p′) on each hemisphere M_{±}^{∗} of M^{∗} = S^{2}(1/2). Our calculation of such an integral formula will be based upon a special type of coordinates; namely, we choose the points {𝔟_{23}, 𝔟_{31}} as a bipolar system whose associated distance functions {σ_{1}, σ_{2}} constitute a coordinate system on each hemisphere M_{±}^{∗}. In our final formula, however, we shall express the distance in terms of I_{1}, I_{2}, and more simply in terms of their translates Ĩ_{i} = I_{i} − m_{i}^{∗}.

To this end, it is convenient to apply the above vector algebra representation, where we use the unit sphere S^{2}(1) representation of M^{∗} rather than the sphere S^{2}(1/2), and the collision points 𝔟_{23}, 𝔟_{31} and variable points p, p′ are replaced by the unit vectors b̂_{1}, b̂_{2}, p, p′, respectively. Thus, σ(p, p′) is half of the spherical distance between p and p′ on the unit sphere.

We shall reduce the calculation of the distance function σ to a simple vector algebra involving determinants and Lagrange’s formula:

det(b̂_{1}, b̂_{2}, p) det(b̂_{1}, b̂_{2}, p′) = | \begin{matrix}1 & b̂_{1} · b̂_{2} & b̂_{1} · p′ \\ b̂_{2} · b̂_{1} & 1 & b̂_{2} · p′ \\ p · b̂_{1} & p · b̂_{2} & p · p′ \end{matrix} | . (142)

By writing the left side as

[det(b̂_{1}, b̂_{2}, p)^{2} det(b̂_{1}, b̂_{2}, p′)^{2}]^{1/2}

and applying Lagrange’s formula to each square, the right side of (142) equals the product

| \begin{matrix}1 & b̂_{1} · b̂_{2} & b̂_{1} · p \\ b̂_{2} · b̂_{1} & 1 & b̂_{2} · p \\ p · b̂_{1} & p · b̂_{2} & 1 \end{matrix} |^{1/2} | \begin{matrix}1 & b̂_{1} · b̂_{2} & b̂_{1} · p′ \\ b̂_{2} · b̂_{1} & 1 & b̂_{2} · p′ \\ p′ · b̂_{1} & p′ · b̂_{2} & 1 \end{matrix} |^{1/2} (143)

and this gives us an identity where b̂_{1} · b̂_{2} = cos β_{3} is a constant, the inner product

p · p′ = cos 2σ(p, p′)

appears linearly, and the other non-constant entries b̂_{i} · p = cos 2σ_{i}, b̂_{i} · p′ = cos 2σ′_{i} in the determinants are of the type (126) (or (140)).

Let D, D′ be the determinants in (143). Solving the above determinant identity with respect to p · p′ gives

cos 2σ(p, p′) = \frac{1}{sin^{2} β_{3}} (\sqrt{DD′} + F),  (144)

where F is a bilinear form of both vectors (Ĩ_{1}, Ĩ_{2}) and (Ĩ′_{1}, Ĩ′_{2}), and moreover, F = 0 and D′ = sin^{2} β_{3} when p′ is the pole P of the hemisphere. The latter observation implies

cos 2σ(p,P) = cos 2r = 4Δ\sqrt{m_{1}m_{2}m_{3}} = \begin{matrix}√ & \overline{D} \\ sin & β & 3 \end{matrix} , cf. (111).

Consequently, by the Ceva-Heron formula (69)

D = \frac{sin^{2} β_{3}}{m_{1}m_{2}m_{3}} Q^{∗} = \frac{sin^{2} β_{3}}{m_{1}m_{2}m_{3}} (m_{1}m_{2}m_{3} − Q_{0}^{∗}(Ĩ_{1}, Ĩ_{2})),

where Q^{∗} = Q|_{M^{∗}} is the restriction of the quadratic form (68), and

Q_{0}^{∗} = (1 − m_{2})^{2}Ĩ_{1}^{2} + (1 − m_{1})^{2}Ĩ_{2}^{2} + 2(m_{3} − m_{1}m_{2})Ĩ_{1}Ĩ_{2}.  (145)

On the other hand, taking p = p′ in (144) gives

\frac{F}{sin^{2} β_{3}} = \frac{m_{1}m_{2}m_{3} − Q^{∗}}{m_{1}m_{2}m_{3}} = \frac{Q_{0}^{∗}}{m_{1}m_{2}m_{3}} ,

and therefore, F as a function of (Ĩ_{1}, Ĩ_{2}, Ĩ′_{1}, Ĩ′_{2}), is a constant times the polarization of Q_{0}^{∗} in (145). This establishes the general spherical distance formula

σ(p, p′) = \frac{1}{2} arccos(\frac{\sqrt{Q^{∗}Q^{′∗}} + \frac{1}{2}P ol(Q_{0}^{∗})}{m_{1}m_{2}m_{3}}),  (146)

where

\frac{1}{2}P ol(Q_{0}^{∗}) = (1 − m_{2})^{2}Ĩ_{1}Ĩ′_{1} + (1 − m_{1})^{2}Ĩ_{2}Ĩ′_{2} + (m_{3} − m_{1}m_{2})(Ĩ_{1}Ĩ′_{2} + Ĩ′_{1}Ĩ_{2}).

Remark 22. By spherical trigonometry it is easy to see that the polarization term in (146) can be expressed in polar coordinates as

sin φ sin φ′ cos(θ − θ′).

5. Motions of m-triangles with conserved angular momentum

In the previous chapters we have investigated the kinematic quantities, their general relationships, and the resulting kinematic identities are valid for any motion of m-triangles, with no explicit assumption on the invariance of the angular momentum vector Ω. In dynamics, however, the motion is governed by a potential function and one is primarily interested in the trajectories of the equations of motion, which is a second order ODE. In these cases the invariance of Ω is generally seen as a consequence of (rotational) symmetry properties of the potential function, as in the Newtonian case discussed in the introductory chapter. From this viewpoint, the linear motions (cf. Section 3.3) are the trajectories in the trivial case of a constant potential function, but still we have found them to be useful in our survey of the kinematic geometry of the moduli space M̄.

In this chapter we turn to the study of virtual m-triangle motions, in full generality except with the explicit assumption that the angular momentum is conserved. Our aim is also to complete the proofs of the Main Theorems B, D, E1, E2 and F stated in Section 2.2.

Remark 23. It is a classical result, dating (at least) back to Weierstrass, that a 3-body motion with Ω = 0 must be planar. There is a simple and purely kinematic proof of this fact. Namely, if (X(t), n(t)) is a motion of oriented m-triangles and X = (a_{1}, a_{2}, a_{3}) is nondegenerate, then ṅ = 0 if and only if q_{1} = q_{2} = 0, where q_{i} = n · ȧ_{i}. Therefore, the proof follows from the identity

±(a_{1} × a_{2}) × Ω = q̃_{1}a_{1} + q̃_{2}a_{2},

where for {i, j} = {1, 2}, q̃_{i} is a linear combination of q_{i} and q_{j} with coefficients 2Δm_{i}(1 − m_{j}) and 2Δm_{i}m_{j}, respectively.

5.1. Moving eigenframe and intrinsic decomposition of velocities. Consider an m-triangle motion X(t) = (a_{1}, a_{2}, a_{3}) with a (continuous) eigenframe 𝔉(t) = (u_{1}, u_{2}, n) of its inertia tensor B_{X} (24). For convenience, let us assume X(t) is nondegenerate (say, for some time interval) and hence X spans a plane Π(X) = lin {u_{1}, u_{2}}. Any vector v in 3-space has an orthogonal splitting, v = v^{τ} + v^{η}, where v^{τ} is the tangential component lying in the plane Π(X) and v^{η} is the normal component.

We will combine the splitting (20) of the velocity of X(t) with its tangential and normal decomposition, namely we decompose the individual velocities ȧ_{i} into their tangential and normal parts and write

Ẋ = Ẋ^{τ} + Ẋ^{η} = (ȧ_{1}^{τ}, ȧ_{2}^{τ}, ȧ_{3}^{τ}) + (ȧ_{1}^{η}, ȧ_{2}^{η}, ȧ_{3}^{η}).  (147)

Recall the roles of the (instantaneous) angular velocity vector ω = ω(t) and the angular momentum Ω, which by the inertia operator (23) essentially determine each other. They are responsible for the purely rotational (or rigid) motion of the m-triangle, and in accordance with (147) the rotational velocity has the splitting

Ẋ^{ω} = ω × X = ω^{η}×X + ω^{τ}×X = (Ẋ^{ω})^{τ} + (Ẋ^{ω})^{η}.  (148)

Lemma 24. The normal velocity component of the m-triangle motion is purely rotational, that is,

Ẋ^{η} = (Ẋ^{ω})^{η} = ω^{τ}×X,

where ω^{τ} is the tangential component of ω, and moreover,

ṅ = ω × n = ω^{τ}×n.

Proof. Consider the 2-parameter family of degenerate m-triangles

(λu_{1} + μu_{2}) × X = (c_{1}n, c_{2}n,c_{3}n),

where the constants c_{i} = c_{i}(λ, μ) are linear combinations of λ and μ with the relation ∑ m_{i}c_{i} = 0. The last identity also expresses the range of the linear transformation (λ, μ) → (c_{1}, c_{2}, c_{3}).

On the other hand, for certain constants k_{i}

Ẋ^{η} = (k_{1}n,k_{2}n,k_{3}n), ∑ m_{i}k_{i} = 0,

and consequently the system of equations c_{i} = k_{i}, i = 1, 2, 3, has a unique solution (λ, μ), that is, there is a unique vector ω′ such that Ẋ^{η} = ω′×X. Clearly, ω′ is just the tangential component ω^{τ} of ω. Finally, n is a multiple of a_{1} × a_{2} and ṅ is a tangential vector, so it is a simple vector algebra calculation to verify that ṅ = ω^{τ}×n.

It follows that the tangential velocity version of (20) reads

Ẋ^{τ} = ω^{η}×X + Ẋ^{h}

and, in particular, the horizontal velocity of an m-triangle motion is always tangential, whereas the rotational velocity (148) in general has a tangential and normal component.

Now, we turn to the angular momentum

Ω = X × Ẋ^{ω} = X × (ω × X)

which we assume is a fixed vector along the z-axis, say, and the expansion

Ω = Ωk = Ω^{τ} + Ω^{η} = (g_{1}u_{1} + g_{2}u_{2}) + g_{3}n  (149)

defines its (time dependent) coordinate vector (g_{1}, g_{2}, g_{3}) relative to the moving frame 𝔉(t). The inner product of Ω with a vector v may be written as

Ω · v = (X × ω) × X · v = (X × ω) · (X × v) = B_{X}(ω, v).

Hence, by letting v = v_{i} be any of the vectors from 𝔉,

g_{i} = Ω · v_{i} = B_{X}(ω, v_{i}) = λ_{i}ω · v_{i}, i = 1, 2, 3,

and we obtain the expansion

ω = ω^{τ} + ω^{η} = (\frac{g_{1}}{λ_{1}} u_{1} + \frac{g_{2}}{λ_{2}} u_{2}) + \frac{g_{3}}{I} n,  (150)

In particular, the rotational kinetic energy can be expressed as

T^{ω} = \frac{1}{2} |Ẋ^{ω}|^{2} = \frac{1}{2}B_{X}(ω, ω) = \frac{1}{2}(\frac{g_{1}^{2}}{λ_{1}} + \frac{g_{2}^{2}}{λ_{2}} + \frac{g_{3}^{2}}{I})  (151)

with tangential and normal parts

(T^{ω})^{τ} = \frac{1}{2} \frac{g_{3}^{2}}{I} , (T^{ω})^{η} = \frac{1}{2}(\frac{g_{1}^{2}}{λ_{1}} + \frac{g_{2}^{2}}{λ_{2}} ).

Now, let us also have a closer look at the individual velocities in (147) and their splitting, namely

ȧ_{i} = ȧ_{i}^{τ} + ȧ_{i}^{η} = (ω_{i}(n × a_{i}) + \begin{matrix}ρ̇ & i \\ ρ & i \end{matrix} a_{i}) + (ω^{τ} × a_{i}), cf. (74), (152)

where ω_{i} is the scalar tangential angular velocity of a_{i}. We claim that

ω_{i} = ω_{i}^{0} + \frac{g_{3}}{I} , cf. Remark 1,  (153)

where ω_{i}^{0} is the scalar angular velocity in the case of vanishing angular momentum, and a formula is given in Theorem C1, see (44). The proof of (153) is really the same as in Section 3.2.1, if we only consider velocity components in the plane Π(X) and replace Ω by its normal component Ω^{η}. The identities in Section 3.2.1, in fact, expresses kinematic relationships valid at each moment t, and there is no need to assume Ω is a constant.

Finally, according to (152) the individual kinetic energy terms expresses as

T_{i} = T_{i}^{τ} + T_{i}^{η} = (\frac{1}{2}ω_{i}^{2}I_{i} + \frac{İ_{i}^{2}}{8I_{i}}) + \frac{1}{2}m_{i} |ω^{τ} × a_{i}|^{2}  (154)

and hence they depend, in fact, only on the moduli curve Γ̄(t) of the motion and the moving frame coordinates g_{i} of the angular momentum.

5.2. Final proof of the Main Theorems D, B, E1,E2.

5.2.1. The Euler equations and proof of Theorem D. Consider the horizontal (i.e. with vanishing angular momentum) m-triangle motion X^{h}(t) = (b_{1}, b_{2}, b_{3}), with the same moduli curve Γ̄(t) as X(t) and with moving eigenframe

𝔉^{h}(t) = (u_{1}^{h}(t), u_{2}^{h}(t), n(t_{0})),

subject to the initial conditions

X^{h}(t_{0}) = X(t_{0}), 𝔉^{h}(t_{0}) = 𝔉(t_{0}).  (155)

The existence of this motion is the statement of Theorem B in the simple case of vanishing angular momentum (cf. Section 2.2.2).

Let Γ^{∗}(t) = (φ(t), θ(t)) be the associated shape curve on M^{∗} = S^{2}, expressed in the usual spherical coordinates, and consider the two nearby points

p = (φ(t_{0}), θ(t_{0})), p′ = (φ(t_{0} + Δt), θ(t_{0} + Δt))

with the longitude difference Δθ. The meridians through p and p′ intersect the equator circle E^{∗} in the points e and e′ respectively, and there is the piecewise geodesic closed path

p → e → e′→ p′ → p

enclosing the shaded region as indicated in Figure 7, whose area (on the unit sphere) is

ΔA ≡ cos φ Δθ

modulo higher orders of Δθ.

It follows from Theorem C2 and Theorem 17 applied to the above path, with a piecewise linear motion in the plane perpendicular to n(t_{0}), that

u_{1}^{h}(t_{0} + Δt) ≡ cos(Δψ)u_{1}(t_{0}) + sin(Δψ)u_{2}(t_{0})  (156)
u_{2}^{h}(t_{0} + Δt) ≡ − sin(Δψ)u_{1}(t_{0}) + cos(Δψ)u_{2}(t_{0})

modulo higher orders of Δθ, where

Δψ = \frac{1}{2} cos φ Δθ.  (157)

Consequently, we infer from (156) and (157)

u̇_{1}^{h}(t_{0}) = (\frac{1}{2}θ̇ cos φ)u_{2}|_{t_{0}}, u̇_{2}^{h}(t_{0}) = (−\frac{1}{2}θ̇ cos φ)u_{1}|_{t_{0}}.  (158)

Next, consider the following intrinsic frame version of (20)

u̇_{i} = ω × u_{i} + u̇_{i}^{h}, i = 1, 2; ṅ = ω × n,  (159)

By taking the inner product with Ω on both sides of these identities, using (149) and (150), we perform the following calculations:

Ω · u̇_{1}(t_{0}) = Ω·(ω × u_{1} + u̇_{1}^{h})|_{t_{0}}

= Ω· ((\frac{g_{1}}{λ_{1}} u_{1} + \frac{g_{2}}{λ_{2}} u_{2} + \frac{g_{3}}{I} n) × u_{1} + (\frac{1}{2}θ̇ cos φ)u_{2}) |_{t_{0}}

= Ω· (−\frac{g_{2}}{λ_{2}} n + (\frac{g_{3}}{I} + \frac{1}{2}θ̇ cos φ)u_{2}) |_{t_{0}}

= g_{2} ((\frac{1}{I} − \frac{1}{λ_{2}} )g_{3} + \frac{1}{2} cos φ θ̇) |_{t_{0}}

Ω · u̇_{2}(t_{0}) = Ω·(ω × u_{2} + u̇_{2}^{h})|_{t_{0}}

= Ω· ((\frac{g_{1}}{λ_{1}} u_{1} + \frac{g_{2}}{λ_{2}} u_{2} + \frac{g_{3}}{I} n) × u_{2} − (\frac{1}{2}θ̇ cos φ)u_{1}) |_{t_{0}}

= Ω· (\frac{g_{1}}{λ_{1}} n + (−\frac{g_{3}}{I} − \frac{1}{2}θ̇ cos φ)u_{1}) |_{t_{0}}

= g_{1} ((\frac{1}{λ_{1}} − \frac{1}{I})g_{3} − \frac{1}{2} cos φ θ̇) |_{t_{0}}

Ω · ṅ = Ω · (ω × n) = Ω· ((\frac{g_{1}}{λ_{1}} u_{1} + \frac{g_{2}}{λ_{2}} u_{2} + \frac{g_{3}}{I} n) × n) |_{t_{0}}

= Ω· (−\frac{g_{1}}{λ_{1}} u_{2} + \frac{g_{2}}{λ_{2}} u_{1}) |_{t_{0}} = (\frac{1}{λ_{2}} − \frac{1}{λ_{1}} )g_{1}g_{2}|_{t_{0}}.

Since Ω is a constant vector and time t_{0} is arbitrary, the above three identities amount precisely to the ODE (53), and this completes the proof of Theorem D.

Finally, we turn to the precession angle χ(t) which records the motion of the normal vector n around the z-axis, that is, the fixed Ω-axis. For example, using spherical coordinates (φ̃, θ̃) on the unit sphere in Euclidean 3-space, with φ̃ = 0 at the north pole k and χ(t) = θ̃(t), it is a simple exercise to deduce the first equality in (54), and by substituting ṅ = ω × n the second expression in (54) follows by calculating cross products in the frame 𝔉.

5.2.2. The lifting problem and proof of Theorem B. To complete the proof in the general case Ω ≠ 0, let us also choose a horizontal lifting X^{h}(t) of Γ̄(t). Then it follows from the identity (20) that the lifting Γ(t) is the motion X(t) = (a_{1}(t), a_{2}(t), a_{3}(t)) determined by the following initial value problem

\frac{d}{dt}X = (ω × X)+\frac{d}{dt}X^{h}, X(t_{0}) = Γ(t_{0}).  (160)

Here the vector ω is a function of X and the constant vector Ω, namely for a fixed and nondegenerate X, ω is found by inverting the inertia operator on 3-space:

𝕀_{X} : ω → X × (ω × X) = Ω.

The matrix of this operator is

B = |X|^{2} Id − ADA^{t} ∼ diag(λ_{1}, λ_{2}, λ_{3}),

where Id is the identity, the vectors a_{i} are the columns of A, and

D = diag(m_{1}, m_{2}, m_{3}).

The eigenvalues λ_{i} of B are listed in Section 2.2.4, and one of them vanishes when X ≠ 0 is degenerate. However, in that case it is easy to check that the indeterminacy of ω is a summand along the line Π(X) and consequently the summand ω × X in (160) is still well defined as a function of X and Ω.

For another proof of Theorem B, more directly related to the construction of a position curve γ(t) in SO(3), we use either Theorem C1 or D. The first theorem applies to planary motions and calculates a position curve γ(t) in SO(2), recording the rotation of the vectors a_{i}. In the nonplanary case the position of the m-triangle is represented by the position of its moving eigenframe 𝔉, and the latter is determined by the coordinate vector (g_{1}, g_{2}, g_{3}) of Ω relative to 𝔉 together with the precession angle χ (of the normal vector n) in the ”invariant” plane perpendicular to Ω. The four functions g_{i} and χ are the solution of an initial value problem depending only on the moduli curve Γ̄(t), as explained by Theorem D and formula (54).

5.2.3. Geometric reduction of Newton’s equation and proof of Theorem E1 and E2. To derive the reduced Newton’s equations from the Newton’s equations (1), we differentiate the kinematic quantities I_{i} up to second order, for example, İ_{1} = 2m_{1}a_{1} · ȧ_{1} and

Ï_{1} = 2m_{1} \left|{ \atop ȧ_{1}}\right|^{2} + 2m_{1}a_{1} · ä_{1}
= 4T_{1} + 2m_{1}a_{1} · (\frac{m_{2}}{r_{12}^{3}} (a_{2} − a_{1}) + \frac{m_{3}}{r_{13}^{3}} (a_{3} − a_{1})) .  (161)

Then we use the Ceva-cosine law (63), stated in the form

a_{i} · a_{j} = \frac{−1}{2m_{i}m_{j}} C_{k} = \frac{1}{2m_{i}m_{j}} (m_{k}I_{k} − m_{i}I_{i} − m_{j}I_{j}),

to replace all inner products in (161) by linear combinations of the I′_{i}s. This procedure leads to the differential equations (55).

In the above differential equations the individual kinetic energies T_{i} are crucial terms, and their actual splitting (154) distinguishes the two cases of planary and non-planary 3-body motions Γ(t). Of course, the actual case is also decided by the initial data Γ(t_{0}), Γ̇(t_{0}). However, it is not decided by Γ(t_{0}) and the angular momentum vector, unless Γ(t_{0}) is nondegenerate. Clearly, the statements of Theorem E1 and E2 must be modified if they should also cover the case where the initial configuration Γ(t_{0}) is collinear.

First, observe that a planary three-body motion is characterized by having all normal kinetic energies T_{i}^{η} = 0, and then its moduli curve Γ̄(t) is a solution of the Ω-reduced equations (55) with T_{i} = T_{i}^{τ} given by the first summand in (154).

Conversely, let γ̄(t) be a moduli curve which is a solution of this ODE, for a given value of Ω. By Theorem B there is a unique lifting X(t), namely a virtual motion in the xy-plane, with angular momentum Ωk and specified initial position X(t_{0}). From this knowledge we may calculate the initial velocity

Ẋ(t_{0}) = ω(t_{0}) × X(t_{0}) + Ẋ^{h}(t_{0}), ω(t_{0}) = \frac{Ω}{I(t_{0})}k,

since the horizontal velocity Ẋ^{h}(t_{0}) is determined by \frac{d}{dt} γ̄(t_{0}). On the other hand, Newton’s equations (1) also has a unique solution Γ(t) with the above initial conditions, namely Γ(t_{0}) = X(t_{0}) and Γ̇(t_{0}) = Ẋ(t_{0}), and clearly the moduli curve of Γ(t) is a solution of the Ω-reduced ODE. By uniqueness of the lifting we conclude that Γ(t) = X(t) for all t, and this completes the proof of Theorem E1.

