On the n-body problem in ${\mathbb{R}}^4$

Abstract

Using geometric mechanics methods, we examine aspects of the dynamics of n mass points in ${\mathbb{R}}^4$ with a general pairwise potential. We investigate the central force problem, set up the n-body problem and discuss certain properties of relative equilibria. We describe regular n-gons in ${\mathbb{R}}^4$, and when the masses are equal we determine the invariant manifold of motions with regular n-gon configurations. In the case n = 3, we reduce the dynamics to a 6 d.f. system and we show that for generic potentials and momenta, relative equilibria with equilateral configuration are unstable.

This article is part of the theme issue ‘Topological and geometrical aspects of mass and vortex dynamics’.

1. Introduction

In the last decades, a good body of work has been devoted to the study of various generalizations of the classical n-body problem. Thus, we find studies of n mass points with modified potentials [1–5], or on configuration spaces with a non-Euclidean structure [6–10], or in higher-dimensional Euclidean spaces [11–18].

In this paper, we investigate aspects of the dynamics in ${\mathbb{R}}^4$ of n mass points with binary interaction depending on distance only. Working from a geometric mechanics perspective, we identify a variety of interesting properties and retrieve some known results.

As a preamble, we review the geometry of the SO(4) action on ${\mathbb{R}}^4$. It is known that any matrix in SO(4) can be expressed as the product of two planar rotations, each in its own invariant plane, with the two planes mutually orthogonal. Thus, any element of SO(4) is conjugate to one in the double planar rotation group, defined in an Oxyzw cartesian system of coordinates as SO(2)_xy × SO(2)_zw, the subgroup of SO(4) leaving the principal planes Oxy and Ozw invariant. This is a normal form for rotations, and can be understood as a