Next, we turn to the general case, described by Theorem E2. The kinetic energies T_{i} in the Ω-reduced ODE have the general form (154), and we \underline{claim} they depend only on the moduli curve and the functions g_{i}. This clearly holds for the tangential summand T_{i}^{τ}, whose expression is even independent of g_{1} and g_{2}. On the other hand, the normal summand is, say, for i = 1:

T_{1}^{η} = \frac{1}{2}m_{1} |ω^{τ} × a_{1}|^{2} = \frac{1}{2}m_{1} |(\frac{g_{1}}{λ_{1}} u_{1} + \frac{g_{2}}{λ_{2}} u_{2}) × (cos ψ_{1}u_{1} + sin ψ_{1}u_{2})|^{2}
= \frac{1}{2}m_{1}(\frac{g_{1}}{λ_{1}} sin ψ_{1} − \frac{g_{2}}{λ_{2}} cos ψ_{1})^{2},  (162)

where ψ_{1} is the angle between u_{1} and a_{1}, satisfying

tan ψ_{1} = \frac{cos φ}{1 + sin φ} tan \frac{θ}{2}, cf. Theorem 18.  (163)

In this formula θ is the longitude angle measured from the binary collision point b̂_{1} and is increasing in the direction towards b̂_{2}. It follows (e.g. by symmetry) that one obtains the corresponding formula for T_{2}^{η} and T_{3}^{η} from (162) when ψ_{1} is replaced by the corresponding angle ψ_{2} and ψ_{3}, determined by the same formula (163) with θ measured from b̂_{2} or b̂_{3}, respectively. This proves the above claim.

Problem 25. It is an interesting task to simplify the expression for the energies T_{i}^{η}, and to express them in terms of coordinates I_{j} as in the case of T_{i}^{τ}. But we leave the topic here.

Thus, in the general case our ”reduced” ODE actually consists of the reduced Newton’s equations (55) together with the Euler equations (53). The initial data will be a given nondegenerate configuration X(t_{0}) and angular momentum vector Ω, and as before, this determines the initial velocity Ẋ(t_{0}) and hence the motion X(t) is unique (by Newton’s equation (1)).

However, to avoid ambiguity in the initial value problem for the ”reduced” ODE we must also specify the initial orientation (i.e. the normal vector n(t_{0})) which decides whether the shape curve starts out on the upper or lower hemisphere of M^{∗}. With this choice the initial eigenframe {u_{1}, u_{2}, n} |_{t=t_{0}}, and hence also the initial values g_{i}(t_{0}), will be unique relative to a fixed convention, say, the angle ψ_{1} between u_{1} and a_{1} is (initially) in the range (114). Now, the remaining part of the proof of Theorem E2 is similar to the proof of Theorem E1.

5.3. Geometric reduction of the least action principles. Recall from Section 1.2 the two classical least action principles with the action integral J_{1} and J_{2}, respectively. The underlying geometric structure naturally associated to the former is Riemannian while that of the latter is symplectic. Thus the two types of least action principles are radically different in their basic geometric setting, although both of them characterize the same motion. Indeed, it is easy to verify that the Euler-Lagrange equations of both variational principles are equivalent to the Newton’s equation of motion (1).

However, the classical approach to the three-body problem is mainly based on J_{2}, namely the least action principle of Hamilton and the Hamilton-Jacobi theory, in the framework of canonical transformations and symplectic geometry. On the other hand, the kinematic geometry of m-triangles is, on the other hand, more naturally associated with the least action principle of Euler-Lagrange-Jacobi and the action integral J_{1}, and it is in this geometric framework that we have established many basic results, such as the universal sphericality, the kinematic Gauss-Bonnet formula (and geometric phase), kinematic moving frames and the generalized Euler equations.

5.3.1. Proof of Theorem F. We consider virtual 3-body motions in the xy-plane, represented by motions δ(t) of oriented m-triangles with k as their common normal vector, and hence a (nondegenerate) m-triangle δ = (a_{1}, a_{2}, a_{3}) is positively oriented if a_{1}×a_{2} points in the direction of k. The corresponding configuration space is ℝ^{4}, see (28), and ℝ^{4}/SO(2) = M̄ is the full moduli space. The SO(2)-orbit δ̄ of an oriented m-triangle δ will be regarded both as a point in M̄ and as a subset (congruence class) of ℝ^{4}. In the sequel, all motions in ℝ^{4} are also assumed to have a constant angular momentum.

The set of differentiable (C^{1}-smooth) curves in M̄ from δ̄ to δ̄_{1} is denoted 𝔓̄_{δ̄,δ̄_{1}}, and similarly 𝔓_{δ,δ̄_{1}}denotes the set of differentiable curves in ℝ^{4} which start at δ and terminate at the orbit δ̄_{1}. For a fixed angular momentum Ω the kinetic energy

T = T_{Ω} = T̄ + T^{ω} = T̄ + \frac{Ω^{2}}{2I} ,

the potential function U, the Lagrange function L_{Ω} = T_{Ω} + U and total energy h = T_{Ω}−U, are functions which are also defined at the level of M̄. Therefore, for fixed value of Ω, h or time interval [0, t_{1}], we may consider corresponding subsets of 𝔓̄_{δ̄,δ̄_{1}}

𝔓̄_{δ̄,δ̄_{1}}(h), 𝔓̄_{δ̄,δ̄_{1}}([t_{0}, t_{1}])  (164)

with the obvious meaning, and similarly subsets of 𝔓_{δ,δ̄_{1}}

𝔓_{δ,δ̄_{1}}(Ω), 𝔓_{δ,δ̄_{1}}(Ω, h), 𝔓_{δ,δ̄_{1}}(Ω, [t_{0}, t_{1}]).  (165)

Clearly, the solution curves of Newton’s equation belong to sets of type (165).

We can also define the reduced action integrals

J̄_{1,Ω} = ∫ T_{Ω}dt , J̄_{2,Ω} = ∫ L_{Ω}dt  (166)

acting on moduli curves, and there is the following commutative diagram

\begin{aligned}𝔓̄_{δ̄,δ̄_{1}} & ⟵^{π_{Ω}} & 𝔓_{δ,δ̄_{1}}(Ω) \\ J̄_{i,Ω} ↘ & & ↙ J_{i,Ω} \\ & ℝ \end{aligned}  (167)

where the map π_{Ω} takes a curve Γ(t) to its moduli curve Γ̄(t) and J_{i,Ω} is the restriction of J_{i} to the subspace of curves with fixed angular momentum Ω. Moreover, for δ a nondegenerate m-triangle the map π_{Ω} is, in fact, a bijection due to the unique lifting property described by Theorem B.

Now, let us turn to the proof of Theorem F, which we restate as follows:

Theorem 26. The solution curves of the planary Ω-reduced Newton’s equation can be characterized as the extremal curves of J̄_{1,Ω} (respectively J̄_{2,Ω}) restricted to the sets

𝔓̄_{δ̄,δ̄_{1}}(h), respectively 𝔓̄_{δ̄,δ̄_{1}}([t_{0}, t_{1}])

of moduli curves, with fixed energy h or time interval [t_{0}, t_{1}], respectively.

Proof. First of all, extremal curves of J̄_{1,Ω} (respectively J̄_{2,Ω}) are solutions of the associated Euler-Lagrange equations, and one checks that these are second order ODE whose solution curves are (as usual) uniquely determined by their initial position and velocity.

On the other hand, the Ω-reduced Newton’s equation is also a second order ODE whose solution curves are uniquely determined by their initial position and velocity. Therefore, to show that the Euler-Lagrange equations and the Ω-reduced Newton’s equation have the same solutions it suffices to verify this locally. More precisely, it suffices to show that any small segment of a solution curve of the Ω-reduced Newton’s equation is also an extremal curve of J̄_{1,Ω} (respectively J̄_{2,Ω}).

Let Γ̄ be a small segment from δ̄ to δ̄_{1} of a solution curve of the Ωreduced Newton’s equation. We may assume the points are sufficiently close to ensure that Γ̄ is the only segment (of a solution) linking them.

Choose δ in the orbit δ̄. Since the actions in (166) are always nonnegative and the orbits δ̄ and δ̄_{1} are sufficiently close, there exists a J_{1}-minimizing (respectively J_{2}-minimizing) curve Γ in ℝ^{4} between δ and the orbit δ̄_{1}, say δ_{1} is its end point. Then Γ is a solution of Newton’s equation and it must, in fact, be the lifting of Γ̄. Moreover, it is the unique curve in 𝔓_{δ,δ̄_{1}}(Ω) with minimal action integral of J_{1} (respectively J_{2}).

Consequently, Γ is, of course, also a small segment of an extremal curve of J_{i,Ω} and therefore by (167) its moduli curve Γ̄ is a small segment of an extremal curve of J̄_{i,Ω}.

Remark 27. Theorem F does not extend to the case of general threebody motions. The reason is that the kinetic energy of a motion δ(t) does not depend only on the moduli curve and the angular momentum vector Ω, but also on the instantaneous configuration δ(t). Hence, one cannot proceed as above, since it is not clear what should be the appropriate action J̄_{i,Ω} at the moduli space level.

6. The Newtonian potential function

In this chapter our primary task is to analyze the Newtonian potential function (2) and its crucial dependence on the mass distribution. The function is naturally defined at moduli space level,

U = ∑_{i=1}^{3} \frac{m̂_{i}}{s_{i}} = ∑_{i=1}^{3} \begin{matrix} & m̂_{i}^{3/2} \\ √ & \overline{(1 − m_{i})I − I_{i}} \end{matrix} ,  (168)

where in each half-space M̄_{±} = ℝ_{±}^{3} the two triples (I_{1}, I_{2}, I_{3}) and (s_{1}, s_{2}, s_{3}), related by (64), are natural coordinate systems. Certainly, U has the simplest possible form when expressed by the mutual distances s_{i}. Even so, sometimes it is also convenient to use Euclidean coordinates or their associated spherical coordinates (I, φ, θ), where I = ρ^{2} = \sqrt{x^{2} + y^{2} + z^{2},} as explained in Section 4.5.

Let U^{∗} be the restriction of U to the ”unit” sphere M^{∗} = S^{2} = (I = 1). As a function of the coordinates I_{j} (respectively s_{j}) U is homogeneous of degree −\frac{1}{2} (respectively −1), namely

U(I_{1}, I_{2}, I_{3}) = \frac{1}{ρ}U^{∗}(\frac{I_{1}}{I} , \frac{I_{2}}{I} , \frac{I_{3}}{I}) = \frac{1}{ρ}U^{∗}(δ^{∗}) = \frac{1}{ρ}U^{∗}(φ, θ),

where δ^{∗} ⟷ (φ, θ) represents the shape of an m-triangle. For the sake of convenience, the formula for U^{∗} in terms of spherical coordinates is

U^{∗} = ∑_{i=1}^{3} \frac{m̂_{i}^{3/2}(m_{i}^{∗})^{−1/2}}{\sqrt{1 − sin φ cos(θ − θ_{i})}},  (169)

where θ_{1}, θ_{2}, θ_{3} are the longitude angles of the binary collision points 𝔟_{23}, 𝔟_{31}, 𝔟_{12}, respectively. This follows by substituting the expression (123) with θ̃_{i} = θ−θ_{i} into (168). In particular, with the convention that θ_{1} = 0 made in Remark 15, we must use

(θ_{1}, θ_{2}, θ_{3}) = (0, β_{3}, −β_{2}),

where the angles β_{i}, described in (131) - (133), measure the longitude differences between the points 𝔟_{kl}.

The analysis of U trivially reduces to that of the restriction U^{∗} = U |_{M^{∗}}. In fact, by symmetry it suffices to investigate U^{∗} on the closed upper hemisphere, 0 ≤ φ ≤ π/2. U^{∗} has no maximum points since U^{∗} tends to ∞ at the singular points 𝔟_{kl} (which are poles of U^{∗}). On the other hand, it is a classical result, dating (at least) back to Lagrange, that U^{∗} has a unique minimum value at the shape of a regular triangle, namely (for ρ = 1)

s_{1} = s_{2} = s_{3} = \frac{1}{\sqrt{m̂}}, or I_{i} = 1 − m_{i} − \frac{m̂_{i}}{m̂} .  (170)

This defines a unique point 𝔭_{0}^{±} on each hemisphere M_{±}^{∗} which we also refer to as the physical center (as opposed to the poles which are the geometric center). It is easy to prove the above statement using the coordinates s_{i} and Lagrange’s multiplier method subject to the constraint 1 = I = ∑ m_{i}^{∗}s_{i}^{2}, cf. (130). Another proof follows from Lemma 30 below. The spherical coordinates of the shape (170) is worked out in Section 8.8, cf. (322), (323).

6.1. Vector algebra analysis of the Newtonian function. The vector algebra representation of M̄ in Section 4.5 is also a convenient setting for the local analysis of U^{∗}, namely the function on the unit sphere of Euclidean 3-space defined by

U^{∗}(p) = ∑_{i=1}^{3} U_{i}^{∗}(p) = ∑_{i=1}^{3} \frac{k_{i}}{|p − b̂_{i}|}, k_{i} = \frac{2m̂_{i}^{3/2}}{\sqrt{1 − m_{i}}} ,  (171)

where

|p − b̂_{i}| = \frac{k_{i}}{m̂_{i}} s_{i} (cf. (141))

is the Euclidean distance from p to the binary collision point b̂_{i}.

Fix a point p on the sphere with z > 0, and consider nearby points p′ = p + x on the sphere, that is, x is subject to the constraint

2p · x + x · x = 0,  (172)

which in turn implies

|p′ − b̂_{i}|^{2} = |p − b̂_{i}|^{2} − 2b̂_{i} · x.

Hence, there is the following expansion at p

U^{∗}(p + x) = ∑_{i=1}^{3} \frac{U_{i}^{∗}(p)}{(1 − z_{i})^{1/2}} = ∑_{i=1}^{3} U_{i}^{∗}(p) ∑_{n=0}^{∞} c_{n}z_{i}^{n} = ∑_{n=0}^{∞} F_{n}(p; x), (173)

where we use the notation

z_{i} = \frac{2b̂_{i} · x}{|p − b̂_{i}|^{2}} = (\frac{m_{i}^{∗}m_{i}}{m̄})\frac{b̂_{i} · x}{s_{i}^{2}}

c_{n} = \frac{1 · 3 · 5 ⋯ (2n − 1)}{2^{n}n!}, c_{0} = 1.

Remark 28. It is easy to see that the convergence condition |z_{i}| < 1 for the series of U_{i}^{∗}(p + x) in (173) is equivalent to the condition

(p + x) · b̂_{i} > 2p · b̂_{i} − 1.

Geometrically, this means that p + x belongs to the hemispherical cap D_{i} centered at b̂_{i} which is cut out by the plane parallel to the tangent plane at b̂_{i} and separated by the distance R_{i} = 2(1−p·b̂_{i}). In particular, D_{i} is the whole hemisphere if p · b̂_{i} ≤ 0, and by Lemma 21, we know p · b̂_{i} ≤ 0 holds for at least one i. The domain of convergence for the series of U^{∗} is the ”polygonal” region ∩D_{i}.

The zero order term of the expansion (173) is, of course, F_{0} = U^{∗}(p), and the first order term is

F_{1}(p; x) = (\frac{k_{1}b̂_{1}}{|p − b̂_{1}|^{3}} + \frac{k_{2}b̂_{2}}{|p − b̂_{2}|^{3}} + \frac{k_{3}b̂_{3}}{|p − b̂_{3}|^{3}} ) · x  (174)

which is essentially the gradient of U^{∗} at p. To make this precise, consider the following function from S^{2}(1) to the xy-plane

p → B(p) = ∑_{i=1}^{3} \frac{k_{i}b̂_{i}}{|p − b̂_{i}|^{3}} ∈ ℝ^{2}.  (175)

Lemma 29. The gradient vector of U^{∗} at p is given by

∇U^{∗}(p) = B(p) − (B(p) · p)p.  (176)

Proof. Since by (174)

∇U^{∗}(p) · x = F_{1}(p; x) = B(p) · x

and x is tangential to p in the limit as x → 0, it follows that

∇U^{∗}(p) · t = B(p) · t

holds for all tangent vectors t. Hence, the tangent vector ∇U^{∗}(p) is the orthogonal projection of B(p) in the direction of p.

Lemma 30. The zero points of B are the two critical points of U^{∗} outside the equator circle z = 0, namely the physical center p̂_{0}^{±} (cf. (170)) on each hemisphere z > 0 or z < 0.

Proof. By Lemma 21, B(p) = 0 if and only if for some constant λ > 0,

m_{1}^{∗} \frac{|p − b̂_{1}|^{3}}{k_{1}} = m_{2}^{∗} \frac{|p − b̂_{2}|^{3}}{k_{2}} = m_{3}^{∗} \frac{|p − b̂_{3}|^{3}}{k_{3}} = λ,

and by (141), this is equivalent to s_{1}^{3} = s_{2}^{3} = s_{3}^{3} = \frac{λ}{2}, namely s_{1} = s_{2} = s_{3}. In particular, p is a point with z ≠ 0, cf. (177c). On the other hand, for z ≠ 0 it is easy to see that B(p) = 0 if and only if ∇U^{∗}(p) = 0.

The identity (176) also implies that the critical points of U^{∗} on the unit circle z = 0 are the ”eigenvectors” of B, in the sense that B(p) = λp for some λ, necessarily equal to B(p) · p ≠ 0. Clearly, B has a pole at b̂_{i} and B(p)/ |B(p)| tends to b̂_{i} as p tends to b̂_{i}. A simple analysis of B will show there are exactly one solution p between each pair b̂_{j}, b̂_{k} of poles. These are the so-called Euler points ê_{i}, i = 1, 2, 3, and they are the saddle points of U^{∗}on the 2-sphere. We omit the proof of this well known fact.

6.1.1. Series expansion of U^{∗} at its minimum point. Henceforth, we shall focus attention on the expansion (173) at the physical center p̂_{0} = (x̂_{,}ŷ, ẑ), ẑ > 0, that is, p̂_{0} is the minimum point of U^{∗} on the hemisphere z > 0. The coordinates of p̂_{0} are the following mass dependent constants

x̂ = p̂_{0} · b̂_{1} = 1 − \frac{m̄}{m̂m_{1}^{∗}m_{1}} ,  (177a)
ŷ = \frac{\sqrt{m̄}}{m̂}(\frac{m_{2} − m_{3}}{m_{2} + m_{3}} ),  (177b)
ẑ = cos 2r_{0} = 4Δ_{0}\sqrt{m_{1}m_{2}m_{3}} = \sqrt{3}\frac{\sqrt{m̄}}{m̂},  (177c)

where Δ_{0} is the area of the regular triangle with I = 1. These expressions follow from (111), (140) and (170). Note, for example, that ẑ becomes arbitrarily small when some mass m_{i} tends to zero, and ẑ = 1, that is, p̂_{0} is the north pole N, precisely when the masses are equal. Concerning the sign of ŷ = ±\sqrt{1 − x̂^{2} − ẑ^{2}}, we used (140) to check, for example, that ŷ > 0 if m_{2} > m_{3}.

The constant term of the series (173) is the minimum value

F_{0} = U^{∗}(p̂_{0}) = ∑ U_{i}^{∗}(p̂_{0}) = ∑ \sqrt{m̂}m̂_{i} = m̂^{3/2},  (178)

and F_{1} = 0, of course. Moreover, by (170), m̂s_{i}^{2} = 1 holds for all i, and therefore the n-th order term can be written as

F_{n}(p̂_{0}; x) = (\frac{c_{n}m̂^{n+ \frac{1}{2}}}{m̄^{n−1}} ) ∑_{i=1}^{3} \frac{1}{m_{i}} (m_{i}m_{i}^{∗})^{n}(b̂_{i} · x)^{n}.  (179)

Now, let us turn to the local analysis of the series, namely U^{∗} developed as a power series in suitable coordinates around p̂_{0}. To this end, consider a positively oriented orthonormal frame of the Euclidean 3-space of type

{t_{1}, t_{2}, p̂_{0}}, t_{i} ∈ Π for i = 1, 2,  (180)

where Π is the tangent space of the sphere at p̂_{0}. By condition (172), the components of x = p − p̂_{0} must satisfy

x = ξt_{1} + ηt_{2} + ζp̂_{0}, 2ζ + ξ^{2} + η^{2} + ζ^{2} = 0,  (181)

and consequently the map

(ξ, η) → (ξ, η, ζ),

where

ζ = −1 + \sqrt{1 − (ξ^{2} + η^{2})} = −\frac{1}{2}(ξ^{2} + η^{2}) − \frac{1}{8}(ξ^{2} + η^{2})^{2} − …, (182)

is a parametrization of the region of the sphere lying above the plane through the origin and parallel to Π. Geometrically, the unit disk of Π is projected down to the sphere in the direction of the normal vector p̂_{0}.

It is natural to choose t_{1} tangent to the meridian through p̂_{0}. Indeed, the intrinsic nature of this condition will lead to a series expansion whose coefficients are essentially symmetric functions of the masses m_{i}. The projection in the xy-plane of such an orthonormal basis {t_{1}, t_{2}} of Π is (up to sign) given by

t′_{1} = \frac{ẑ}{\sqrt{1 − ẑ^{2}}} (x̂, ŷ), t′_{2} = \frac{1}{\sqrt{1 − ẑ^{2}}} (−ŷ, x̂) = t_{2}.  (183)

In order to express the n-th term (179) of the U^{∗}-series in terms of ξ, η, we proceed as follows. Expand the ”variables” in (179)

b̂_{i} · x = a′_{i}ξ + b′_{i}η + c′_{i}ζ, i = 1, 2, 3  (184)

where the three triples (a′_{i}, b′_{i}, c′_{i}) of coefficients are specific functions of the parameters m_{i} determined by

a′_{i} = b̂_{i} · t_{1} = b̂_{i} · t′_{1}, b′_{i} = b̂_{i} · t_{2}, c′_{i} = b̂_{i} · p̂_{0}.  (185)

For example, by (139) and (183),

a′_{1} = \begin{matrix}x̂ & ẑ \\ √ & \overline{1 − ẑ^{2}} \end{matrix} , b′_{1} = \frac{−ŷ}{\sqrt{1 − ẑ^{2}}} , c′_{1} = x̂,

where x̂, ŷ, ẑ are the known functions in (177a) - (177c), and the three triples in (185) permute covariantly with the parameters m_{i}.

In view of (179), it is slightly more convenient to replace (184) by

(m_{i}m_{i}^{∗})b̂_{i} · x = a_{i}ξ + b_{i}η + c_{i}ζ, i = 1, 2, 3

and hence we calculate the modified coefficients, namely

a_{i} = \begin{matrix}√ & \overline{3m̄} \\ √ & \overline{m̂^{2} − 3m̄} \end{matrix}(m_{i}m_{i}^{∗} − \frac{m̄}{m̂}),
b_{i} = \frac{−\sqrt{m̄}m_{i}}{2\sqrt{m̂^{2} − 3m̄}}(m_{i+1} − m_{i+2}) (i mod 3)  (186)
c_{i} = m_{i}m_{i}^{∗} − \frac{m̄}{m̂}.

Thus, with the coordinate system (ξ, η), we finally arrive at the following symmetrization of the power series in (173):

U^{∗}(ξ, η) = ∑_{n=0}^{∞} K_{n} ∑_{i=1}^{3} \frac{1}{m_{i}} (a_{i}ξ + b_{i}η + c_{i}ζ)^{n}

= ∑_{n=0}^{∞} K_{n} ∑_{j+k=n} ( \begin{matrix}n \\ j \end{matrix} ) A_{j,k}ξ^{j}η^{k}, (187)

where

K_{n} = \frac{1 · 3 · 5 ⋯ (2n − 1)}{2^{n}n!} \begin{matrix} & m̂^{n+ \frac{1}{2}} \\ & m̄^{n−1} \end{matrix} ,

and A_{j,k}, k even, are symmetric functions of the masses m_{i}, whereas A_{j,k} is alternating symmetric when k is odd.

Remark 31. The Newton sums S_{k} = ∑ m_{i}^{k} are, of course, polynomials of m̂ and m̄, cf. (15). For example,

S_{4} = 1 − 4m̂ + 4m̄ + 2m̂^{2}.

In terms of the S_{k} it is rather straightforward to obtain explicit expressions for the above symmetric functions A_{j,k}. For k odd, the alternating function A_{j,k} is a product of a symmetric function and the basic alternating function

𝔄 = (m_{1} − m_{2})(m_{2} − m_{3})(m_{3} − m_{1})  (188)

= ∑_{i mod 3} m_{i}(1 − m_{i})(m_{i+1} − m_{i+2}).

6.1.2. The quadratic term of U^{∗}. We shall work out explicitly the quadratic term of the function U^{∗} expanded at the physical center p̂_{0}, namely the n = 2 term of (173) or (187)

F_{2} = \frac{3m̂^{5/2}}{8m̄} (Aξ^{2} + 2Bξη + Cη^{2}) = κ(λ̃_{1}ξ̃^{2} + λ̃_{2}η̃^{2}),  (189)

where (ξ̃, η̃) is the coordinate system of a diagonalizing frame {t̃_{1}, t̃_{2}} of the tangent plane Π at p̂_{0}.

First, let us determine the coefficients A = A_{2,0}, B = A_{1,1}, C = A_{0,2} in (187) as symmetric or alternating functions of the symbols m_{i}. By using (188), the identities

∑ m_{i}(1 − m_{i})^{2} = m̂ + 3m̄, ∑ m_{i}(m_{i+1} − m_{i+2})^{2} = m̂ − 9m̄

and the expressions (186), we find

A = ∑ \frac{1}{m_{i}} a_{i}^{2} = \frac{1}{4} \frac{3m̄}{m̂}(\frac{m̂^{2} + 3m̂m̄ − 4m̄}{m̂^{2} − 3m̄})
C = ∑ \frac{1}{m_{i}} b_{i}^{2} = \frac{1}{4} \frac{m̄(m̂ − 9m̄)}{m̂^{2} − 3m̄}
B = ∑ \frac{1}{m_{i}} a_{i}b_{i} = \frac{1}{4}(\frac{−\sqrt{3}m̄}{m̂^{2} − 3m̄})𝔄.

Hence, the eigenvalues of the quadratic in (189) are determined from the equations

λ̃_{1} + λ̃_{2} = A + C = \frac{m̄}{m̂}, λ̃_{1}λ̃_{2} = AC − B^{2} = \frac{3}{4} \frac{m̄^{2}}{m̂} ,

which yield

λ̃_{i} = \frac{m̄}{2m̂}(1 ± \sqrt{1 − 3m̂}).  (190)

Consequently,

F_{2} = \frac{3}{16}m̂^{3/2} ((1 ± μ) ξ̃^{2} + (1 ∓ μ)η̃^{2}) , μ = \sqrt{1 − 3m̂},  (191)

where (ξ̃, η̃) are coordinates with respect to the eigenvectors

t̃_{1} = cos α̃ t_{1} + sin α̃ t_{2}, t̃_{2} = − sin α̃ t_{1} + cos α̃ t_{2}

obtained by rotating the intrinsic frame {t_{1}, t_{2}} of the plane Π_{0}. Similar to (117), the angle α̃ is given by

tan 2α̃ = \frac{2B}{A − C} = \frac{−\sqrt{3}m̂(m_{1} − m_{2})(m_{2} − m_{3})(m_{3} − m_{1})}{m̂^{2} + 9m̄m̂ − 6m̄}. (192)

Then it also follows from (119) that the largest eigenvalue λ̃_{1} ∼ 1 + μ corresponds to the vector t̃_{1} if and only if sin 2α̃ and B have the same sign.

Remark 32. In the simplest case of uniform mass distribution, m_{i} = 1/3, the analysis of U^{∗} is much simpler than in the general case. In this case, where the geometrical and physical center coincide, some of the above expressions such as (192), are of indeterminate type. However, if m_{i} = m_{j} ≠ m_{k} then tan 2α̃ = 0, and hence the frame {t_{1}, t_{2}} is already diagonalizing.

7. A geometric setting for the study of triple collisions

Recall the well known fact, stated by Weierstrass and proved by Sundman (cf. [14], [15]), that three-body motions leading to triple collision must have vanishing angular momentum, Ω = 0, and consequently they are also planary. Thus we shall focus attention on planary virtual motions X(t), namely curves in the configuration space M_{0} ≃ ℝ^{4}, with zero angular momentum, and we continue to use the vector algebra representation (cf. Section 4.5) of the moduli space M̄ = M_{0}/SO(2) = ℝ^{3}, where the (equator) xy-plane Ē = ℝ^{2} represents congruence classes of eclipse (i.e. collinear) configurations.

The total kinetic energy T = T̄ can be expressed as a positive definite quadratic differential form on M̄ ∖ {O}, namely the kinematic Riemannian metric

ds̄^{2} = 2T dt^{2} = dρ^{2} + ρ^{2}dσ^{2} = dρ^{2} + \frac{ρ^{2}}{4} ds^{2} , cf. (138),  (193)

which describes M̄ as the Riemannian cone over the shape space M^{∗} = S^{2}(1/2), and

ds^{2} = 4dσ^{2} = dφ^{2} + (sin^{2} φ)dθ^{2}  (194)

is the metric of the magnified sphere S^{2}(1).

Following Jacobi, we introduce the following conformal modification of the metric (193) for each energy level h, namely

ds̄_{h}^{2} = (U + h)ds̄^{2},  (195)

which we refer to as the physical metric, and transform the action integral J̄_{1,0} of (166) into the arc-length integral in the Riemannian space

(M̄_{h}, ds̄_{h}^{2}) : M̄_{h} = {p ∈ M̄; U(p) + h ≥ 0} .  (196)

Consequently, the trajectories of three-body motions with total energy h are mapped to curves in M̄ which are geodesics in the space (M̄_{h}, ds̄_{h}^{2}). Notice that reflection in the equator plane Ē ⊂ M̄, that is, the transformation φ → π − φ, restricts to an involutive isometry of the Riemannian space (196) with the eclipse subspace Ē_{h} = Ē ∩M̄_{h} as fixed point set, and hence this is a totally geodesic submanifold of M̄_{h}. In particular, a geodesic curve in (M̄_{h}, ds̄_{h}^{2}) is transversal to Ē_{h}, unless it lies entirely in Ē_{h}. Moreover, for a simple geometrical reason, a shortest geodesic in (M̄_{h}, ds̄_{h}^{2}) between a point outside Ē_{h} and the origin O cannot have any intermediate intersection with Ē_{h}.

Finally, we note that Newton’s equation (1) has a 1-parameter group of space-time scaling symmetries {g_{s}}, where g_{s} sends a solution X(t) to a solution

Y(t) = e^{2s/3}X(e^{−s}t)  (197)

and changes the energy level from h to e^{−2s/3}h. Hence, all the Riemannian structures in (196) with energy h of the same sign are mutually homothetic, and consequently there are essentially only three distinct cases, namely when the total energy h is negative, zero or positive.

7.1. Geodesic rays and distance estimates. Clearly, for h ≥ 0 the variety M̄_{h} is the whole moduli space M̄, whereas for h < 0 it is the star-shaped union of all ray segments

[O, \frac{U^{∗}(p)}{|h|}p], p ∈ M^{∗}  (198)

from the origin O to the point where the ray through p intersects the boundary ∂M̄_{h}, that is, the level surface U = −h. By definition, the physical metric (195) vanishes on the boundary, meaning that the distance between any two boundary points is zero.

For h < 0 the length of any ray segment (198) is

L_{h}(p) = ∫ ds̄_{h} = ∫_{0}^{\frac{U^{∗}(p)}{|h|}}\sqrt{\frac{U^{∗}(p)}{ρ} + h} dρ,  (199)

and therefore there is a unique pair of shortest length

L_{h} = L_{h}(p̂_{0}^{±}) = ∫_{0}^{\frac{μ_{0}}{|h|}} \sqrt{\frac{μ_{0}}{ρ} + h} dρ,  (200)

where the points p̂_{0}^{±} on the hemispheres z > 0 and z < 0 represent the shape of a regular triangle and hence U^{∗} has the minimal value

μ_{0} = U^{∗}(p̂_{0}^{±}) = m̂^{3/2}, cf. (178).

The following is a useful fact in Riemannian geometry which follows from general analysis of the first variation of arc-length.

Lemma 33. Let ds^{2} and ds̃^{2} be two Riemannian metrics on a given manifold such that

ds̃^{2} = f^{2}ds^{2},

where f is a smooth and positive function, that is, ds̃^{2} is a conformal modification of ds^{2}. Let Γ be a C^{2}-smooth curve and let n denote a normal vector at a given point on Γ. Then the geodesic curvatures of Γ in (the direction of n) with respect to the two metrics are related by

K̃(n) = K(n)−\frac{d}{dn} ln f.

We shall apply the lemma to the kinematic and physical metric, namely the metrics ds̄^{2} and ds̄_{h}^{2} on M̄, cf. (195). Thus, a moduli curve Γ̄ is a geodesic with respect to ds̄_{h}^{2} if and only if its geodesic curvature K̃(n) with respect to ds̄_{h}^{2} vanishes for all n, or equivalently

K(n) = \frac{1}{2} \frac{d}{dn} ln(U + h),  (201)

where K(n) is the geodesic curvature in the normal direction n, with respect to ds̄^{2}.

The simplest type of 3-body motions are the shape invariant ones, that is, the shape curve is a single point on M^{∗} = S^{2} and hence the moduli curve is confined to a ray emanating from O in the cone M̄. Rays are, of course, geodesics with respect to the kinematic metric ds̄^{2}, consequently a ray through p ∈ S^{2} (or a ray segment (198) if h < 0) is also a geodesic of the metric ds̄_{h}^{2} if and only if the normal derivative vanishes in all directions n normal to the ray, that is,

\frac{d}{dn} ln(\frac{1}{ρ}U^{∗} + h) = 0.

This condition is independent of the radial coordinate ρ, and for ρ = 1 the vectors n span the tangent plane of S^{2} at the point p. Consequently, the solutions are the five critical points p of U^{∗}, namely the three saddle points (called Euler points) ê_{i} on the equator circle ρ = 1 in the xyplane, and the pair p̂_{0}^{±} of minimumspoints (also called Lagrange points). Thus, there are altogether exactly five geodesic rays (or ray segments) in (M̄_{h}, ds̄_{h}^{2}).

Lemma 34. For h < 0, the two ray segments [O, \frac{μ_{0}}{|h|}p̂_{0}^{±}] are the unique shortest geodesic curves in (M̄_{h}, ds̄_{h}^{2}) linking a boundary point and the base point O (ignoring curve pieces of zero length along ∂M̄_{h}). In particular, the distance from ∂M̄_{h} to O is the number L_{h} in (200).

Proof. In (M̄, ds̄^{2}), let B_{h} be the geodesic ball of radius \frac{μ_{0}}{|h|} centered at O. It lies inside M̄_{h} and touches ∂M̄_{h} at the two points \frac{μ_{0}}{|h|}p̂_{0}^{±}. If Γ̄ is any curve between O and a point q on ∂M̄_{h}, let Γ̄_{1} be the portion of Γ̄ between O and the first point q_{1} on ∂M̄_{h}. From the calculation

L(Γ̄) ≥ L(Γ̄_{1}) = ∫ \sqrt{h + U}ds̄ ≥ ∫ \sqrt{h + \frac{μ_{0}}{ρ}} ds̄ ≥ ∫0^{\frac{μ_{0}}{|h|}} \sqrt{h + \frac{μ_{0}}{ρ}} dρ = L_{h}

it is clear that the two ray segments of length L_{h} are, indeed, the shortest curves, and they are unique (modulo a portion along the boundary).

Remark 35. For h ≥ 0, the geodesic rays through p̂_{0}^{±} are still length minimizing, whereas for any h this fails for the three geodesic rays (or segments) in the eclipse plane Ē.

7.2. Existence of triple collision motions with minimal action. We shall combine the above differential geometric setting, Theorem F and Hilbert’s direct method to study the existence problem of three-body motions leading to triple collision, starting from a given non-degenerate m-triangle and with minimal action integral (7), say. When this problem is pushed down to the level of M̄, at a given energy level h, it can be reduced to the problem of existence of a shortest geodesic, with respect to the metric ds̄_{h}^{2}, between a given point in M̄_{h} − Ē and the base point O.

Remark 36. For any energy h there are Newtonian motions, with the constant shape of a regular triangle or an Euler configuration, through which the m-triangle shrinks homothetically to a triple collision in finite time. To find the time parametrization of such a three-body motion, in fact, amounts to solve a two-body (or Kepler) problem, and this leads to the classical solutions found by Lagrange and Euler, see [2], [7], [13]. For these motions minimal action is achieved for the Lagrange motions (regular triangle), but not for the Euler motions (which are collinear).

More generally, let us first consider the case h < 0, and define the variety

D̄_{h} = {p ∈ M̄_{h}; d(p, O) ≤ d(p, ∂M̄_{h}) + L_{h}} ,  (202)

where d(p, O) (respectively d(p, ∂M̄_{h})) is the distance between O and p (respectively ∂M̄_{h}) in M̄_{h} with the metric ds̄_{h}^{2}. The interior D_{h} (respectively boundary ∂D̄_{h}) of D̄_{h} is defined by strict inequality (respectively equality) in (202).

Remark 37. The two surfaces ∂M̄_{h} and ∂D̄_{h} are interesting geometric objects in the study of triple collision orbits. They touch each other at the two points \frac{μ_{0}}{|h|}p̂_{0}^{±} on ∂M̄_{h} closest to the cone vertex O.

We will prove the following existence result:

Theorem 38. (cf. [3], Theorem 5) Starting from a given oriented mtriangle δ whose congruence class δ̄ belongs to D̄_{h}, h < 0, there is a three-body motion with total energy h and minimal action integral which leads to a triple collision.

Proof. As a consequence of Theorem F, the proof reduces to the existence of a curve with minimal length in M̄_{h} linking δ̄ to O. For δ̄ in D_{h} such a curve is necessarily a geodesic.

Assume first δ̄ ∈ D_{h}, and let \left\{{ \atop Γ̄_{i}}\right\} be a sequence of curves in M̄_{h} between δ̄ and O whose lengths satisfy

L(Γ̄_{i}) < d(δ̄, ∂M̄_{h}) + L_{h}, lim_{i→∞} L(Γ̄_{i}) = d(δ̄, O).

In particular, each Γ̄_{i} is disjoint from the boundary ∂D̄_{h}.

Let us divide Γ̄_{i} into m2^{i} segments of equal length and replace each segment by the unique shortest geodesic between its end points. Then it is quite straightforward to apply the direct method of Hilbert to find a suitable subsequence of \left\{{ \atop Γ̄_{i}}\right\} with a limiting curve Γ̄, and this is necessarily a geodesic curve in M̄_{h} between δ̄ and O with minimal length L(Γ̄) = d(δ̄, O).

Assume next δ̄ ∈ ∂D̄_{h}, and let {δ̄_{k}} be as sequence of points in D_{h} with δ̄ as its limit. Moreover, let \left\{{ \atop Γ̄_{k}}\right\} be a sequence of curves, where Γ̄_{k} is a shortest geodesic between δ̄_{k} and O, that is, L(Γ̄_{k}) = d(δ̄_{k}, O). It follows that

lim_{k→∞} d(δ̄_{k}, O) = d(δ̄, O)  (203)

and it is not difficult so see that there is a suitable subsequence of \left\{{ \atop Γ̄_{k}}\right\} with a limiting curve Γ̄ whose length is the limit (203), and moreover, Γ̄ is a geodesic between δ̄ and O.

Finally, we consider the case h ≥ 0, namely when M̄_{h} = M̄, and then we have the following analogue of the above theorem .

Theorem 39 (cf. [3], Theorem 5’). Starting from a given oriented mtriangle δ, for a given energy level h ≥ 0 there is always a three-body motion leading to triple collision and with minimal action integral.

Proof. This is similar to the case δ̄ ∈ D_{h} of the previous proof, and the application of Hilbert’s direct method will give the existence of the curve we seek.

Remark 40. The direct method of Hilbert can, of course, also be applied to study the existence problem of a geodesic curve Γ̄ realizing the minimal distance between two given points δ̄_{1} and δ̄_{2} of (M̄_{h}, ds̄_{h}^{2}). Then, by Theorem B, there are liftings Γ of Γ̄ which are planary three-body motions (with specified angular momentum) starting from a given oriented m-triangle δ_{1} belonging to the congruence class δ̄_{1}. However, the end point configuration δ_{2} of Γ is already determined by Γ̄ and δ_{1}, according to Theorem C2. Consequently, only three-body motions with specific relative positions of their initial and terminal m-triangles δ_{i} can have minimal action integrals. In the above two theorems there is no such relative position constraint since the triple collision configuration δ_{2} = O consists of a single congruence class.

7.3. The uniqueness problem for triple collision motions with minimal action. In view of the above existence theorems, it is natural to investigate the following uniqueness problem for motions starting from a given configuration at a sufficiently large energy level.

Problem 41. To a given non-degenerate oriented m-triangle δ and energy level h above a lower bound, say h ≥ 𝔥(δ̄), is there a unique threebody motion from δ leading to a triple collision with minimal action integral?

This problem requires a considerable amount of in-depth analysis of the geodesic equation of (M̄_{h}, ds̄_{h}^{2}), and here we shall leave it as an open problem.

However, to facilitate future analytical studies of the above problem and related problems we shall discuss a geometric reduction technique which reflects some useful feature of the Riemannian structure of M̄ = C(M^{∗}) as a cone over the subspace M^{∗} = (ρ = 1). In M̄ the integral curves of the vector field \frac{∂}{∂ρ} are the rays emanating from O, and they define the radial (i.e. a natural ”vertical”) direction at every point ≠ O.

The above problem is, indeed, simple and has an optimal solution in the special case mentioned in Remark 36, namely for the shape invariant motions of Lagrange type. A 3-body motion is shape invariant if its moduli curve Γ̄ is confined to a ray, that is, the associated shape curve Γ^{∗} is a single point. In fact, if a Newtonian motion is shape invariant over some time interval of length > 0, then for all time Γ^{∗} is a single point (necessarily a critical point of U^{∗}). However, in view of Remark 36, the collinear solutions with the shape of an Euler point are not even action minimizing.

In general, let Γ̄ be a smooth curve in M̄ − {O}. Since the associated shape curve Γ^{∗} is the radial projection of Γ̄ onto the transversal subspace M^{∗}, Γ^{∗} will be smooth as long as Γ̄ is transversal to the radial direction, whereas a cusp may occur at points where this fails. Hence, in the long run the typical shape curves of 3-body motions are rather piecewise smooth, but still they can be parametrized by arc-length. Moreover, unless Γ^{∗} is a single point, Γ̄ may also be parametrized by the arc-length parameter of Γ^{∗}.

Definition 42. Let Γ̄ be a curve in the moduli space M̄ and let Γ^{∗} be the associated shape curve. The cone consisting of all rays emanating from O and passing through points on Γ̄ (or Γ^{∗}) is called the cone surface of Γ̄ (or Γ^{∗}), and it is denoted either C(Γ̄) or C(Γ^{∗}).

We assume Γ̄ (and hence also Γ^{∗}) has a given orientation. Since the metric on M^{∗} = S^{2}(1/2) is denoted by dσ^{2} (cf. e.g. 38), σ also denotes the arc-length parameter of Γ^{∗}. For σ ranging over some interval [σ_{0}, σ_{1}], the corresponding surface C(Γ^{∗}) is immersed in (M̄, ds̄^{2}) with the induced kinematic metric

ds̄^{2}|_{C(Γ^{∗})} = dρ^{2} + ρ^{2}dσ^{2}; σ_{0} ≤ ϑ ≤ σ_{1},  (204)

and hence it is isometric to a flat Euclidean sector of angular width σ_{1} − σ_{0}, with (ρ, σ) as polar coordinates centered at the origin O.

The moduli space M̄ ≈ ℝ^{3} has the standard (right handed) orientation and, in particular, the 2-sphere M^{∗} has the induced orientation with \frac{∂}{∂ρ} as positive normal vector field. The surface C(Γ^{∗}) is naturally oriented with the positive orthonormal frame

\{\frac{∂}{∂ρ}, \frac{1}{ρ} \frac{∂}{∂σ} \} ,  (205)

and we choose its normal vector field ν so that ν followed by the frame (205) is a positive orthonormal frame in M̄.

In (204) the curve Γ^{∗} becomes the circular arc of radius ρ = 1, whereas the (original) moduli curve Γ̄ appears as a radial deformation of Γ^{∗}. When s̄ in (204 is viewed as the arc-length parameter of

Γ̄ : s̄ → (ρ(s̄), σ(s̄)),

(204) becomes an identity along the curve.

The extrinsic geometry of C(Γ^{∗}) ⊂ M̄ is completely determined by the extrinsic geometry of Γ^{∗} ⊂ M^{∗}. Indeed, the lines of curvature are the two families of coordinate curves, namely the rays (σ constant) and the ”circles” (ρ constant) in C(Γ^{∗}). The principal curvature of C(Γ^{∗}) at a point δ̄ is zero in the ray direction and is equal to K_{g}^{∗}/ρ in the direction of \frac{∂}{∂σ}, where K_{g}^{∗} is the geodesic curvature of Γ^{∗} in M^{∗} at the corresponding point δ^{∗}.

Along the curve Γ̄ we will also consider the positive orthonormal moving frame {τ , η, ν}, where τ is the unit tangent vector in the (chosen) positive direction of Γ̄, and hence the frame {τ , η} of C(Γ^{∗}) differs from the stationary frame (205) by a certain rotation angle α. Namely, we define the (radial) inclination angle α of Γ̄ by writing

τ = cos α\frac{∂}{∂ρ} + sin α\frac{1}{ρ} \frac{∂}{∂σ} , η = − sin α\frac{∂}{∂ρ} + cos α\frac{1}{ρ} \frac{∂}{∂σ} ,  (206)
cos α = \frac{dρ}{ds̄}, sin α = ρ\frac{dσ}{ds̄} , cot α = \frac{1}{ρ} \frac{dρ}{dσ} = \frac{d}{dσ} ln ρ.  (207)

Briefly, α is the angle between the ray direction and the tangent direction, and 0 ≤ α ≤ π since sin α in (207) is not negative. The extreme values α = 0, π occur when Γ̄ is not transversal to the radial direction, in which case α and Γ^{∗} (as functions of σ or time) may encounter a singularity, namely Γ^{∗} encounters a cusp.

Remark 43. The angle α(σ) and the radial distance ρ(σ) are mutually dependent according to (207). For example, we have for ρ(σ_{0}) ≠ 0

ρ(σ) = ρ(σ_{0}) exp(∫_{σ_{0}}^{σ} cot α(σ)dσ).  (208)

The geodesic condition for a curve Γ̄ in (M̄_{h}, ds̄_{h}^{2}) is evidently equivalent to two identities of type (201), namely for two linearly independent normal vectors n to Γ̄. Thus, we shall consider the two cases

(i) n = η : tangential to C(Γ̄)  (209)
(ii) n = ν : perpendicular to C(Γ̄).

The first case amounts to the characterization of Γ̄ as a geodesic in the (truncated) cone surface C(Γ̄) ∩ M̄_{h} with the metric ds̄_{h}^{2}, as follows:

Lemma 44. Let u(σ) be the restriction of the potential function U^{∗} along the shape curve Γ^{∗}(σ). Then the geodesic equation for the moduli curve Γ̄ in the cone surface C(Γ̄) with the metric

ds̄_{h}^{2} = (\frac{1}{ρ}U^{∗} + h)(dρ^{2} + ρ^{2}dσ^{2})

is equivalent to the equation

\frac{dα}{ds̄} + \frac{dσ}{ds̄} = \frac{1}{2ρ} (sin α\frac{u(σ)}{u(σ) + hρ} + cos α\frac{u′(σ)}{u(σ) + hρ}) .  (210)

Proof. Let η be the normal vector in (206). Then the geodesic condition is by (201)

K(η) =\frac{1}{2} \frac{d}{dη} ln(\frac{u(σ)}{ρ} + h),  (211)

where K(η) is the (geodesic) curvature of Γ̄ in the Euclidean sector (204). However, in a Euclidean plane it is easy to see that K(η) can be expressed as dζ/ds̄, where ζ is the angle between a fixed reference ray σ = 0 (say, the positive x-axis) and the tangent line, in fact, ζ = α + σ, see Figure 8. Finally, calculation of the normal derivative on the right side of the identity (211), using the orthonormal frame (205), leads to the formula (210).

In the second case of (209) the geodesic condition is the identity (201) with n equal to the normal ν of the surface. In this case K(ν) equals the normal sectional curvature of C(Γ̄) in the direction of Γ̄, namely the value Π(τ , τ) of the second fundamental form. The latter has the frame (205) as eigenvectors, with eigenvalues 0 and K_{g}^{∗}/ρ respectively, and hence by (206) and Euler’s classical formula for the decomposition of normal geodesic curvature

K_{g}^{∗} sin^{2} α = ρK(ν) = \frac{ρ}{2} \frac{d}{dν} ln(\frac{1}{ρ}U^{∗} + h).  (212)

As a summary we now state the following theorem, valid as long as the quantities involved are well defined.

Theorem 45 (cf. [3], Theorem 6). In the moduli space M̄ with the kinematic Riemannian metric ds̄^{2}, let Γ̄ be the oriented moduli curve of a three-body motion with total energy h and vanishing angular momentum, and let Γ^{∗} be the corresponding shape curve on the sphere M^{∗} = S^{2}(1/2) with unit tangent (respectively normal) vector τ^{∗} (respectively ν^{∗}) so that {τ^{∗}, ν^{∗}} is a positive frame on the sphere. Then Γ̄ = (ρ, Γ^{∗}) can be characterized as a solution of the following system of ODE

(i) : \frac{dα}{dσ} = −1 + \frac{1}{2} \frac{u(σ)}{u(σ) + hρ} (1 + cot α\frac{d}{dτ^{∗}} ln(U^{∗})) ,  (213)
(ii) : K_{g}^{∗} sin^{2} α = \frac{1}{2} \frac{u(σ)}{u(σ) + hρ} \frac{d}{dν^{∗}} ln(U^{∗}),

where K_{g}^{∗} is the geodesic curvature of Γ^{∗} in M^{∗}, σ is the arc-length parameter of Γ^{∗}, U^{∗} is the restriction of U to M^{∗} and u(σ) is its further restriction along Γ^{∗}, and α ∈ [0, π] is the angle between the (outgoing) ray direction and Γ̄ in M̄.

Proof. By using the expression for sin α in (207), equation (210) can be stated as

\frac{dα}{ds̄} = (−1 + \frac{1}{2} \frac{u(σ)}{u(σ) + hρ})\frac{dσ}{ds̄} + \frac{1}{2ρ} cos α\frac{u′(σ)}{u(σ) + hρ}.

When we replace the arc-length parameter s̄ of Γ̄ by σ, using a formula from (207), this equation reads

\frac{dα}{dσ} = (−1 + \frac{1}{2} \frac{u(σ)}{u(σ) + hρ}) + \frac{1}{2} cot α\frac{u′(σ)}{u(σ) + hρ},

and by viewing u′(σ)/u(σ) as the tangential derivative of ln(U^{∗}) we obtain the first equation (213).

The second equation of (213) is merely a reformulation of (212), whose right side may be expressed as

\frac{1}{2} \frac{d}{dν^{∗}} ln(\frac{1}{ρ}U^{∗} + h) = \frac{2^{−1}}{u(σ) + hρ} \frac{d}{dν^{∗}} U^{∗}.

Here we use the fact that the normal vector ν along Γ̄ may be identified with the scaling of ν^{∗} by the factor 1/ρ, that is, ν = ν^{∗}/ρ, and moreover, differentiation in the direction of ν commutes with the scaling.

Remark 46. The above system (213) is easily seen to be scaling invariant. Namely, when the size function ρ is multiplied by a fixed constant k > 0, the energy level changes as h → h/k and hence the product hρ stays invariant. In particular, since the energy level h = 0 is invariant with respect to scaling of solutions, the explicit dependence on ρ in (213) disappears in this case. Moreover, the angle α, geometrically interpreted in (206) as the inclination angle of the moduli curve Γ̄, is a neat scaling invariant which together with Γ^{∗} represents Γ̄ uniquely up to scaling. In fact, ρ is generally obtained from α by ”quadrature” along Γ^{∗}, cf. (208). This explains the following result.

Corollary 47. Let the pair (α, Γ^{∗}) represent the moduli curve Γ̄ of a three-body motion with vanishing angular momentum and vanishing total energy, where the shape curve Γ^{∗} is not a single point and is viewed as a curve on the standard sphere S^{2} of radius 1. Then (α, Γ^{∗}) is a solution of the following system of ODE

\{ \begin{matrix}\frac{dα}{ds} = −\frac{1}{4} + \frac{1}{2} cot α\frac{d}{dτ^{∗}} ln(U^{∗}) \\ K_{g}^{∗} sin^{2} α = \frac{1}{2} \frac{d}{dν^{∗}} ln(U^{∗}) \end{matrix}  (214)

where s = 2σ is the arc-length parameter of Γ^{∗} on S^{2} and K_{g}^{∗} is its geodesic curvature. Moreover, a solution (α, Γ^{∗}) can only encounter a singularity (cusp) when α = 0 or π, or when Γ^{∗} reaches a collision point.

We are particularly interested in applying the system (213) or (214) to the study of triple collision motions. A triple collision is simply expressed by the condition ρ = 0, but this singular event is not explicitly visible in (214) since the variable ρ is eliminated. However, the term hρ in (213) also disappears when ρ → 0, so the two systems should behave ”similarly” in the limit. Hence, the system (214) is likely to be significant also when h ≠ 0.

One of the major results of Sundman and Siegel in their work on the local analysis of triple collisions prove the existence of both a limiting shape, necessarily a critical point of U^{∗}, and a limiting position, cf. [14], [15], [11], [12]. The existence of a limiting position is the statement that the 3-body motion Γ(t) has a ”size normalized” limit,

\frac{Γ(t)}{|Γ(t)|} → δ,  (215)

at the configuration space level, and we shall express the statement concerning the limiting shape (due to Sundman) by saying the pair (α, Γ^{∗}) approaches a specific pair (α̂, δ^{∗}), namely

α̂ ∈ {0, π} = ∂[0, π] , δ^{∗} ∈ {ê_{1}, ê_{2}, ê_{3}, p̂_{0}^{+}, p̂_{0}^{−}} ⊂ S^{2}.  (216)

It is also known (cf. e.g. Siegel-Moser[13], p. 89) that an Euler point ê_{i} can only be the limiting shape of a triple collision motion confined to a fixed line. (However, ê_{i} may well be the limiting shape of a non-collinear motion as t → ±∞ ).

The two ”boundary” values of α in (216) actually distinguish between the two events triple explosion and triple collision, as follows: α = 0 when Γ̄ starts (or ”explodes”) out from the cone vertex O of M̄ , and lim α = π when Γ̄ is oriented towards O and terminates with a ”total collapse”. Anyhow, we are free to run a three-body motion in either directions, and the associated initial value problem for (213) or (214) is (a priori) of singular type in the above case since sin α = 0. In fact, a solution (α, Γ^{∗}) of (214) may also encounter another type of singularity (called cusp) when sin α = 0, but with ρ ≠ 0. In Chapter 8 these events and related problems will be further investigated in selected testing cases.

8. Case study of triple collision motions with zero energy

8.1. The basic setting and statement of Theorem G. Due to the simplicity of the system (214), the special case of vanishing total energy, h = 0, lends itself as the simplest testing case of three-body motions leading to a triple collision. We shall investigate this case more carefully, and for convenience, let us also restrict ourselves to the case of equal masses, m_{i} = 1/3, which largely simplifies the series expansions of the potential function and its derivatives.

We shall address the triple collision problem as an initial value problem, namely as a triple explosion, although we usually write ”triple collision” motions. Let us first recall the so-called Lagrange-Jacobi equation which is the result of differentiating I = ρ^{2} twice with respect to time t, using the homogeneity of U and conservation of energy h = T − U, namely in our case

\frac{d^{2}}{dt^{2}} I = 2(T + h) = 2T > 0.  (217)

It follows that I is a nonnegative convex function of time t, and starting from a triple collision (say, I(0) = 0) it is strictly increasing and tends to ∞ as t → ∞.

Following the setup from Section 7.3, we seek a description of the moduli curves of triple collision motions, valid for some appropriate time interval [0, t_{1}]. For this purpose it is convenient to use the coordinates (ρ, φ, θ) in the cone M̄ = C(M^{∗}), where as before ρ = \sqrt{I} and (φ, θ) are spherical coordinates on the unit sphere S^{2} = M^{∗} centered at the north pole N. In this setting a moduli curve Γ̄ and the associated shape curve Γ^{∗} have coordinate representations

Γ̄(s) = (ρ(s), φ(s), θ(s)), Γ^{∗}(s) = (φ(s), θ(s)),  (218)

where s = s(t) is the arc-length parameter s = s(t) of Γ^{∗} and is an increasing function of time t. Here we must exclude, of course, the well understood shape invariant motions, namely the trivial case that Γ^{∗} is a single point (in which case ρ(t) is the solution of a 1-dimensional Kepler problem).

Thus, we seek a description of all those moduli curves Γ̄(s) emanating from a triple collision, at s = 0 say. According to Remark 46 it suffices to consider the class of Γ̄ modulo scaling, represented by the pair (α, Γ^{∗}) where α(s) ∈ [0, π] is the (radial) inclination angle of Γ̄. In fact, recall from (208) that the size function ρ of Γ̄ is recovered from (α, Γ^{∗}) by the general quadrature formula

ρ(s) = ρ(s_{0})e^{\frac{1}{2} ∫_{s_{0}}^{s} cot(α)ds}, ρ(s_{0}) ≠ 0.  (219)

Remark 48. Using the parameter s rather than time t is, of course, crucial for our geometric approach below. The relationship between s and t is follows from the kinematic metric, using e.g. (204), (207), (219), (264). Namely, in the case of zero angular momentum there are the identities

2(U + h)dt^{2} = 2T dt^{2} = ds̄^{2} = dρ^{2} + \frac{ρ^{2}}{4} ds^{2} = (cos^{2} α)ds̄^{2} + \frac{ρ^{2}}{4} ds^{2},

from which we deduce the relationship

dt = \frac{ρ(s)}{2^{3/2} sin α(s)\sqrt{u(s)/ρ(s) + h}}ds,  (220)

where u(s) = U^{∗}(Γ^{∗}(s)). Moreover, by switching over to t it is, in fact, not difficult to see that ρ (at any time t_{0}) can be determined solely from the time parametrized shape curve Γ^{∗}(t) and the normal derivative of U^{∗} (near t = t_{0}).

Now, resuming the assumption h = 0, our approach is to determine the above pairs (α, Γ^{∗}) by solving the system

ODE^{∗} : \{ \begin{matrix}\frac{dα}{ds} = −\frac{1}{4} + \frac{1}{2} cot α\frac{d}{dτ^{∗}} ln(U^{∗}) \\ K_{g}^{∗} sin^{2} α = \frac{1}{2} \frac{d}{dν^{∗}} ln(U^{∗}) \\ 1 = (\frac{dφ}{ds})^{2} + (sin^{2} φ)(\frac{dθ}{ds})^{2} \end{matrix}  (221)

as an appropriate initial value problem which represents a triple collision, see (242). The system (221) is a copy of (214) since the third equation merely expresses the constraint that Γ^{∗} is a curve on the unit sphere. We will refer to the first and second equation of (221) as the inclination and curvature equation respectively. The first one relates the growth of the inclination angle α (of the moduli curve Γ̄) with the tangential derivative of ln(U^{∗}) along Γ^{∗}, whereas the second one - which is of order two- relates α to the geodesic curvature of Γ^{∗} and the normal derivative of ln(U^{∗}) along Γ^{∗}.

The following theorem summarizes the main result of this chapter. It describes the family 𝔖(p̂_{0}) of all shape curves Γ^{∗}(s) representing triple collision motions, with the limiting shape of p̂_{0} at the collision.

Theorem G_{1} In the case of uniform mass distribution and zero total energy, consider the family 𝔖(p̂_{0}) of arc-length parametrized shape curves Γ^{∗}(s), s ≥ 0, which emanate from the north pole p̂_{0} = Γ^{∗}(0) of the 2sphere S^{2} and represent 3-body motions with a triple collision at s = 0. This family has the following properties:

(i) There is a unique curve Γ_{θ_{0}}^{∗} for each initial longitude direction θ_{0}.

(ii) The family is invariant under the induced action of the dihedral isometry group 𝔇_{3} of S^{2} which fixes p̂_{0} and permutes the three Euler points ê_{i}. In particular, Γ_{θ_{0}+2π/3}^{∗} is obtained from Γ_{θ_{0}}^{∗} by rotating the sphere, θ → θ + 2π/3.

(iii) Each curve stays within a sector of angular width π/3 and bounded by meridians representing the shape of isosceles triangles, at least until the first eclipse (i.e. crossing the equator circle).

(iv) Each curve extends analytically through s = 0 and Γ_{θ_{0}+π}^{∗}(s) = Γ_{θ_{0}}^{∗}(−s), and it has no singularity before the first eclipse.

(v) For each θ_{0} the associated inclination angle function α_{θ_{0}}(s) is the unique analytic solution of (221) along Γ_{θ_{0}}^{∗}, with the singular initial condition α(0) = 0, α′(0) > 0. Moreover, α_{θ_{0}+π}(s) = −α_{θ_{0}}(−s).

(vi) The sign of the curvature of the curves in 𝔖(p̂_{0}) depends only on the sector, and in neighboring sectors the sign is opposite.

Remark 49. In the case of rectilinear three-body motions, the corresponding sets 𝔖(ê_{i}) are obviously ”congruent”, each consisting of the pair Γ_{±}^{∗} of arcs of the equator circle, in opposite directions and starting at the Euler point ê_{i}. The associated inclination angle function α_{±}(s) is defined by a unique analytic function α(s) so that α_{±}(s) = α(s) for s ≥ 0, and α(−s) = −α(s). We refer to Section 8.4.

We also refer to Section 8.6.2 for more information concerning the geometric behavior of the curves in 𝔖(p̂_{0}). Indeed, we are actually close to a stronger version of Theorem G_{1}, but the proof needs more elaboration, so we formulate the following conjecture as an open problem.

Conjecture 50. The different triple collision shape curves Γ^{∗}(s), s ≥ 0, intersect the equator circle the first time at different points, and moreover, each point on the circle is reached by a unique curve. The curves do not intersect each other, except possibly after the first eclipse.

Corollary 51. Under the current hypothesis of uniform mass distribution and zero total energy, consider the ”moduli space” consisting of all three-body motions in 3-space which start from a triple explosion at time t = 0, and moreover, the motion is neither rectilinear nor shape invariant (i.e. homographic). This space can be naturally identified with the manifold

SO(3) × SO(2) × ℝ^{+}.

In particular, the ”moduli space” for those triple collision three-body motions confined to a fixed plane is

O(2) × SO(2) × ℝ^{+}.

Indeed, starting with the space 𝔖(p̂_{0}) ≃ SO(2) of shape curves described in Theorem G_{1}, SO(2)×ℝ^{+} is the space of their associated curves Γ̄(s) in the moduli space M̄ since the size function ρ(s) can be scaled by any positive number λ ∈ ℝ^{+} without affecting the shape curve. Next, the possible liftings Γ(s) of Γ̄(s) to three-body motions with Γ(0) = 0 are distinguished by the normalized limit δ in (215), which is an oriented, regular m-triangle Δ of unit size. We may identify the various positions of such an oriented triangle with the rotation group SO(3) which measures its ”deviation” from a fixed reference position. In terms of the fibration

O(2) → SO(3) → SO(3)/O(2) ≃ ℝP^{2}

we can say that the projective plane ℝP^{2} represents the choices of 2planes containing Δ (and the motion), whereas O(2) represents the possible positions of an oriented regular triangle in a given plane. However, we mention that there is no global ”field” of reference positions (or gauge) for all the planes, since this would imply the fibration is trivial, which is certainly not true.

8.2. Analysis of the potential function for equal masses. In this chapter we shall choose the zero meridian θ = 0 for the polar coordinate system (φ, θ) of S^{2} different from the convention in Remark 15. Namely, the three binary collision points b̂_{i}, i = 1, 2, 3, which are now equally spaced along on the equator circle φ = π/2, will have the longitude angles

θ_{1} = −\frac{π}{3} , θ_{2} = \frac{π}{3} , θ_{3} = π (cf. Figure 9).  (222)

Note, for example, the antipodal point of b̂_{i} is the Euler point ê_{i}, and now θ = 0 at ê_{3}. It is also convenient to use negative values of φ, with the usual interpretation so that (φ, θ) and (−φ, θ + π) is the same point on the 2-sphere. This is consistent with our trigonometric formulae for U^{∗}(φ, θ) below, see (224) and (246).

For convenience, let us normalize U^{∗} by a constant factor to make its minimum value U^{∗}(p̂_{0}) = 1. In fact, scaling of U^{∗} has no effect on the system (221). Thus, for p ∈S^{2}

U^{∗}(p) = \frac{\sqrt{2}}{3} ∑_{i=1}^{3} \frac{1}{|p − b̂_{i}|} = \frac{1}{3} ∑_{i=1}^{3} \frac{1}{1 − z_{i})^{1/2}} ,  (223)

where (for any mass distribution, indeed)

|p − b̂_{i}| = \sqrt{2}(1 − z_{i})^{1/2}, with z_{i} = sin φ cos(θ − θ_{i}),  (224)

is the usual Euclidean distance from p to the binary collision point b̂_{i} and θ_{i} is the longitude angle of b̂_{i}, cf. (141), (169), (171), 222.

Consequences of the invariance of U^{∗} with respect to permutation of the points b̂_{i} will be analyzed and exploited later (cf. Section 8.3.1). At the algebraic level, however, the following symmetrization technique will facilitate the analysis of U^{∗} and related series expansions. For each integer k ≥ 0, define

S_{k} = ∑_{i=1}^{3} cos^{k}(θ − θ_{i})

and write

f(x) = ∏_{i=1}^{3}(x − cos(θ − θ_{i})) = x^{3} − \frac{3}{4}x + \frac{1}{4} cos(3θ).

Logarithmic differentiation of f(x) leads to the formal identity

(∑_{k=0}^{∞} \frac{S_{k}}{x^{k+1}} ) f(x) = f′(x) = 3x^{2} − \frac{3}{4}

from which we deduce the recursive formula

S_{k+3} = \frac{3}{4}S_{k+1} − \frac{1}{4} cos(3θ)S_{k}, k ≥ 0.  (225)

For convenience, the first few S_{k} are listed as follows:

S_{0} = 3, S_{1} = 0, S_{2} = \frac{3}{2}, S_{3} = −\frac{3}{4} cos(3θ), S_{4} = \frac{9}{8}
S_{5} = −\frac{15}{16} cos(3θ), S_{6} = \frac{27}{32} + \frac{3}{16} cos^{2}(3θ), S_{7} = −\frac{63}{64} cos(3θ).

It follows, for example, that S_{k} as a polynomial in cos(3θ) has all its nonzero coefficients positive (respectively negative) when k is even (respectively k is odd). By expanding U^{∗} as a sum of binomial series in the variables z_{i} and using the identity

∑_{i=1}^{3} z_{i}^{k} = (sin^{k} φ)S_{k},

we arrive at the following trigonometric series

U^{∗} = U^{∗}(φ, θ) = \frac{1}{3} ∑_{k=0}^{∞} \binom{−\frac{1}{2}}{k}(−1)^{k}(sin^{k} φ)S_{k}  (226)
= 1 + \frac{3}{16} sin^{2} φ − \frac{5}{64}(sin^{3} φ)(cos 3θ) + … + .

For later use we also introduce the following functions on the sphere

F (p) = \frac{\sqrt{2}}{3} ∑_{i=1}^{3} \frac{sin(θ − θ_{i})}{|p − b̂_{i}|^{3}}, G(p) = \frac{\sqrt{2}}{3} ∑_{i=1}^{3} \frac{cos(θ − θ_{i})}{|p − b̂_{i}|^{3}}. (227)

It follows that

\frac{∂U^{∗}}{∂θ} = −F sin φ, \frac{∂U^{∗}}{∂φ} = G cos φ.  (228)

Remark 52. It is easy to check that F vanishes precisely along the six meridians

Γ_{k}^{∗} : θ = k \frac{π}{3} ,  0 ≤ k ≤ 5  (229)

passing through either a binary collision point b̂_{i} or an Euler point ê_{i} (= −b̂_{i}). Hence, F changes its sign by crossing these meridians, but on the other hand, the function G is positive (except undefined at p̂_{0}). For example, F is negative for 0 < θ < π/3, and in this sector the gradient flow of U^{∗} is depicted in Figure 9.

8.3. Reduction, regularity and singularity. Here we shall describe a finite group acting on moduli curves and, in particular, it is a symmetry group of the space of solutions s → (α(s), Γ^{∗}(s)) of (221). Moreover, regularity and singularity aspects of the solutions we seek are also briefly discussed.

8.3.1. Discrete symmetries and reduction. In addition to time translation and space-time scaling symmetries which transform solutions of the general 3-body problem as in (197)), there is also an additional symmetry group of order 4 which we denote by

𝔇_{1} × ℤ_{2} = ⟨σ̄, τ̄⟩ ≃ ℤ_{2} × ℤ_{2}.  (230)

The involution τ̄ represents reversal of time, t → −t , and its induced action on oriented curves in the moduli space is expressed by

τ̄ : (s, α) → (s̃, α̃) = (−s, π − α) (reversal of direction)  (231)

which takes a solution (α(s), Γ^{∗}(s)) of the system (221) on the interval s_{1} < s < s_{2} to the ”reverse” solution in the opposite direction and defined on the interval −s_{2} < s̃ < −s_{1}(or any translation of this interval). The other involution σ̄ is a purely geometric symmetry, arising from the reversal of orientation of m-triangles. At the moduli space level, the latter is the reflection of M̄ ≃ ℝ^{3} in the (equator) xy-plane, that is, the map φ → π − φ in the coordinates (ρ, φ, θ).

On the other hand, under the present assumption of equal masses, there is also the (order 6) dihedral isometry group 𝔇_{3} ⊂ O(2) of S^{2} which fixes the poles p̂_{0}^{±} and permutes the Euler points ê_{i}. It is a symmetry group of the 3-body problem since it leaves U^{∗} invariant, cf. Section 8.2. The action is generated by the rotation θ → θ + 2π/3 and the reflection θ → −θ, and altogether we have a symmetry group of order 24,

𝔊 = 𝔇_{6} × ℤ_{2} = (𝔇_{3} × 𝔇_{1}) × ℤ_{2},  (232)

where we also regard 𝔇_{6} ⊂ O(3) as a dihedral isometry group of S^{2} generated by reflections. Thus, the action of 𝔇_{6} divides the sphere into 12 congruent spherical triangles called chambers, and we choose one of them to be our fundamental chamber, namely the following geodesic triangle on the upper hemisphere

ℭ_{0} = \{(φ, θ) ∈ S^{2}; 0 ≤ φ ≤ \frac{π}{2} , 0 ≤ θ ≤ \frac{π}{3} \}  (233)

(cf. Figure 9) with the vertices

p̂_{0} = (φ = 0), ê_{3} = (φ = π/2, θ = 0),  (234)
b̂_{2} = (φ = π/2, θ = π/3).

In particular, the action of 𝔇_{6} on solutions (α, Γ^{∗}) of (221) reduces the study of solutions to the study of ”solution segments” Γ^{∗} inside ℭ_{0}. In particular, we may restrict the study of triple collision solutions Γ^{∗} to those emanating from the vertex p̂_{0} or ê_{3} with initial direction leading into the chamber ℭ_{0}. The natural first step of this program is to look for solutions whose curvature equation in (221) is trivially satisfied, and Section 8.4 is devoted to this preliminary study.

8.3.2. Cusps and other singularities. It is well known that a 3-body motion Γ(t) can be ”regularized” through a binary collision (cf. [13]), and the only ”real” singularity must be a triple collision. Away from collisions the moduli curve Γ̄ and the shape curve Γ^{∗} are also analytic functions when we parametrize by time t or the arc-length s̄ of Γ̄. However, we also want to parametrize by the arc-length s of Γ^{∗}, and then singularities may occur at specific instants where the time derivative ṡ(t) ≥ 0 vanishes. Although this type of ”singularity” is rather artificial, it has geometric significance which explains the possible cusps of the embedded curve Γ^{∗} on the 2-sphere. These are also singularities of the system (221), and in this subsection we shall discuss them in some detail.

Since \frac{ds̄}{dt} > 0 for all t, the event ṡ(t_{1}) = 0 is equivalent to the condition α(t_{1}) = 0 or π. In this case, \frac{d}{dt}Γ^{∗}(t_{1}) = 0 and we say p = Γ^{∗}(t_{1}) is a halting point for the shape curve. Geometrically, this can be a singular point for Γ^{∗}, namely it is the type of singularity that may occur when a regular curve ψ̄(t) in 3-space with vertical tangent at t = t_{1} is projected to a curve ψ^{∗}(t) in the xy-plane, cf. also (218). On the other hand, when Γ^{∗} is parametrized by s and s_{1} = s(t_{1}), the unit tangent vector \frac{d}{ds}Γ^{∗}(s_{1}) still exists (as a one-sided limit) at the halting point p. Moreover, the (one-sided) geodesic curvature K_{g}^{∗} of Γ^{∗} will be bounded near p. In fact, for a ”thin” region 0 < ρ_{1} ≤ ρ ≤ ρ_{1}+δρ of the moduli space (M̄, ds̄^{2}) with the kinematic metric (193), the projection to the shape space (M^{∗}, ds^{2}) may be viewed as a Riemannian submersion modulo an almost constant scaling. Restricting to the above region, the curvature of the moduli curve Γ̄ is certainly bounded, and its image curve in M^{∗} will also have bounded curvature.

Now, consider a pair (α(s), Γ^{∗}(s)) which is a solution of the system (221). The pair is regular on the interval (s_{1}, s_{2}) if the three functions α(s), φ(s), θ(s) are analytic and α(s) ≠ 0, π, and a singularity is encountered at s = s_{i} if we cannot extend the functions regularly beyond this point. In that case it follows from (221) that either sin α(s_{i}) = 0, in which case we call p = Γ^{∗}(s_{i}) a cusp, or p is a binary collision point b̂_{j} (in which case α(s_{i}) = π/2 and α′(s_{i}) = ∞). On the interval (s_{1}, s_{2}) the growth of α(s) is governed by the inclination angle equation (cf. (221))

\frac{dα}{ds} = −\frac{1}{4} + \frac{1}{2} cot α(s)D(s)  (235)

whose dependence on the shape curve is solely through the tangential logarithmic derivative

D(s) = ∇(ln U^{∗}) · τ^{∗} = \frac{u′(s)}{u(s)}, u(s) = U^{∗}(Γ^{∗}(s)).  (236)

Assume there is a halting point at s = s_{i}, say α(s_{i}) = 0 and write p = Γ^{∗}(s_{i}). If D(s_{i}) ≠ 0, then by (235) α′(s_{i}) = ±∞ and p is a cusp. On the other hand, if D(s_{i}) = 0 and p is not a critical point of U^{∗}, then the curvature (i.e. second) equation of (221), whose right side is nonzero at s = s_{i}, would force the geodesic curvature of Γ^{∗} to become infinitely large towards p. However, as observed above, such a behavior of the shape curve is not possible. Hence, D(s_{i}) = 0 is only possible when the halting point p of Γ^{∗} is also a critical point of U^{∗}.

Finally, assume the halting point p is also a critical point of U^{∗}. For s close to s_{i} we have cot α ∼ 1/α, and the local behavior of α(s) near s = s_{i} is largely governed by equation (235), from which we can show α′(s) is bounded in a neighborhood of s_{i}. Choose some s_{0} so that s_{1} < s_{0} < s_{2} and consider the two cases depending on whether s is approaching s_{i} from above or below:

i) s_{i} = s_{1} : ∫_{s_{0}}^{s_{1}} cot α(s) ds ≈ ∫_{s_{0}}^{s_{1}} \frac{ds}{α(s)} = −∞  (237)
ii) s_{i} = s_{2} : ∫_{s_{0}}^{s_{2}} cot α(s) ds = ∞.  (238)

In particular, in case i) an associated 3-body motion must necessarily encounter a triple collision at s = s_{1} since formula (219) implies ρ(s_{1}) = 0, whereas in case ii) ρ(s_{2}) = ∞ and p = Γ^{∗}(s_{2}) is the limiting shape of the 3-body motion as t → ∞. In contrast to this, for a cusp singularity with 0 < ρ(s_{i}) < ∞, we have α(s) → 0 and α′(s) → ±∞ as s → s_{i}, in such a way that the integrals (237) or (238) will converge. For example, this would be the case if α(s) ∼ k(s − s_{i})^{p} for some constant k and p < 1.

We claim, in fact, that (α(s), Γ^{∗}(s)) is regularizable at the above halting point p = Γ^{∗}(s_{i}) which is also a critical point of U^{∗}, that is, as a solution of (221) the functions can be extended analytically beyond s_{i}. Indeed, once the derivative α′(s_{i}) exists, the power series developments at s = s_{i} of the functions α(s), φ(s), θ(s) are recursively determined from the system (221). The recursive scheme is worked out in detail in Section 8.4 for the case (237) with a triple collision at the north pole p̂_{0} = Γ^{∗}(s_{1}), and case ii) is similar.

The collinear type of triple collision is at an Euler point such as p = ê_{3}, in which case Γ^{∗}(s) moves along the equator. In particular, u(s) in (236) has a minimum at s = s_{1}. In fact, approaching ê_{3} from other directions (not tangential to the equator) would force α′(s) to become complex valued. It is remarkable that α′(s_{1}) = a_{0} (respectively b_{0}) turns out to be the same constant for all possible triple collision curves emanating from p̂_{0}^{±} (respectively ê_{i}), see (250).

In order to describe analytically the regularization of the triple collision motions it is convenient to extend the domain of the angle α to negative values as well. Indeed, −α should be identified with π − α, and therefore we introduce the α-circle

\frac{[0, π]}{(0 ∼ π)} ≃ S^{1} ⊂ ℂ : α → z_{α} = e^{2iα}  (239)

as the new domain for α. Then the continuous motion z_{α}(s) on the circle illustrates the qualitative behavior of the solution (α(s), Γ^{∗}(s)) and hence also the moduli curve Γ̄(s) of an associated 3-body motion. For example, ρ(s) is increasing (respectively decreasing) with s when z_{α} lies on the upper (respectively lower) semicircle. A halting point is characterized by z_{α} = 1, and it represents either a cusp, a triple collision (ρ → 0), or an escape (ρ → ∞).

The only way z_{α}(s) enters the other semicircle is at z_{α} = 1, but for ρ → 0 or ∞, since at a cusp z_{α} ”bounces back” on the same semicircle. Similarly, z_{α} reaches the value −1 at a binary collision point b̂_{i}, but again z_{α} ”bounces back” (if the moduli curve is continued via regularization with ρ increasing ).

Summary 53. The solution (z_{α}(s), Γ^{∗}(s)) has a singularity at s = s_{i} if either i) the pair takes the value (1, p) where p does not belong to the set

{±p̂_{0}, ê_{1}, ê_{2}, ê_{3}} ∪ {b̂_{1}, b̂_{2}, b̂_{3}} ,  (240)

or ii) it takes the value (−1, b̂_{i}), i > 0. The singularity is a cusp (respectively a binary collision) in the first (respectively second) case.

8.3.3. The initial value problem for triple collision solutions. For later reference we introduce the set 𝔖(p) of all shape curves Γ^{∗}(s), s ≥ 0, representing a triple collision motion with the limiting shape p = Γ^{∗}(0) at the collision. Here p can be any of the five points of the first subset in (240). In fact, each Γ^{∗}(s) is associated with a unique (inclination angle) function α(s) so that the pair (α, Γ^{∗}) is a solution of the system ODE^{∗} in (221) with the additional and singular initial condition

i) α(0) = 0,  ii) α′(0) ≥ 0.  (241)

In particular, for a fixed Γ^{∗}(s), α(s) is the unique solution of (221) and (241). Thus we may as well consider the totality of pairs (α, Γ^{∗}) and define

𝔖(p) = {(α(s), Γ^{∗}(s)); s ≥ 0, Γ^{∗}(0) = p, α(0) = 0, α′(0) ≥ 0} (242)

as the solutions of a specific initial value problem for the system ODE^{∗}, as indicated.

For the calculation of the sets (242) we may assume (by symmetry) the initial shape p belongs to the fundamental chamber ℭ_{0}, namely p is either its vertex ê_{3} or p̂_{0}, see (234). Observe that the set (242) has the induced symmetry group 𝔊_{p} which is the ”isotropy” subgroup at p of 𝔊 (cf. (232))

𝔊ê_{3} = {1, r̄} × {1, τ̄} ≃ 𝔇_{1} × ℤ_{2}, 𝔊p̂_{0} = 𝔇_{3} × ℤ_{2},  (243)

where r̄ ∈ 𝔇_{3} is the reflection θ → −θ.

Remark 54. Contrary to the above, the initial value problem for (221) at a cusp or binary collision is not well defined since we would have α′(s_{i}) = ±∞. Therefore, one cannot continue a solution (α(s), Γ^{∗}(s)) across the singularity using only the system ODE^{∗}. To circumvent the problem one should, for example, turn to the moduli curve Γ̄ itself, which is regular in any case. In Section 8.4.2 below we shall use Newton’s equation of motion more directly to develop in time a specific solution through several cusps.

8.4. Isosceles and collinear triple collision motions. In this section we take the opportunity to illustrate the above approach applied to the ”simplest” type of triple collision motions apart from the shape invariant ones. We shall also supply with numerical calculations, for comparison reasons and illustration of examples only.

Namely, consider the possibility that the shape curve Γ^{∗} of a triple collision motion is confined to a great circle on the sphere, that is, K_{g}^{∗}(s) = 0 for all s. Then the curvature equation in (221) forces the normal derivative of U^{∗} along Γ^{∗} to vanish. Hence, by (228) and Remark 52, Γ^{∗} ”moves” either on the equator circle (φ = π/2) or on one of the six meridians (229) passing through an Euler point ê_{i} or a binary collision point b̂_{i}. These meridians represent the shape of isosceles triangles, so the associated 3-body motions are either of collinear type or isosceles triangle type.

By the symmetry reduction explained in Section 8.3.1 it suffices to consider three separate cases, namely Γ^{∗} is (initially) confined to one of the boundary arcs of the fundamental chamber (233). We list the starting point (at the triple collision) and a choice of arc-length parameter s to be used (up to the first cusp or binary collision point) in each case:

(1) p̂_{0} : θ = 0, s = φ ≥ 0,
(2) ê_{3} : φ = π/2, s = θ ≥ 0,  (244)
(3) p̂_{0} : θ = π/3, s = φ ≥ 0.

8.4.1. The inclination angle α and its ODE. We shall investigate the first equation of (221)

\frac{dα}{ds} = −\frac{1}{4} + \frac{1}{2} cot(α)D_{i}(s) , i = 1, 2, 3  (245)

for each of the three cases (244), where D_{i}(s) is calculated from the appropriate expression of U^{∗} = U^{∗}(φ, θ) = U^{∗}(φ, θ + 2π/3), namely (cf. (224))

U^{∗}(φ, 0) = \frac{1}{3}(\frac{2}{\sqrt{1 − \frac{1}{2} sin φ}} + \frac{1}{\sqrt{1 + sin φ}}),  (246)

U^{∗}(π/2, θ) = \frac{1}{3}(\frac{1}{\sqrt{1 − cos(θ − \frac{π}{3})}} + \frac{1}{\sqrt{1 − cos(θ + \frac{π}{3})}} + \frac{1}{\sqrt{1 + cos θ}})
U^{∗}(φ, π/3) = U^{∗}(−φ, 0) = U^{∗}(φ, π).

As will be demonstrated below, there is a unique solution of the initial value problem (241).

The derivatives D_{i} are the following analytic functions expanded at the point p̂_{0} in case (1) and (3), and ê_{3} in case (2):

D_{1}(φ) = \frac{\frac{d}{dφ}U^{∗}(φ, 0)}{U^{∗}(φ, 0)} = φ(\frac{3}{8} − \frac{15}{64}φ + \frac{23}{256}φ^{2} − …),
D_{2}(θ) = \frac{\frac{d}{dθ}U^{∗}(π/2, θ)}{U^{∗}(π/2, θ)} = θ(\frac{29}{20} + \frac{1801}{1200}θ^{2} + \frac{17569}{12000}θ^{4} + …), (247)
D_{3}(φ) = \frac{\frac{d}{dφ}U^{∗}(φ, π/3)}{U^{∗}(φ, π/3)} = φ(\frac{3}{8} + \frac{15}{64}φ + \frac{23}{256}φ^{2} + …).

It follows that D_{1}(φ) = −D_{3}(−φ) and hence case (3) of (244) can be subsumed under case (1) by using the range φ < 0 and moreover, α < 0 interpreted in accordance with (239). In fact, if α_{i}(φ) is the solution of (245) for i = 1, 3, then α_{3}(φ) = −α_{1}(−φ). Hence, we need only consider the cases (1) and (2) of (245).

To investigate the nature of the singularity α = 0, let us first approximate the functions D_{1}(φ), D_{2}(θ) and

cot α = α^{−1} − \frac{1}{3}α − \frac{1}{45}α^{3} + …,

by their first term \frac{3}{8}φ, \frac{29}{20}θ and α^{−1} respectively. Then the initial value problem α(0) = 0 for the simplified version of (245) has the two straight line solutions

(1) : α = a_{0}φ, a_{0} = \frac{±\sqrt{13} − 1}{8}  (248)
(2) : α = b_{0}θ, b_{0} = \frac{±\frac{1}{5} \sqrt{1185} − 1}{8}

found by solving a second order polynomial, namely

a_{0} = −\frac{1}{4} + \frac{3}{16} \frac{1}{a_{0}} , b_{0} = −\frac{1}{4} + \frac{29}{40} \frac{1}{b_{0}} .  (249)

However, the extended initial value condition (241) demands α to be initially increasing and hence selects the positive solution in (248) as the leading coefficient for the two types of triple collision, namely

Lagrange type: a_{0} = \frac{\sqrt{13} − 1}{8} ≈ 0.326,  (250)
Euler type: b_{0} = \frac{\frac{1}{5} \sqrt{1185} − 1}{8} ≈ 0.736.

Returning to the original equation (245) we can determine recursively the power series expansion of α,

(1) α = a_{0}φ(1 + a_{1}φ + a_{2}φ^{2} + …),  (251)
(2) α = b_{0}θ(1 + b_{1}θ + b_{2}θ^{2} + …)

which depends solely on the initial coefficients a_{0}, b_{0} in (250). For convenience we list (with a few decimals only) the first terms of the expansions:

(1) α ≈ φ(0.3257 − 0.0955φ + 0.0129φ^{2} − 0.0233φ^{3} + …),
(2) α = θ(0.7356 + 0.1941θ^{2} + 0.0487θ^{4} + …).

As indicated, in case (1) the series is alternating and in case (2) α(s) is an odd function since b_{1} = b_{3} = b_{5} = … = 0.

The above series can be easily developed and used with high accuracy for small φ (or θ). However, we shall only use it to calculate an initial value of α for some small φ (or θ) and then solve equation (245) by standard numerical procedures, on the maximal interval bounded by the first singularity in each direction. Namely, in case (1) we find

lim_{φ→−π/2} α(φ) = −π/2,  lim_{φ→−π/2} α′(φ) = ∞,  (252)

lim_{φ→φ_{1}}α(φ) = 0, lim_{φ→φ_{1}} α′(φ) = −∞, _{1} ≈ 1.876 ≈ 107.5^{∘}.

On the interval [0, φ_{1}], α(φ) increases up to its maximum at φ ≈ 1.2 and is thereafter decreasing, see Figure 10.

On the other hand, if we rotate by 180^{∘} the graph of α(φ) over the interval [−π/2, 0], then we obtain the graph of α(φ) on [0, π/2] for case (3) of (244). Finally, the graph of α(θ) in case (2) on the interval [0, π/3] is quite similar to that of case (3), see Figure 11. Its graph over [−π/3, π/3] is symmetric with respect to the origin.

As calculated above, the shape curve Γ^{∗} along the meridian θ = 0 reaches the first cusp at φ = φ_{1}, which is beyond ê_{3}. At the cusp the motion Γ^{∗}(s) changes its direction and continues northward and across ê_{3}. See Section 8.4.2 and the time development of this motion.

Here is a brief summary of the analysis of the preliminary cases (244) and the solution sets (242) to which they belong:

• The set 𝔖(ê_{3}) has only two solutions (α_{+}, Γ_{+}^{∗}), (α_{−}, Γ_{−}^{∗}), and they are equivalent modulo the isometric reflection r̄: θ → −θ belonging to 𝔇_{3} ∩ 𝔊ê_{3}. Let Γ^{∗}(θ), θ ∈ (−π/3, π/3), be the arc-length parametrization of the equator circle from b̂_{1} to b̂_{2}. There is an analytic function α(θ) so that the pair (α, Γ^{∗}) is a solution of (221) and moreover,

α_{−}(s) = −α(−s) for s = −θ ≥ 0; α_{+}(s) = α(s) for s = θ ≥ 0;
Γ_{−}^{∗}(s) = Γ^{∗}(−s) for s = −θ ≥ 0; Γ_{+}^{∗}(s) = Γ^{∗}(s) for s = θ ≥ 0.

• The set 𝔖(p̂_{0}) has a unique solution (α_{0}, Γ_{0}^{∗}) and (α_{π/3}, Γ_{π/3}^{∗}) in case (1) and (3) of (244), respectively. Consider also the solution

μ̄(α_{π/3}, Γ_{π/3}^{∗}) = (α_{π}, Γ_{π}^{∗}) ∈ 𝔖(p̂_{0})

along the meridian θ = π, obtained by applying the rotation μ̄ ∈ 𝔇_{3} : θ → θ + 2π/3, as indicated, and let Γ^{∗}(φ), φ ∈ (−π/2, π/2), be the arc-length parametrization of the half-circle (b̂_{3} → p̂_{0} → ê_{3}). There is an analytic function α(φ) so that the pair (α, Γ^{∗}) is a solution of (221) and moreover,

α_{π}(s) = −α(−s) for s = −φ ≥ 0; α_{0}(s) = α(s) for s = φ ≥ 0;
Γ_{π}^{∗}(s) = Γ^{∗}(−s) for s = −φ ≥ 0; Γ_{0}^{∗}(s) = Γ^{∗}(s) for s = φ ≥ 0.

8.4.2. Time dependence and Newton’s equation. We shall choose case (1) of (244) and compare the above approach using the system (221) with the time parametrized motion Γ(t) using Newton’s equation (1). Namely, in the xy-plane we consider the motion of three point masses of mass 1/3, symmetric with respect to the y-axis and with position vectors

a_{1} = (−x, y), a_{2} = (x, y), a_{3} = (0, −2y).

Newton’s equation (1) reads

ẍ = −\frac{1}{12} \frac{|x|}{x^{3}} − \frac{1}{3} \frac{x}{(x^{2} + 9y^{2})^{3/2}} , ÿ = −\frac{y}{(x^{2} + 9y^{2})^{3/2}} .  (253)

As initial condition at time t = 0, assume y = 0 (i.e. Γ^{∗} is at the Euler point ê_{3}) and moment of inertia I = ρ^{2} = 1. Moreover, let β denote the oriented angle from the positive x-axis to the initial velocity vector ȧ_{2}, whose length is denoted v. Assuming the total energy h = T − U vanishes, the initial condition now reads

(x, y)|_{t=0} = (\sqrt{\frac{3}{2}}, 0),  (ẋ, ẏ)|_{t=0} = v(cos β, sin β)  (254)

v = \sqrt{\frac{\frac{5}{6} \sqrt{\frac{2}{3}}}{1 + 2 sin^{2} β}} .

Finally, let us assume 0 < β < π, which means the shape curve Γ^{∗} is heading southwards from ê_{3} for small t > 0.

The above data specify a 1-parameter family (parametrized by β) of isosceles 3-body motions Γ(t) = (x(t), y(t)) with total energy h = 0, with normalized size I = 1 and collinear shape ê_{3} at time t = 0. We shall use the equation (253) to investigate the time dependence of the various geometric and kinematic quantities of the motion, such as I, T, φ, α, where

I = \frac{2}{3}x^{2} + 2y^{2}, cos φ = ±\frac{4}{3\sqrt{3}} \frac{Δ}{I} = \frac{−4}{\sqrt{3}} \frac{xy}{\frac{2}{3}x^{2} + 2y^{2}}  (255)
T = U = \frac{1}{9}(\frac{2}{|x|} + \frac{2}{\sqrt{x^{2} + 9y^{2}}} ) = \frac{1}{2}(ρ̇^{2} + \frac{ρ^{2}}{4} φ̇^{2}).

By definition of the inclination angle α,

cos α = \frac{\frac{∂}{∂ρ} · \frac{d}{dt} Γ̄(t)}{|\frac{d}{dt} Γ̄(t)|} = \frac{ρ̇}{\sqrt{2T}} = \frac{\frac{2}{3}xẋ + 2yẏ}{\sqrt{2IU}} ,  (256)

and denoting the angle at t = 0 by α_{0} we deduce

cos α_{0} = \frac{3}{\sqrt{5}}(\frac{2}{3})^{1/4}v cos β = \begin{matrix}cos & β \\ √ & \overline{1 + 2 sin^{2} β} \end{matrix} .  (257)

Thus, the correspondence β ⟷ α_{0} is a bijection of the interval (0, π) such that π − β corresponds to π − α_{0}.

We claim there are exactly two values of β leading to a triple collision motion, namely the pair β_{0} and π − β_{0} for some β_{0} < π/2. This choice of β = β_{0} yields ρ̇(0) > 0, and hence the triple collision occurred in the past, namely at some negative time t_{0} < 0, with the shape curve Γ^{∗} at the north pole p̂_{0}. Similarly, using β = π − β_{0} the triple collision is reached at time −t_{0} > 0 with Γ^{∗} at the south pole.

The angle β_{0} is calculated using the formula (257), where α_{0} = α(π/2) and α(φ) is the solution of the equation

\frac{dα}{dφ} = −1/4 + \frac{φ}{2} cot(α)(\frac{3}{8} − \frac{15}{64}φ + \frac{23}{256}φ^{2} − …)  (258)

with initial condition α(0) = 0, cf. case (1) of (245) and (247). The solution found by the approach in Section 8.4.1, is approximately

α_{0} = α(π/2) ≈ 0.18673 …, β_{0} = 0.10865 ….

Thus, we know the initial data (254) corresponding to triple collision motions. As a test, by running the system (253) backwards in time one will find that the triple collision occurs approximately at t_{0} = −1.0228 …

It is also interesting to follow the shape curve Γ^{∗} of Γ(t) for t > 0, for example, using the ratio y(t)/x(t) or calculating φ(t) directly from (255). The solution Γ(t) can be continued in time t through the cusps since they are not singularities for Newton’s equation, and only truncation errors or numerical instability may invalidate the calculation in the long run. At t_{1} ≈ 10.4, y/x is maximal and φ(t_{1}) = φ_{1} (cf. (252)) is the colatitude of the first cusp - here α = 0 and Γ^{∗}(t) turns northward. After passing ê_{3} there is a second cusp where Γ^{∗} turns southward again and crosses ê_{3}, but the next cusp is closer to ê_{3}, and so on. Concerning the possible asymptotic behavior of Γ^{∗}(t), at triple collision or as t → ∞, see also Section 8.6.3.

Finally, we consider the time dependent size function ρ(t) = \sqrt{I(t)} and compare it with the integral formula (219). We have, by assumption, ρ = 1 at time t = 0, and Newton’s equation (253) yields, for example,

ρ(−0.5) ≈ 0.634726, φ(−0.5) = φ̂ ≈ 1.381793.

On the other hand, numerical integration of the solution α(φ) of (258) yields

ρ|_{φ=φ̂} = e^{\frac{1}{2} ∫ φ̂ ; 1 cot αdφ} ≈ 0.634725.  (259)

Alternatively, let us also evaluate this integral using the time parametrized function α(t) calculated by (256) and developed via Newton’s equation. Thus we change the variable φ in (259) to t using (255), namely

dφ = φ′(t)dt = (\frac{d}{dt} arccos(−\frac{4x(t)y(t)}{\sqrt{3}(\frac{2}{3}x(t)^{2} + 2y(t)^{2})}))dt,

and then numerical integration similar to the case (259) yields

ρ(−0.5) = e^{\frac{1}{2} ∫_{0}^{−0.5} cot(α)φ′(t)dt} ≈ 0.634726.

8.5. Analytic uniqueness of triple collision motions . In this section we turn to the full family 𝔖(p̂_{0}) of triple collision solutions (α, Γ^{∗}) of the system ODE^{∗}, with Γ^{∗} starting out from p̂_{0}. Due to the symmetry group 𝔇_{3} we may assume the shape curve Γ^{∗} enters the fundamental chamber ℭ_{0}, which limits the initial direction θ_{0} to the range [0, π/3]. Therefore, in terms of spherical coordinates (φ, θ) the appropriate and complete initial condition (242) now reads

α(0) = 0, α′(0) ≥ 0; φ(0) = 0, θ(0) = θ_{0}, 0 ≤ θ_{0} ≤ π/3.

The border cases θ_{0} = 0, π/3 are the cases (1) and (3) of (244) already investigated in Section 8.4, and now it is natural to generalize the procedure used there to the whole range of angles θ_{0}. This time, however, the strength of the curvature equation of (221) must be fully utilized. At this point, we assume (tentatively) that the functions α, φ, θ have power series expansions at p̂_{0}, necessarily of type

φ = s(c_{0} + c_{1}s + c_{2}s^{2} + c_{3}s^{3} + … +)
θ = θ_{0} + s(d_{0} + d_{1}s + d_{2}s^{2} + … +), 0 ≤ θ_{0} ≤ \frac{π}{3} ,  (260)
α = a_{0}s(1 + a_{1}s + a_{2}s^{2} + … +).

Recall from Section 8.1, we have excluded the trivial case of constant shape, and hence α does not vanish identically.To justify the notation in the third line, it will be demonstrated below that the leading term is a_{0}s with a_{0} ≠ 0.

By considering the leading coefficients of the series for φ and θ the third equation of ODE^{∗} implies

c_{0} = 1, c_{1} = 0, c_{2} = −\frac{1}{6}d_{0}^{2} ≤ 0  (261)

and clearly

(sin φ)θ′ = ε(1 − φ^{′2})^{1/2} = ε(−6c_{2}s^{2} + … +)^{1/2}, ε = ±1.  (262)

The calculation of a_{0} in the expansion α = a_{0}s + … is really the same as in Section 8.4.1 and gives the same value (250) independent of θ_{0}. To see this, we write for clarity the first equation of ODE^{∗} as

α(2α′ + \frac{1}{2}) = (α cot α)D(s),  (263)

where

α cot α = 1 − \frac{1}{3}α^{2} − \frac{1}{45}α^{4} − \frac{2}{945}α^{6} − \frac{1}{4725}α^{8} + O(α^{10}),

and the potential function (226) and its logarithmic derivative along Γ^{∗} have the expansions

u(s) = U^{∗}(Γ^{∗}(s)) = ∑_{i=0}^{∞} u_{i}s^{i} = 1 + \frac{3}{16}s^{2} − \frac{5}{64}(cos 3θ_{0})s^{3} + …, (264)

D(s) = \frac{d}{ds} ln(u) = \frac{u′(s)}{u(s)} = \frac{1}{u}(\frac{∂U^{∗}}{∂φ} φ′ + \frac{∂U^{∗}}{∂θ} θ′)
= s ∑_{i=0}^{∞} μ_{i}s^{i} = s(\frac{3}{8} − (\frac{15}{64} cos 3θ_{0})s + …).  (265)

In particular, the first order term in (265) is independent of θ_{0} and the leading terms of (263) yield the single condition

4a_{0}^{2} + a_{0} − 3/4 = 0 , with positive root : a_{0} = \frac{\sqrt{13} − 1}{8}.  (266)

The identity (263) also provides recursive relations for the calculation of a_{k}, k > 0, expressed in terms of the coefficients μ_{i}, i ≤ k, see (277) below.

8.5.1. The method of undetermined coefficients. The proof of Theorem G_{1}, concerning the existence and uniqueness of the curves, is based upon formal power series substitution for the three functions (260) involved in ODE^{∗}. This leads to a recursive procedure - the method of undetermined coefficients - which is consistent and determines successively the higher order coefficients in terms of c_{2}, d_{0} and θ_{0}. By (261) c_{2} is already determined by d_{0}, and at the final stage we shall find that d_{0} is actually determined by θ_{0}. Consequently, the expansions in (260) are, indeed, determined by the initial angle θ_{0} alone and hence θ_{0} parametrizes the whole solution set 𝔖(p̂_{0}).

On the 2-sphere there is the positive, orthonormal frame \{\frac{∂}{∂φ}, \frac{1}{sin φ} \frac{∂}{∂θ}\} associated with the coordinates φ, θ. Along the oriented shape curve Γ^{∗} we also have the positive, orthonormal moving frame {τ^{∗}, ν^{∗}}, where τ^{∗} is the tangent vector. The latter frame differs from the stationary frame by a rotation angle β, namely in analogy with (206) - (207),

τ^{∗} = cos β \frac{∂}{∂φ} + \begin{matrix}sin & β \\ sin & φ \end{matrix} \frac{∂}{∂θ}, ν^{∗} = − sin β \frac{∂}{∂φ} + \begin{matrix}cos & β \\ sin & φ \end{matrix} \frac{∂}{∂θ}  (267)
cos β = φ′, sin β = (sin φ)θ′ = ε(1 − φ′^{2})^{1/2}, cf. (262).  (268)

Now, we turn to the second equation of (221), namely the curvature equation written as an identity

LHS = RHS  (269)

between the left hand and right hand side

LHS = (1 − cos 2α)K_{g}^{∗}U^{∗}  (270)
RHS = U^{∗}(∇U^{∗} · ν^{∗}) = − sin β \frac{∂U^{∗}}{∂φ} + \begin{matrix}cos & β \\ sin & φ \end{matrix} \frac{∂U^{∗}}{∂θ} .

In LHS the geodesic curvature term decomposes as

K_{g}^{∗} = \frac{dβ}{ds} + cos φ\frac{dθ}{ds} = −ε(1 − φ^{′2})^{−1/2}φ′′ + (cos φ)θ′,  (271)

where β′ is calculated using (267). We have actually ε = 1, see the Remark below.

By substituting the first expression for sin β in (267) into RHS and comparing the leading terms of the expansions of LHS and RHS, it follows that either c_{2} ≠ 0 or all c_{i} = 0 for i > 1, and that c_{2} = 0 implies sin 3θ_{0} = 0. Thus, c_{2} = 0 means θ_{0} = 0 or π/3, that is, the two meridian solutions of isosceles triangle type already discussed in Section 8.4.

Henceforth, we shall assume c_{2} ≠ 0. Comparison of the leading terms (of order 2) in (269) yields

4a_{0}^{2}d_{0} = −\frac{3}{8}d_{0} + \frac{15}{64} sin 3θ_{0}

which combined with (261) gives

d_{0} = \frac{15}{16} \frac{sin 3θ_{0}}{(16a_{0}^{2} + \frac{3}{2})}, c_{2} = −\frac{75}{512} \frac{sin^{2} 3θ_{0}}{(16a_{0}^{2} + \frac{3}{2})^{2}}  (272)

with the approximate values

d_{0} ≈ 0.293 sin 3θ_{0}, c_{2} ≈ −0.014 sin^{2} 3θ_{0}.

Remark 55. In particular, in (262) we have ε = sgn(d_{0}) = 1, and the expressions in (272) are, in fact, valid in the whole closed chamber ℭ_{0} : 0 ≤ θ_{0} ≤ π/3. However, ε = ±1 actually changes sign across the border meridian of two neighboring chambers.

In view of (267), (271) we also need the expansions

(1 − φ^{′2})^{1/2} = d_{0}s(1 + b_{1}s + b_{2}s^{2} + …),
(1 − φ^{′2})^{−1/2} = \frac{1}{d_{0}s}(1 + b̄_{1}s + b̄_{2}s^{2} + …),

where by writing c̃_{k} = c_{k}/c_{2} for k ≥ 3 we have by simple inspection

b_{n} = \frac{n + 3}{6}c̃_{n+2} + B_{n}(c̃_{3}, …, c̃_{n+1}), n ≥ 1  (273)
b̄_{n} = −\frac{n + 3}{6}c̃_{n+2} + B̄_{n}(c̃_{3}, …, c̃_{n+1}), n ≥ 1

where B_{n} and B̄_{n} are polynomials and B_{1} = B̄_{1} = 0.

For simplicity, let P (y_{1}, y_{2}, …) denote any (unspecified) polynomial in the variables y_{i}, except that y_{1} = θ_{0} means it is polynomial in sin 3θ_{0} and cos 3θ_{0}. Using the notation

sin φ = s(1 + g_{2}s^{2} + g_{3}s^{3} + …), g_{k} = c_{k} + P (c_{2}, …, c_{k−1}) (274)
cos φ = 1 − \frac{1}{2}s^{2} + h_{4}s^{4} + h_{5}s^{5} + …, h_{k} = P (c_{2}, …, c_{k−2})

we derive from (262) the following formula for d_{n}

(n + 1)d_{n} = d_{0}b_{n} − (d_{0}g_{n} + 2d_{1}g_{n−1} + … + (n − 1)d_{n−2}g_{2}), n ≥ 1, (275)

and from (271) we calculate the curvature expansion

K_{g}^{∗} = 2d_{0}(1 + k_{1}s + k_{2}s^{2} + …),

where

2d_{0}k_{n} = (d_{0}b̄_{n} + (n + 1)d_{n} − (n + 3)(n + 2)\frac{c_{n+2}}{d_{0}} )  (276)

− \frac{1}{d_{0}} ∑_{k=3}^{n+1}(k + 1)kc_{k}b̄_{n+2−k} + ∑_{k=0}^{n−1}(k + 1)d_{k}h_{n−k}.

Next, let us have a closer look at the coefficients of u(s) and u′(s)/u(s), using (226), (264), (265), and also at the recursive generation of the coefficients of α(s) using (263). It follows that they are of type

u_{n} = P (θ_{0}, c_{2}, …, c_{n−2}, d_{0}, …, d_{n−4}), n ≥ 4
μ_{n} = P (θ_{0}, c_{2}, …, c_{n}, d_{0}, …, d_{n−2}), n ≥ 2  (277)
a_{n} = P (a_{1}, a_{2}, …, a_{n−1}, μ_{1}, …, μ_{n}), n ≥ 1.

For example,

a_{1} = \frac{2μ_{1}}{12a_{0}^{2} + a_{0}} = \frac{8μ_{1}}{10 − \sqrt{13}} = −\frac{15/8}{10 − \sqrt{13}} cos 3θ_{0}.

For convenience, write

1 − cos 2α = 2a_{0}^{2}s^{2}(1 + A_{1}s + A_{2}s^{2} + …),  (278)

where

A_{1} = 2a_{1}, A_{2} = 2a_{2} + a_{1}^{2} − \frac{1}{3}a_{0}^{2}, …, A_{n} = P (a_{1}, a_{2}, …, a_{n}),

and thus we arrive at the following presentation of LHS as a product of series

LHS = 4a_{0}^{2}d_{0}s^{2}(1 + A_{1}s + …)(1 + k_{1}s + …)(1 + u_{1}s + …). (279)

From the structure of U^{∗} as a trigonometric series (226) in the variables sin φ and cos 3θ, we can write

\frac{∂U^{∗}}{∂φ} = (sin φ cos φ)R_{1}, \frac{∂U^{∗}}{∂θ} = (sin^{3} φ sin 3θ)R_{2},  (280)

where

R_{1} = \frac{3}{8} − \frac{15}{64} sin φ cos 3θ + …, R_{2} = \frac{15}{64} + \frac{945}{4096} sin^{2} φ + …

are again polynomial series in the variables sin φ, cos 3θ. Hence, by substituting the series of sin φ, cos φ, θ′, φ′, cos 3θ, sin 3θ into the expression

RHS = (sin^{2} φ) (−θ′ cos φR_{1} + φ′ sin 3θR_{2})  (281)

equation (269) renders a recursive procedure for the calculation of c_{n} and d_{n−2}, n ≥ 2, starting from c_{2} and d_{0} (272).

Lemma 56. The coefficients c_{m} and d_{m−2} can be expressed as

c_{m} = P (θ_{0})c_{2}, d_{m−2} = P (θ_{0})d_{0}; m ≥ 2  (282)

where P (θ_{0}) denotes some polynomial of sin 3θ_{0} and cos 3θ_{0} (generally different for each coefficient).

Proof. The first step is to compare terms of order 3 in (269), which renders the identity

4a_{0}^{2}d_{0}(A_{1} + k_{1} + u_{1}) = −\frac{3}{4}d_{1} + \frac{15}{64}d_{0} cos 3θ_{0},

where

A_{1} = 2a_{1}, k_{1} = \frac{c_{3}}{c_{2}} , u_{1} = 0, d_{1} = \frac{d_{0}c_{3}}{3c_{2}} .

Consequently,

c_{3} = \frac{15(16a_{0} + 1)}{4(16a_{0}^{2} + 1)(12a_{0} + 1)}(cos 3θ_{0})c_{2},  (283)
d_{1} = \frac{15(16a_{0} + 1)}{12(16a_{0}^{2} + 1)(12a_{0} + 1)}(cos 3θ_{0})d_{0}.

We proceed by induction and assume that (282) holds for m in the range 2 ≤ m ≤ n. By (273) - (278), we infer that b_{m−2}, b̄_{m−2}, k_{m−2}, g_{m}, h_{m}, u_{m}, μ_{m} and a_{m} are all of type P (θ_{0}) for m ≤ n. Furthermore,

nd_{n−1} = (\frac{n + 2}{6c_{2}}c_{n+1} + P (θ_{0})) d_{0},  (284)
k_{n−1} = \frac{(n + 1)(n + 2)}{12c_{2}}c_{n+1} + P (θ_{0}).

Consider the terms of order n + 1 in equation (269). By equating the coefficients of s^{n+1} in LHS and RHS we deduce

4a_{0}^{2}d_{0}k_{n−1} + P (θ_{0})d_{0} = −\frac{3}{8}nd_{n−1} + P (θ_{0})d_{0}.  (285)

Here, k_{n−1} and d_{n−1} are the only coefficients depending on c_{n+1}, and by substituting their expressions from (284) into the identity (285), we deduce the identity

(\frac{4a_{0}^{2}(n + 1)(n + 2)}{12} + \frac{n + 2}{16}) c_{n+1} = P (θ_{0})c_{2},

and consequently,

c_{n+1} = P (θ_{0})c_{2}, d_{n−1} = P (θ_{0})d_{0}.

This completes the induction step, and hence (282) holds for all m ≥ 2.

This settles the existence and uniqueness question for the series expansions (260), for each initial longitude angle θ_{0}. Their radius of convergence is certainly positive (e.g. by an inductive argument showing the coefficients are bounded), but we shall not try to estimate the radius here. Clearly, for θ_{0} = kπ/3 the radius is at most π/2.

8.5.2. Symmetries of the solution set 𝔖(p̂_{0}). In the previous subsection it was established that the triple collision solution set (242) for p = p̂_{0} is naturally parametrized by angles θ_{0}, namely

𝔖(p̂_{0}) = {(α_{θ_{0}}, Γ_{θ_{0}}^{∗}); 0 ≤ θ_{0} < 2π}  (286)

is in 1-1 correspondence with points on a circle and therefore inherits ”symmetries” of a circle. However, the actual symmetry group should act with orbits representing the various ”species’ or ”congruence” classes of solutions, and moreover, knowledge of each class suffices to generate all solutions by a straightforward transformation procedure. We contend that 𝔊p̂_{0}defined in (243) is, in fact, the appropriate group.

First of all, 𝔊p̂_{0} contains the group 𝔇_{3} which acts on the 2-sphere and represents the purely geometric symmetries. On the other hand, we have also seen that each solution curve Γ^{∗} = Γ_{θ_{0}}^{∗} corresponds to three analytic functions (α(s), φ(s), θ(s)) in a neighborhood of s = 0, and for s < 0 these functions also describe a motion approaching a triple collision as s → 0^{−}. Hence, by inverting its direction we should obtain a triple collision motion emanating with initial longitude angle θ_{0} + π, which by uniqueness must be the solution in (286) labeled by θ_{0} + π. Consequently, in agreement with the summary of Section 8.4.1, for each ”antipodal” pair (α_{θ_{0}}, Γ_{θ_{0}}^{∗}), (α_{θ_{0}+π}, Γ_{θ_{0}+π}^{∗}) in (286) there is an analytic curve Γ^{∗}(s) passing through p̂_{0} and an analytic function α(s) so that (α(s), Γ^{∗}(s)) is a solution of the system (221), and moreover,

α_{θ_{0}+π}(s) = −α(−s) and α_{θ_{0}}(s) = α(s) for s ≥ 0,
Γ_{θ_{0}+π}^{∗}(s) = Γ^{∗}(−s) and Γ_{θ_{0}}^{∗}(s) = Γ^{∗}(s) for s ≥ 0.

In particular, the inversion operator τ̄ applied to (α(s), Γ^{∗}(s)) induces an involution

τ̄ : (α_{θ_{0}}, Γ_{θ_{0}}^{∗}) → (α_{θ_{0}+π}, Γ_{θ_{0}+π}^{∗})

of the set 𝔖(p̂_{0}) which commutes with the action of 𝔇_{3}, and together they generate the dihedral symmetry group

𝔊p̂_{0} = 𝔇_{3} × {1, τ̄} ≃ 𝔇_{6}  (287)

which may be viewed as an ”isotropy” subgroup of 𝔊 in (232). In effect, this divides the fundamental chamber ℭ_{0} (233) in two sectors of angular width π/6, say

ℭ̃_{0} : 0 ≤ θ_{0} ≤ π/6

is our reduced fundamental chamber. However, we remark that 𝔊p̂_{0} does not act on S^{2}, so ℭ̄_{0} is not a fundamental domain in the geometric sense. But solutions starting out in this region suffice to generate the whole solution set (286) using analytic continuation.

The power series developments (260) of the three functions α(s), φ(s), θ(s), where Γ_{θ_{0}}^{∗}(s) = (φ(s), θ(s)) is the shape curve with initial direction θ_{0} at the north pole, also exhibit a specific symmetry pattern which reflects the 𝔇_{6}-symmetry of their coefficients. For example, consider the ”reflection” in 𝔊p̂_{0}

θ_{0} → π/3 − θ_{0}

which divides ℭ_{0} into two reduced chambers, and write

X = cos 3θ_{0}, Y = sin 3θ_{0}
a_{k} = A_{k}(X, Y) c_{k} = C_{k}(X, Y)c_{2}, d_{k} = D_{k}(X, Y)d_{0},

where A_{k},C_{k},D_{k} are polynomials of two variables (not unique, of course, since X, Y are algebraic dependent). Let a_{k}, c_{k}, d_{k} and ā_{k}, c̄_{k}, d̄_{k} be the coefficients of the solutions (α, φ, θ), (ᾱ, φ̄, θ̄) corresponding to initial angles θ_{0} and π/3 − θ_{0}, respectively. Then we have

\{ \begin{matrix}a_{k} = ā_{k}, c_{k} = c̄_{k}, d_{k} = d̄_{k} for k even \\ a_{k} = −ā_{k}, c_{k} = −c̄_{k}, d_{k} = −d̄_{k} for k odd \end{matrix}

Equivalently, as functions of X the polynomials A_{k},C_{k},D_{k} are odd (respectively even) functions for k odd (respectively even). This is due to the fact that Y is invariant whereas X changes sign under the substitution θ_{0} → π/3 − θ_{0}.

8.5.3. Symbolic manipulations and numerical calculation of power series. The recursive scheme used in Section 8.5.1 will generate all higher order coefficients as polynomials of Y = sin 3θ_{0} and X = cos 3θ_{0}, but the explicit calculations involve a substantial amount of symbolic manipulations. For example, various types of algebraic operations, together with composition, are applied to power series.

In principle, calculations involving elementary functions of power series, such as sin(∑ p_{i}x^{i}), can be reduced to symbolic manipulations on power series of the type

(p_{0} + p_{1}x + p_{2}x^{2} + …)^{n} = ∑_{k=0}^{∞} P_{k}^{(n)}x^{k},

where the n-th multinomial polynomial P_{k}^{(n)} records the n-partitions and associated multinomial coefficients which can be calculated recursively with some effort.

On the other hand, available computer software developed for symbolic computation have built-in procedures which effectively generate the intermediate power series expansions as well as recursive formulas. We have employed such symbolic software for the calculation^{1} of a_{k}, c_{k}, d_{k} in (260), for small k, see also Section 8.7. For convenience, we list the first of them below (omitting the already known c_{2}, d_{0}, a_{0}), and we remark that the exact (or symbolic) expressions are growing fast in complexity as k increases:

1The symbolic and numerical calculations were performed in 1995 by Chee-Whye Chin, an undergraduate student at U.C. Berkeley, using Mathematica software.

c_{3} = c_{2} \frac{15(10 + \sqrt{13})}{116}(cos 3θ_{0}),  (288)

c_{4} = c_{2} (\frac{2040762505 + 136353812\sqrt{13}}{2891425280} ,

+\frac{20984375(113 + 20\sqrt{13})}{2891425280}(cos 6θ_{0})) ,

d_{1} = d_{0} \frac{5(10 + \sqrt{13})}{116}(cos 3θ_{0}),

d_{2} = d_{0} (\frac{2(19733316 + 84347\sqrt{13})}{216856896} + \frac{302175(113 + 20\sqrt{13})}{216856896}(cos 6θ_{0})) ,

a_{1} = \frac{−5(10 + \sqrt{13})}{232}(cos 3θ_{0}),

a_{2} = \frac{3(28004 + 4175\sqrt{13})}{2745024} + \frac{−25(644 + 47\sqrt{13})}{2745024}(cos 6θ_{0}),

+ \frac{4350(9 + \sqrt{13})}{2745024}\sqrt{113 + 20\sqrt{13}}(sin^{2} 3θ_{0}).

8.6. Global behavior of the shape of triple collision motions. First we shall investigate the curvature properties of the flow consisting of the gradient lines of the potential function U^{∗} on the sphere S^{2}(1). This information will be related to the curvature properties of the ”flow” consisting of those curves Γ^{∗} belonging to the set (286), that is, the triple collision shape curves emanating from the north pole p̂_{0}.

8.6.1. Differential geometry of the gradient flow of U^{∗}. We start with the following elementary result about curves on the unit sphere S^{2} in Euclidean 3-space.

Lemma 57. Let t → p(t) be a parametrized curve on S^{2}. Then its geodesic curvature is given by the following triple product

K_{g}(t) = (\frac{dt}{ds})^{3}p × ṗ · p̈ = p × p′ · p′′,  (289)

where (as usual) s is arc-length, ṗ =\frac{d}{dt}p and p′ = \frac{d}{ds}p.

Proof. The unit tangent vector τ^{∗} = p′ points in the positive direction of the curve, and p′′ is the curvature vector in 3-space. With ν^{∗} = p × τ^{∗} as the normal vector field along the curve, {τ^{∗}, ν^{∗}} is a positively oriented frame of the sphere. By definition, the geodesic curvature vector in the sphere is the orthogonal projection of p′′ into the tangent plane, namely

p′′ − (p′′ · p)p = K_{g}ν^{∗},

where the coefficient K_{g} is the (scalar) geodesic curvature. Clearly, K_{g} equals the triple product in (289).

Next, we turn to the gradient field ∇U^{∗} on the sphere, whose integral curves will be referred to as the gradient lines of U^{∗}. Their geodesic curvature will be denoted by K_{g}. Until further notice there is no restriction on the mass distribution, and we use the expressions (171), (175) and (176) for U^{∗}, the vector function B(p) and the gradient ∇U^{∗} field, respectively.

Lemma 58. The geodesic curvature of the gradient line of U^{∗} passing through p is given by the following triple product

K_{g}(p) = \frac{3}{|∇U^{∗}(p)|^{3}} B × C · p,  (290)

where B = B(p) and

C = C(p) = ∑_{i=1}^{3} \frac{k_{i}}{|p − b̂_{i}|^{5}} [p × B · (p × b̂_{i})] b̂_{i}.  (291)

Proof. Consider a gradient line parametrized by arc-length, s → p(s). We can write

∇U^{∗}(p) = f(s)p′(s),

where f(s) = |∇U^{∗}(p(s))|, and hence by (176)

p′ = \frac{B − (B · p)p}{|∇U^{∗}|}.

On the other hand, 0 = p × p′ · ∇U^{∗} = p × p′ · B and by differentiation

p × p′′ · B = −p × p′ · \frac{d}{ds}B.  (292)

Hence, by substituting the expression

B = |∇U^{∗}| p′ + (B · p)p

into the left side of (292), we obtain

|∇U^{∗}| p × p′ · p′′ = p × p′ · \frac{d}{ds}B = \frac{p × B}{|∇U^{∗}|} · \frac{d}{ds}B.  (293)

Finally, we calculate from (175)

\frac{d}{ds}B = 3 ∑_{i=1}^{3} k_{i} \frac{(b̂_{i} · p′)}{|p − b̂_{i}|^{5}} b̂_{i} = \frac{3}{|∇U^{∗}|} ∑ k_{i} \frac{(b̂_{i} × p) · (B × p)}{|p − b̂_{i}|^{5}} b̂_{i}

and by substituting this into (293) it follows from Lemma 57

K_{g} = p × p′ · p′′ = \frac{3}{|∇U^{∗}|^{3}} ∑ k_{i} \frac{(p × B) · (p × b̂_{i})}{|p − b̂_{i}|^{5}}(p × B · b̂_{i}).

This expression can be rewritten as (290).

Problem 59. Regarding K_{g} as a function on S^{2}, determine the curves defined by the condition K_{g}(p) = 0.

Remark 60. Note that B(p) and C(p) are vectors in the xy-plane in the Euclidean model M̄ = ℝ^{3}, and the function K_{g} on the sphere S^{2} : x^{2} +y^{2} +z^{2} = 1 is undefined precisely at the critical points of U^{∗}, namely for z ≥ 0 these are the points b̂_{i}, 0 ≤ i ≤ 3, and the minimumspoint (physical center) p̂_{0}^{+}. From the triple product formula (290) it follows that K_{g}(p) vanishes on the eclipse circle (z = 0), whereas for z > 0 K_{g}(p) vanishes if and only if B(p) and C(p) are linearly dependent.

Henceforth, we shall retain our assumption of uniform mass distribution, and a deeper understanding of the function K_{g} will be achieved. In fact, in this case the above problem has a simple solution, as explained at the end of this subsection.

By assumption,

m_{i} = \frac{1}{3}, k_{i} = \frac{\sqrt{2}}{3}, b̂_{i} · b̂_{j} = −\frac{1}{2} (i, j ≥ 1 and i ≠ j)

and we introduce the three distance functions

δ_{i} = δ_{i}(p) = |p − b̂_{i}| = \sqrt{2}\sqrt{1 − p · b̂_{i}}, i ≥ 1, cf. Section 6.1 (294)

which are algebraically related by

δ_{1}^{2} + δ_{2}^{2} + δ_{3}^{2} = 6,  (295)

due to the identity ∑ p · b̂_{i} = p· ∑ b̂_{i} = 0 and

p · b̂_{i} = 1 − \frac{1}{2}δ_{i}^{2}.  (296)

Let us write

B(p) = \frac{\sqrt{2}}{3} ∑_{i=1}^{3} e_{i}b̂_{i} = \frac{\sqrt{2}}{3} ((e_{1} − e_{3})b̂_{1} + (e_{2} − e_{3})b̂_{2}) ,  (297)

C(p) = \frac{1}{9} ∑_{i=1}^{3} f_{i}b̂_{i} = \frac{1}{9} ((f_{1} − f_{3})b̂_{1} + (f_{2} − f_{3})b̂_{2}) ,  (298)

where

e_{i} = \frac{1}{δ_{i}^{3}} , f_{i} = \frac{3\sqrt{2}}{δ_{i}^{5}}(p × B(p)) · (p × b̂_{i}).  (299)

To simplify our notation we denote products (monomials) of the functions δ_{i} by

δ_{a,b,c} = δ_{1}^{a}δ_{2}^{b}δ_{3}^{c},  (300)

where a, b, c are nonnegative integers, and the alternating polynomial generated by the monomial (300) is

A_{a,b,c} = | \begin{matrix}δ_{1}^{a} & δ_{2}^{a} & δ_{3}^{a} \\ δ_{1}^{b} & δ_{2}^{b} & δ_{3}^{b} \\ δ_{1}^{c} & δ_{2}^{c} & δ_{3}^{c} \end{matrix} | = ∑_{σ∈S_{3}} sgn(σ)δ_{a,b,c}^{σ},  (301)

where S_{3} is the permutation group of {δ_{1}, δ_{2}, δ_{3}} acting on monomials in the obvious way. In particular, the basic alternating polynomial is

A = A_{0,1,2} = | \begin{matrix}1 & 1 & 1 \\ δ_{1} & δ_{2} & δ_{3} \\ δ_{1}^{2} & δ_{2}^{2} & δ_{3}^{2} \end{matrix} | = (δ_{1} − δ_{2})(δ_{2} − δ_{3})(δ_{3} − δ_{1}).  (302)

On the other hand, the symmetric function generated by δ_{a,b,c}, where we may assume a ≥ b ≥ c, is the smallest S_{3}-invariant sum

S_{a,b,c} = ∑ δ_{i}^{a}δ_{j}^{b}δ_{k}^{c} = ∑_{σ∈S_{3}} δ_{a,b,c}^{σ}  (303)

containing δ_{a,b,c}. In particular, S_{2,0,0} = 6, by (295).

Note that S_{a,b,c} is unchanged, whereas A_{a,b,c} may change sign, when a, b, c are permuted. The alternating polynomials can be decomposed as a product of the basic alternating function (302) and a symmetric function, for example

A_{0,2,4} = (S_{2,1,0} + 2S_{1,1,1})A,
A_{0,1,5} = (S_{3,0,0} + S_{2,1,0} + S_{1,1,1})A,  (304)
A_{0,1,6} = (S_{4,0,0} + S_{3,1,0} + S_{2,2,0} + S_{2,1,1})A,
A_{0,3,6} = (S_{4,2,0} + S_{4,1,1} + 2S_{3,2,1} + 3S_{2,2,2})A.

The induced action of S_{3} on the triples {e_{1}, e_{2}, e_{3}} and {f_{1}, f_{2}, f_{3}} is covariant with the action on {δ_{1}, δ_{2}, δ_{3}}. Certainly, the above coefficients e_{i} in (299) are simple rational functions of the δ′_{j}s, and now we prove a similar statement for the f′_{i}s, as follows.

Lemma 61. As a rational function of δ_{1}, δ_{2}, δ_{3}

f_{1} = \frac{1}{δ_{6,6,6}} (−3δ_{1,3,6} − 3δ_{1,6,3} + 2δ_{0,6,6} + δ_{3,6,3} + δ_{3,3,6} + δ_{1,5,6}
+δ_{1,6,5} − \frac{1}{2}δ_{2,6,6} − \frac{1}{2}δ_{3,5,6} − \frac{1}{2}δ_{3,6,5}),

and f_{2} (respectively f_{3}) is obtained from f_{1} (respectively f_{2}) by cyclic permutation, a → b → c → a, of the indices of each monomial δ_{a,b,c}.

Proof. Use the identities (296) and substitute the expression (297) for B into the formula (299) for f_{1}, namely

f_{1} = \frac{3\sqrt{2}}{δ_{1}^{5}} (B · b̂_{1} − (p · b̂_{1})(B · p)) .

Then one obtains the above rational expression for f_{1} by straightforward calculations, and by symmetry it is also clear that f_{2} and f_{3} are obtained from f_{1} as claimed.

Now, turning to the formula (290) for the curvature function K_{g} and inserting the expressions (297), (298), we write

B(p) × C(p) · k =\frac{\sqrt{6}}{54}Ã,

where k is the unit normal vector of the xy-plane and Ã is, by definition, the function

Ã(δ_{1}, δ_{2}, δ_{3}) = (e_{1}f_{2} − e_{2}f_{1}) + (e_{2}f_{3} − e_{3}f_{2}) + (e_{3}f_{1} − e_{1}f_{3}). (305)

When the products e_{i}f_{j} are calculated using the above lemma, for example

e_{1}f_{2} = \frac{1}{δ_{6,6,6}} (−3δ_{3,1,3} − 3δ_{0,1,6} + 2δ_{3,0,6} + δ_{3,3,3} + δ_{3,1,5} + δ_{0,3,6}

+ δ_{2,1,6} − \frac{1}{2}δ_{3,2,6} − \frac{1}{2}δ_{3,3,5} − \frac{1}{2}δ_{2,3,6}),

the expression (305) may be written as

Ã = −\frac{1}{δ_{6,6,6}} (3A_{0,1,6} + A_{0,3,6} + A_{1,3,5} + A_{1,2,6}).  (306)

Finally, substitution of expressions from (304) into (306) yields

Ã(δ_{1}, δ_{2}, δ_{3}) = −S̃(δ_{1}, δ_{2}, δ_{3})A,

where

S̃(δ_{1}, δ_{2}, δ_{3}) = \frac{1}{(S_{1,1,1})^{6}} [3(S_{4,0,0} + S_{3,1,0} + S_{2,2,0} + S_{2,1,1}) + S_{4,2,0} + S_{4,1,1}
+ 2S_{3,2,1} + 3(S_{1,1,1})^{2} + 2S_{1,1,1}S_{2,1,0} + 3(S_{1,1,1})^{2} + S_{1,1,1}S_{3,0,0}].

In summary, we have established the following proposition, where the factor S̃ of K_{g} is always positive!

Proposition 62. The geodesic curvature function of the gradient lines of U^{∗} on the unit sphere S^{2} is given by the product

K_{g}(p) = −\frac{\sqrt{6} z S̃}{18 |∇U^{∗}(p)|^{3}} A  p = (x, y, z), z ≥ 0  (307)

Observe that the 𝔇_{3}-chambers of the (upper) hemisphere of S^{2} are defined by inequalities δ_{i} ≤ δ_{j} ≤ δ_{k}, and therefore A, and hence K_{g} as well, has constant sign in each chamber. For example, the fundamental chamber (233) is given by

ℭ_{0} : δ_{2} ≤ δ_{1} ≤ δ_{3}

and here K_{g} ≥ 0. The meridians which are the walls of the 𝔇_{3}-chambers are defined by relations of type δ_{i} = δ_{j}; these are the zero set of the function A. Together with the equator circle they are the curves where K_{g} vanishes (or is undefined), and this also solves Problem 59 (in the case of uniform mass distribution).

It is easy to visualize the gradient flow on the 2-sphere. For example, in the interior of the spherical triangle ℭ_{0} the flow has the vertex p̂_{0} as source and converges towards the vertex b̂_{2}, with positive curvature everywhere. The flow in ℭ_{0} is illustrated in Figure 9.

8.6.2. Geometry of the triple collision shape curves. It is possible to draw qualitative information about the family 𝔖(p̂_{0}) of shape curves by relating it with the geometry of the gradient flow of U^{∗}. By symmetry it suffices to consider those curves Γ_{θ_{0}}^{∗} starting out in the chamber ℭ_{0}, that is, 0 ≤ θ_{0} ≤ \frac{π}{3}, and we observe that the boundary meridians θ_{0} = 0 and θ_{0} = π/3, emanating from the north pole p̂_{0} towards the equator, are themselves both shape curves and gradient lines. So, the question is what one can say about the shape curves in the interior of ℭ_{0}?

A rough description goes as follows. In ℭ_{0} there are three different ”flows” of curves emanating from p̂_{0}, namely the shape curves, the gradient lines and the meridians (θ constant). At each point p = Γ^{∗}(s_{1}), s_{1} > 0, the shape curve Γ^{∗}(s), s > s_{1}, is ”trapped” between the gradient line and the meridian through p. Being positively curved one may imagine the shape curves arising by gradually bending the meridians towards the gradient lines by means of a ”force” field directed eastward, see Figure 12.

To be more precise, we shall focus on five properties as stated below. For this purpose we introduce two angular functions β(s), γ(s) as follows. Namely, β is the angle between the meridian and Γ^{∗}, as defined in (267). It is the oriented angle from \frac{∂}{∂φ} to the velocity vector τ^{∗} = \frac{d}{ds}Γ^{∗}, and γ is the angle from \frac{∂}{∂φ} to the gradient vector

∇U^{∗} = \frac{∂U^{∗}}{∂φ} \frac{∂}{∂φ} + \frac{1}{sin^{2} φ} \frac{∂U^{∗}}{∂θ} \frac{∂}{∂θ}

= |∇U^{∗}| (cos γ \frac{∂}{∂φ} + sin γ \frac{1}{sin φ} \frac{∂}{∂θ}).

We also recall the role of the inclination angle α(s), which is not directly related to the geometry of the spherical curve Γ^{∗} itself, but to the associated moduli curve Γ̄. However, the curvature equation from (221) can now be stated as

2U^{∗}K_{g}^{∗} sin^{2} α = |∇(U^{∗})| cos(\frac{π}{2} + β − γ)  (308)

and hence relates all three angles with the curvature of Γ^{∗}.

Now, we contend that the following properties are valid for Γ^{∗} = Γ_{θ_{0}}^{∗}, 0 < θ_{0} < \frac{π}{3}, at least until Γ^{∗} leaves ℭ_{0} the first time (but not necessarily later):

• (i) 0 < α < \frac{π}{2}, for s > 0. In particular, the curve s → Γ^{∗}(s) has no cusp singularity for s < π/2.

• (ii) The spherical coordinates φ(s), θ(s) of Γ^{∗}(s) are strictly increasing functions of s.

• (iii) 0 < β ≤ γ < \frac{π}{2}, for s > 0.

• (iv) The geodesic curvature K_{g}^{∗} of Γ^{∗}(s), s ≥ 0, is nonnegative.

• (v) Γ^{∗} leaves the chamber ℭ_{0} by crossing its boundary arc (ê_{3}, b̂_{2})

on the equator circle. In order to verify these statements one may proceed as follows. First,

note that 0 < γ < π/2 follows from the fact, due to (228) and Remark 52, that \frac{∂U^{∗}}{∂φ} > 0 and \frac{∂U^{∗}}{∂θ} > 0 inside ℭ_{0}, and moreover, the gradient lines emanate from p̂_{0} with γ = 0 and approach the binary collision point b̂_{2} with γ = π/2 in the limit. This also explains why property (v) follows from (iii), and using (267 and (308) we also readily deduce properties (ii) and (iv) from (iii). Thus, we are left with the statements (i) and (iii), and let us first establish property (iii) (using property (i) if necessary).

Observe that β ≥ 0 by (267) and Remark 55. But β = 0 for some s would imply θ′ = 0, φ′ = 1, and hence by (271) K_{g}^{∗}(s) would vanish. However, with p in the interior of ℭ_{0} we also have \frac{∂U^{∗}}{∂θ} (p) ≠ 0 and then γ > 0 in the right side of the identity (308). Consequently, K_{g}^{∗}(s) ≠ 0 and this contradiction shows β > 0 must hold. Again by (308), K_{g}^{∗} is positive as long as β < γ, and this certainly holds for small s since

K_{g}^{∗}(0) = d_{0} = \frac{15}{16} \frac{sin 3θ_{0}}{(16a_{0}^{2} + \frac{3}{2})}.

We claim that β ≤ γ holds (at least) until Γ^{∗}(s) leaves the chamber. To see this, suppose we had β = γ for s = s_{1}, β < γ (respectively β > γ) for s < s_{1} (respectively s > s_{1})and s close to s_{1}. Then Γ^{∗} would be tangent to the gradient line at p = Γ^{∗}(s_{1}) and is (locally) lying on the ”upper” side of it, hence K_{g}^{∗}(s_{1}) ≥ K_{g}(p) > 0. This contradicts the fact that K_{g}^{∗}(s_{1}) = 0, by (308).

Remark 63. Property (iii) implies that U^{∗} increases along the curve Γ^{∗}(s), s ≥ 0. In general, the event β = γ means Γ^{∗} is perpendicular to the level curve of U^{∗}, and by (308) this can happen for two reasons, namely i) K_{g}^{∗} vanishes or ii) Γ^{∗} reaches a cusp. We can rule out the second case due to property (i), but only up to the first crossing of the equator.

Finally, we turn to property (i). From the relations

\frac{dρ}{ds} \frac{ds}{dt} = \frac{dρ}{dt} > 0, \frac{dρ}{ds} = ρ(s) cot α, cf. (219),

we deduce 0 ≤ α < π/2. In fact, α = π/2 (and ds/dt = ∞) only at the vertex b̂_{2}, and our claim is that α = 0 only holds at p̂_{0} = Γ^{∗}(0).

It is certainly evident from the numerical analysis of the shape curves (cf. Table 1) that α > 0 for s > 0, at least until Γ^{∗} leaves ℭ_{0}. Indeed, using numerical data and a continuity argument we can establish the uniform lower bound α ≥ 0.18 for π/4 ≤ φ ≤ π/2. However, we shall also explain an alternative and more qualitative approach to settle the problem.

To show α > 0 holds, let assume the contrary and recall the geometric arguments in the setting in Section 7.3, where we regarded Γ^{∗}(σ), σ = s/2, as a curve on the sphere S^{2}(1/2) and C(Γ^{∗}) ⊂ M̄ denotes the cone surface with the induced Euclidean metric (204). The associated moduli curve s̄ → Γ̄(s̄) lies in this surface, and assuming the first cusp occurs at σ = σ_{1} we consider the Euclidean sector 0 ≤ σ ≤ σ_{1} bounded by the rays σ = 0 and σ = σ_{1}. By our assumptions, there is a bijective correspondence [0, s̄_{1}] ⟷ [0, σ_{1}] between the arc-length parametrizations of the moduli curve Γ̄ and Γ^{∗}, and α > 0 for 0 < s̄ < s̄_{1}.

The curve Γ̄(s̄) starts out from the origin and its radial distance ρ is increasing. It has the positively oriented moving frame {τ , η} of (206), where τ (respectively η) is the unit tangent (respectively normal) vector. By (211) and a well known formula for the curvature of curves in the Euclidean plane, the curvature of Γ̄ in the above sector can be expressed as

\frac{dζ}{ds̄} = \frac{1}{2} \frac{d}{dη} (ln U), ζ = σ + α, cf. Figure 8.  (309)

Towards the point Γ̄(s̄_{1}) the curve Γ̄ becomes tangential to the boundary ray σ = σ_{1}, that is, the angle α decreases to zero. Therefore \frac{dσ}{ds̄} vanishes, by (207), and hence

\frac{dζ}{ds̄} ≤ 0 as s̄ → s̄_{1}.

Moreover, the frame \{τ , η\} approaches \{\frac{∂}{∂ρ},\frac{1}{ρ_{1}} \frac{∂}{∂σ}\} as s̄ → s̄_{1}, so we also deduce

\frac{d}{dη} (ln U) → \frac{u′(σ)}{u(σ)}|_{σ=σ_{1}} ≤ 0

and hence u(σ) = U^{∗}(Γ^{∗}(σ)) is decreasing at σ_{1}, or possibly u′(σ_{1}) = 0 and hence Γ^{∗} is tangential to the level curve of U^{∗} on the sphere. However, U^{∗} is actually increasing towards the point Γ^{∗}(σ_{1}) by Remark 63 (which applies here since there is no cusp for σ < σ_{1}). This contradiction rules out any occurrence of cusps inside ℭ_{0}.

8.6.3. Final escape limiting behavior of the shape curves. As an interesting example, recall the time parametrized meridian solution Γ_{0}^{∗}(t) from Section 8.4.2. The first cusp appears at t_{1} ≈ 10.4, with colatitude φ_{1} ≈ 107.5^{∘}. The next cusps occur roughly at times t_{2} ≈ 435, t_{3} ≈ 162400, t_{4} ≈ 5^{.}10^{7}, t_{5} ≈ 1.24^{.}10^{9} (with due regard to numerical instability) and they appear to be approaching ê_{3} as a final limit of the shape curve. The behavior of the curve resembles a damped oscillation converging to its ”stability” point ê_{3} as t → ∞. Let us refer to this limiting behavior as irregular, namely the limit shape p is reached through converging cusps and α′(s) has no limit, as in the above example. Such a behavior at the final escape at infinity is, however, not necessarily related to the fact that the shape curve is a triple collision curve in the other direction.

Thus, one may consider more generally the limiting behavior of moduli curves Γ̄(t) of three-body motions as t → ∞, assuming the limit shape p exists. In the irregular case, however, we do not claim that p is necessarily a central configuration (that is, a critical point of U^{∗}), although this is rather likely. On the other hand, if p is a central configuration, we claim that it is an Euler points ê_{i}, and moreover, Γ^{∗} is not confined to the equator circle.

We define the final limiting behavior to be regular if the final shape p = Γ^{∗}(s_{2}) is a central configuration, where α(s) > 0 for s_{2} − ϵ < s < s_{2} and α(s_{2}) = 0. Moreover, Γ^{∗} is confined to the equator circle (and hence the 3-body motion is collinear) if p = ê_{i}. As in the case of triple collisions, a useful tool in the study of such limiting behavior is again the system ODE^{∗} (221), whose solutions are pairs (α(s), Γ^{∗}(s)). Then the fact that ρ → ∞ as s → s_{2} is expressed by the divergence of the integral (238), and moreover, the system (221) itself imposes the condition that p must be a critical point of U^{∗}.

A closer study of the above regular solutions (α(s), Γ^{∗}(s)) near the final limit may proceed in the same way as we studied triple collision shape curves in Section 8.5. For convenience, let us translate the arc-length parameter, s → s − s_{2}, and consider the power series expansions of α(s) and Γ^{∗}(s) at s = 0. The calculations are similar to the triple collision case worked out in Section 8.4.1 and 8.5.1, but this time α′(s) converges to the negative root of the polynomial in (249), namely

a_{0} = −\frac{\sqrt{13} + 1}{8} ≈ −0.575 69, b_{0} = − \frac{\frac{1}{5} \sqrt{1185} + 1}{8} ≈ −0. 985 6. (310)

Thus, in the collinear (Euler) case with the limit shape Γ^{∗}(0) = ê_{3}, with Γ^{∗}(s) an arc-length parametrization of the equator circle near ê_{3}, the differential equation (245) in the case i = 2 has a unique solution α(s) with α(0) = 0, α′(0) = b_{0}.

Next, for the final limit shape of Lagrange type we consider the following (singular) initial conditions

Γ^{∗}(0) = p̂_{0}; α(0) = 0, α′(0) = a_{0} (hence α(s) > 0 for s < 0)

which define a family of analytic solutions (α(s), Γ^{∗}(s)) of the system (221). The calculations are similar to the triple collision case, with a_{0} equal to the positive number in (250), worked out in Section 8.5.1. Therefore, we leave it to the reader to modify these calculations and perhaps establish the same kind of analytic uniqueness, namely that solutions are parametrized by the terminal angular (longitude) direction θ_{0} of Γ^{∗}(s) at p̂_{0}, cf. (260). In particular, by reversing the direction of the curve segment {Γ^{∗}(s), s ≥ 0} one obtains the shape curve with terminal direction θ_{0} + π.

8.6.4. More about the asymptotic behavior at triple collision. Finally, we turn to the asymptotic behavior, in terms of the time parameter t, of a triple collision taking place at t = 0. Let Γ̄(t) = (ρ(t), Γ^{∗}(t)) be the moduli curve of such a motion, with Γ^{∗}(0) = p̂_{0} or ê_{i}, and write μ = U^{∗}(Γ^{∗}(0)). Then it is a classical result, dating back to the work of Sundman and Siegel, that (for any energy level h and mass distribution)

ρ(t) ∼ κt^{2/3} as t → 0, κ = (\frac{9}{2}μ)^{1/3}  (311)

and moreover, the total kinetic energy is asymptotically dominated by the ”change of size”, in the sense that

T (t) ∼ T^{ρ} = \frac{1}{2}ρ̇(t)^{2} ∼ \frac{2}{9}κ^{2}t^{−2/3} ∼ \frac{μ}{ρ(t)}.  (312)

In particular, the residual kinetic energy

T^{σ} = \frac{1}{8}ρ(t)^{2}(\frac{ds}{dt})^{2} = T − T^{ρ}  (313)

due to the ”change of shape” must be of lower order in t than that of T^{ρ}, in the sense that T^{σ}/T^{ρ} → 0. However, this does not exclude the possibility that T^{σ} → ∞ as t → 0. We claim, however, that T^{σ} → 0, and we shall present the following (rather heuristic) argument for this.

Using the relationship (219) between ρ(s) and α(s) and the series expansion of α(s) we obtain an expansion of type

ρ(s) = ρ_{0}s^{p}(1 + r_{1}s + r_{2}s^{2} + …), p = 1/2α′(0).

Hence, for suitable nonzero constants κ_{0} and κ_{s}

s(t) ∼ κ_{0}ρ(t)^{1/p} = κ_{0}ρ(t)^{2α′(0)} ∼ κ_{s}t^{4α′(0)/3} = κ_{s}t^{e},

where the exponent e depends on the two types of triple collision, namely by (250)

(i) Γ^{∗}(0) = p̂_{0} : e = \frac{4}{3}a_{0} = \frac{1}{6}(\sqrt{13} − 1) ≈ 0. 434 26  (314)
(ii) Γ^{∗}(0) = ê_{i} : e = \frac{4}{3}b_{0} = \frac{1}{6}(\frac{1}{5}\sqrt{1185} − 1) ≈ 0. 98079.

Now, as is the case of ρ(t), let us assume differentiation of s(t) also commutes with taking asymptotic limit, namely ṡ(t) ∼ eκ_{s}t^{e−1}. Then by (313)

T^{σ}(t) ∼ \frac{1}{8}(κκ_{s}e)^{2}t^{4/3+2(e−1)} = κ_{σ}t^{ϵ}, \{ \begin{matrix}Case (i) : ϵ ≈ 0.201 85 \\ Case (ii) : ϵ ≈ 1. 294 9 \end{matrix} (315)

8.7. Numerical solutions of triple collision motions. We shall describe a modified approach to provide numerical C^{1}-data for the 1-pa-rameter family 𝔖(p̂_{0}) of shape curves representing non-collinear triple collision motions, under the standing assumption of equal masses m_{i} = 1/3 and zero total energy. Recall that these curves Γ^{∗}(s) arise from solutions (α, Γ^{∗}) of the system (221) of ordinary differential equations, where Γ^{∗} = Γ_{θ_{0}}^{∗} starts from the north pole p̂_{0} on the 2-sphere with initial (longitude) direction θ_{0}, and the (radial inclination) angle α ≥ 0 has the initial value α = 0. The basic idea is to obtain numerical data close to the initial point, by the analytical method, and use them as the initial data for the remaining integration by means of a Runge-Kutta method. With some more efforts we believe it is possible to settle Conjecture 50 by carefully combining numerical analysis and theory along these lines. The following numerical analysis serves at least to illustrate the geometry of those shape curves.

8.7.1. Outline of a numerical approach. As usual, (θ, φ) are the spherical coordinates of the 2-sphere M^{∗} ≃ S^{2}, with φ = 0 at the initial point p̂_{0}. By symmetry (as explained earlier) we need only consider curves whose initial direction θ_{0} lies in the interval 0 < θ_{0} < π/3, and moreover, we may as well use φ to parametrize each curve since φ increases with the arc-length s. Thus, let

Γ_{θ_{0}}^{∗} = {(φ, θ_{θ_{0}}(φ)), φ ≥ 0; lim_{φ→0} θ_{θ_{0}}(φ) = θ_{0}}

denote the shape curve with initial angle θ_{0}, and let α_{θ_{0}}(φ) denote the corresponding angle α as a function of φ. We shall compute the values of θ_{θ_{0}}, \frac{d}{dφ}θ_{θ_{0}} and α_{θ_{0}} for 0 ≤ φ ≤ π/2 and 0 ≤ θ_{0} ≤ π/3.

Elimination of the arc-length parameter s in the system (221) is achieved by using the third equation to rewrite the first two equations with φ as the independent variable. In standard (explicit) form the new system reads:

α′ = −\frac{1}{4}\sqrt{1 + (sin^{2} φ)θ^{′2}} + \frac{cot α}{2U^{∗}}( \frac{∂U^{∗}}{∂φ} + \frac{∂U^{∗}}{∂θ} θ′)
θ′′ = \frac{1}{2U^{∗}} [\frac{∂U^{∗}}{∂θ} csc^{2} α csc^{2} φ − (4U^{∗} cot φ + \frac{∂U^{∗}}{∂φ} csc^{2} α)θ′  (316)

+ (\frac{∂U^{∗}}{∂θ} csc^{2} α)θ^{′2} − (U^{∗} sin 2φ + \frac{∂U^{∗}}{∂φ} csc^{2} α sin^{2} φ)θ^{′3}],

where α′, θ′, θ′′ means differentiation with respect to φ.

We consider the power series expansions at φ = 0 :

θ_{θ_{0}}(φ) = θ_{0} + φ(f_{0} + f_{1}φ + f_{2}φ^{2} + …)  (317)
α_{θ_{0}}(φ) = φ(g_{0} + g_{1}φ + g_{2}φ^{2} + …)

whose coefficients can be calculated recursively as functions of θ_{0} by the method of undetermined coefficients. This is similar to the calculation of the expansions in (260) using the system ODE^{∗}. Let

θ_{θ_{0}}^{[n]}(φ), α_{θ_{0}}^{[n]}(φ), \frac{d}{dφ}θ_{θ_{0}}^{[n]}(φ)  (318)

be the polynomials in φ of degree n + 1 (respectively n for the third polynomial), where the first two are obtained by substituting the calculated expressions for f_{i}, g_{i} (as functions of θ_{0}), 0 ≤ i ≤ n, into (317) and truncating higher order terms. The last polynomial is the derivative of θ_{θ_{0}}^{[n]}(φ). Then for sufficiently small φ the ”true” functions θ_{θ_{0}}(φ), α_{θ_{0}}(φ) and \frac{d}{dφ}θ_{θ_{0}}(φ) = θ′_{θ_{0}}(φ) will be closely approximated by the polynomials (318), and the approximations can be made arbitrarily accurate by taking n sufficiently large, i.e. by computing enough coefficients f_{i}, g_{i}.

Given a value of θ_{0} with 0 < θ_{0} < π/3, we fix a small φ_{0} and compute the polynomials (318), to be regarded as approximations of θ_{θ_{0}}(φ), α_{θ_{0}}(φ) and θ′_{θ_{0}}(φ) on the interval 0 ≤ φ ≤ φ_{0}. In particular, the values of the three polynomials at φ = φ_{0} will serve as (approximate) initial values for the functions α, θ and θ′, whose further development on the interval φ_{0} ≤ φ ≤ π/2 is governed by the system (316). This allows us to obtain numerical solutions for θ and α on this interval, using any of the standard iterative methods. Pieced together, these data furnish us with the C^{1}-data of the triple collision shape curves Γ_{θ_{0}}^{∗} within the interval 0 ≤ φ ≤ π/2.

8.7.2. C^{1}-data for a selection of triple collision motions. As in Section 8.5.3 we perform symbolic computations to calculate successively the coefficients f_{i} and g_{i} of the expansions (317). As before, these are trigonometric polynomials of 3θ_{0}, namely polynomials of sin 3θ_{0} and cos 3θ_{0}, and we have calculated the exact expressions for i ≤ 9. Beyond that they tend to be rather untractable in their exact form. The exact expressions for the first few coefficients are listed below for the sake of reference:

f_{0} = \frac{5(10 + \sqrt{13})}{232} sin 3θ_{0}, f_{1} = \frac{25(113 + 20\sqrt{13})}{53824} sin 6θ_{0}
f_{2} = \frac{15(544702206 + 58374421\sqrt{13})}{201243199488} sin 3θ_{0}
+ \frac{5875625(1390 + 313\sqrt{13})}{201243199488} sin 3θ_{0} cos 6θ_{0}  (319)
f_{3} = \frac{2(1909168577687 − 51730231240\sqrt{13})}{1318947929444352} sin 6θ_{0}

+ \frac{1422740625(17969 + 4520\sqrt{13})}{1318947929444352} sin 6θ_{0} cos 6θ_{0}

g_{1} = \frac{−15(1 + 3\sqrt{13})}{1856} cos 3θ_{0},
g_{2} = \frac{543509 + 352091\sqrt{13}}{29280256} + \frac{−8925(49 + 31\sqrt{13})}{29280256} cos 6θ_{0} (320)
g_{3} = \frac{10(253093537 + 56946315\sqrt{13})}{134162132992} cos 3θ_{0}

+ \frac{−3525375(893 + 359\sqrt{13})}{134162132992} cos 3θ_{0} cos 6θ_{0}

The coefficient g_{0} equals the constant a_{0} in (266) and hence is omitted here. To illustrate the (decreasing) magnitude of the coefficients of the trigonometric polynomials f_{i}, g_{i} we list a few approximate expressions

f_{1} ≈ (8.6)10^{−2} sin 6θ_{0} g_{1} ≈ −(9.6)10^{−2} cos 3θ_{0},
f_{3} ≈ (2.6)10^{−3} sin 6θ_{0} + (3.7)10^{−2} sin 6θ_{0} cos 6θ_{0},
g_{3} ≈ (3.1)10^{−2} cos 3θ_{0} − (5.7)10^{−2} cos 3θ_{0} cos 6θ_{0}.

In this way one obtains the approximating polynomials (318) for n = 9, say. Hence, to obtain approximate numerical data for the solutions Γ_{θ_{0}}^{∗} of the system (316), we have chosen φ_{0} = 0.05 and θ_{0} = kπ/300, k = 0, 1, …, 100, and we have computed the numerical solutions using the Runge-Kutta method. These 101 solution curves outline the general behavior of the shape curve Γ_{θ_{0}}^{∗} of triple collision motions parametrized by the initial longitude angle θ_{0}. We refer to Table 1, Table 2 and Table 3 which list the calculated values of α_{θ_{0}}(φ), θ_{θ_{0}}(φ) and \frac{d}{dφ}θ_{θ_{0}}(φ) for θ_{0} = kπ/30, k = 0, 1, .., 10, and for 6 different values of φ. All angles are measured in radians.

Table 1 : Inclination angle α_{θ_{0}}(φ), for θ_{0} = kπ/30

\begin{matrix}k \ φ & \frac{π}{4} & \frac{3π}{8} & \frac{15π}{32} & \frac{63π}{128} & \frac{255π}{512} & \frac{1023π}{2048} \\ 0 & .1947 & .2292 & .2072 & .1926 & .1883 & .1871 \\ 1 & .2132 & .3470 & .5938 & .7243 & .7705 & .7835 \\ 2 & .2464 & .4554 & .7917 & .9694 & 1.0379 & 1.0586 \\ 3 & .2742 & .5170 & .8895 & 1.0914 & 1.1747 & 1.2014 \\ 4 & .2943 & .5542 & .9465 & 1.1643 & 1.2597 & 1.2922 \\ 5 & .3083 & .5780 & .9827 & 1.2121 & 1.3179 & 1.3566 \\ 6 & .3181 & .5938 & 1.0067 & 1.2247 & 1.3599 & 1.4054 \\ 7 & .3248 & .6043 & 1.0226 & 1.267? & 1.3904 & 1.4433 \\ 8 & .3292 & .6110 & 1.0328 & 1.2816 & 1.4116 & 1.4723 \\ 9 & .3317 & .6147 & 1.0385 & 1.2898 & 1.4243 & 1.4917 \\ 10 & .3325 & .6159 & 1.0403 & 1.2925 & 1.4285 & 1.4989 \end{matrix}

Table 2 : Longitude angle θ_{θ_{0}}(φ), for θ_{0} = kπ/30

\begin{matrix}k \ φ & \frac{π}{4} & \frac{3π}{8} & \frac{15π}{32} & \frac{63π}{128} & \frac{255π}{512} & \frac{1023π}{2048} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & .23558 & .40978 & .66916 & .77595 & .80954 & .81866 \\ 2 & .42158 & .61693 & .83930 & .92403 & .95180 & .95968 \\ 3 & .55924 & .73215 & .90859 & .97447 & .99697 & 1.0037 \\ 4 & .66448 & .80830 & .94747 & .99907 & 1.0173 & 1.0231 \\ 5 & .74914 & .86481 & .97349 & 1.0137 & 1.0283 & 1.0333 \\ 6 & .82059 & .91025 & .99302 & 1.0236 & 1.0350 & 1.0391 \\ 7 & .88344 & .94901 & 1.0089 & 1.0310 & 1.0394 & 1.0426 \\ 8 & .94075 & .98370 & 1.0226 & 1.0370 & 1.0425 & 1.0448 \\ 9 & .99475 & 1.0160 & 1.0352 & 1.0423 & 1.0450 & 1.0462 \\ 10 & 1.0472 & 1.0472 & 1.0472 & 1.0472 & 1.0472 & 1.0472 \end{matrix}

Table 3 : Values of \frac{d}{dφ}θ_{θ_{0}}(φ), for θ_{0} = kπ/30

\begin{matrix}k \ φ & \frac{π}{4} & \frac{3π}{8} & \frac{15π}{32} & \frac{63π}{128} & \frac{255π}{512} & \frac{1023π}{2048} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & .29933 & .63143 & 1.2517 & 1.7182 & 1.9466 & 2.0210 \\ 2 & .40322 & .60768 & .99427 & 1.3887 & 1.6602 & 1.7708 \\ 3 & .39183 & .50135 & .76959 & 1.0998 & 1.3930 & 1.5427 \\ 4 & .34216 & .40251 & .60029 & .87390 & 1.1710 & 1.3623 \\ 5 & .28289 & .31737 & .46599 & .68801 & .97142 & 1.2023 \\ 6 & .22295 & .24306 & .35361 & .52754 & .78004 & 1.0406 \\ 7 & .16476 & .17645 & .25526 & .38365 & .58941 & .85626 \\ 8 & .10857 & .11502 & .16580 & .25043 & .39595 & .62716 \\ 9 & .05389 & .05675 & .08166 & .12369 & .19908 & .33781 \\ 10 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}

8.8. An outlook on the general case. Finally, let us briefly consider the more general case of non-equal masses and/or non-vanishing total energy h. Namely, by Theorem 45, the system (221) must be replaced by

\{ \begin{matrix}\frac{dα}{ds} = −\frac{1}{2} + \frac{1}{4} \frac{u(s)}{u(s)+hρ} (1 + 2 cot α\frac{d}{dτ^{∗}} ln(U^{∗})) \\ K_{g}^{∗} sin^{2} α = \frac{1}{2} \frac{u(s)}{u(s)+hρ} \frac{d}{dν^{∗}} ln(U^{∗}) \\ 1 = (\frac{dφ}{ds})^{2} + (sin^{2} φ)(\frac{dθ}{ds})^{2} \end{matrix}  (321)

where u(s) = U^{∗}(Γ^{∗}(s)) and Γ^{∗}(s) is the arc-length parametrized shape curve (218) on the unit sphere S^{2}(1). Now the size function ρ in the moduli space M̄ appears explicitly, so the system involves the two ”auxiliary” functions α(s), ρ(s) which are still related by (207) and (208). Their initial value at triple collision is α(0) = ρ(0) = 0, and the shape curve Γ^{∗} emanates from the physical center p̂_{0} = Γ^{∗}(0), namely the minimumspoint of U^{∗} on the northern hemisphere. Due to the space-time scaling symmetries of the Newtonian equation (cf. Chapter 7), for h ≠ 0 there are essentially only two cases, h > 0 and h < 0, and our case h = 0 may be viewed as the limiting case between negative and positive energies. However, for h ≠ 0 we may scale and assume h = ±1.

We point out the open problem of finding the appropriate version of Theorem G (or G_{1}) in the two cases h = ±1 or when the masses m_{i} are unequal. For this purpose, it is natural to try first the following two special cases.

• h = ±1 and equal masses (i.e., m_{i} = 1/3). Many results in Section 8.5 still apply and there are symmetries as before, e.g. it suffices to consider shape curves whose angular direction at the north pole is in the range 0 ≤ θ_{0} ≤ π/3. However, the initial value problem is ”essentially” singular, in the sense that the solutions of (321) are singular at s = 0. For example, if we assume a series expansion

α = a_{0}s^{q}(1 + a_{1}s + ..), ρ = ρ_{0}s^{p}(1 + r_{1}s + ..)

then we would have q = 1 and ρ would have the leading exponent p = 1/2a_{0}. Moreover, from the first equation of (321) it follows that a_{0} has the value from (266), but this equation also tells us that Γ^{∗}(s) is singular at s = 0.

• h = 0 and the masses are not equal. The system (321) is the same as (221). Note that the mass distribution {m_{i}} affects the system (321) solely via the potential function U^{∗}, but the trigonometric series development of U^{∗}, similar to that in Section 8.2, remains to be done for non-equal masses. We also seek a convenient coordinate system on the sphere near the point p̂_{0}. In Section 6.6.1 we actually worked out a series expansion of U^{∗} centered at p̂_{0}, involving coefficients which are symmetric functions of the masses, but here we rather need its spherical polar coordinate version.

On the other hand, in Chapter 6 there are expressions for U^{∗} in terms of spherical polar coordinates (φ, θ) centered at the north pole N. Therefore, one approach is to find the coordinates (φ̂, θ̂) of p̂_{0} in this coordinate system and then use spherical trigonometry to determine the transformation from (φ, θ) to spherical polar coordinates (φ̄, θ̄) centered at p̂_{0}. For this purpose, we recall

cos φ̂ = \frac{\sqrt{3}\sqrt{m̄}}{m̂}, cf. (177c)  (322)

and the value of θ̂ (which depends on the choice of zero meridian, θ = 0) can be determined from the longitude differences ω̂_{i} = ±(θ̂ − θ_{i}), with 0 ≤ ω̂_{i} ≤ π, i = 1, 2, 3, where θ_{i} denotes the longitude angle of the binary collision points b̂_{i}. To this end, consider the spherical triangle with vertices N, p̂_{0} and b̂_{i} and note that ω̂_{i} is the angle at the vertex N. The arc opposite to N, connecting p̂_{0} and b̂_{i}, has length

2σ_{i} = 2 arccos \sqrt{\frac{I_{i}}{1 − m_{i}}} = 2 arccos \sqrt{1 − \frac{m̂_{i}}{(1 − m_{i})m̂}},

cf. (127), (170), and the two other sides have length π/2 and φ̂. Hence, by the spherical cosine law we deduce the formula

cos ω̂_{i} = \frac{cos(2σ_{i})}{sin φ̂} = \frac{1 − \frac{2m̂_{i}}{(1−m_{i})m̂}}{\sqrt{1 − \frac{3m̄}{m̂^{2}}}} .  (323)

Knowing the expansion of U^{∗} as a trigonometric series in terms of the angles φ̄, θ̄, we believe the initial value problem at s = 0 for the system (221) can be investigated in the same way as we did for equal masses. But this time we expect regularity issues to depend crucially on the mass distribution.

124 WU-YI HSIANG AND ELDAR STRAUME KINEMATIC GEOMETRY OF TRIANGLES 125 126 WU-YI HSIANG AND ELDAR STRAUME KINEMATIC GEOMETRY OF TRIANGLES 127 128 WU-YI HSIANG AND ELDAR STRAUME KINEMATIC GEOMETRY OF TRIANGLES 129 130 WU-YI HSIANG AND ELDAR STRAUME

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Department of mathematics, University of California, Berkeley, USA

Department of mathematical sciences, Norwegian University of Science and Technology, Trondheim, Norway E-mail address: eldars@math.ntnu.no

Received October 10, 2006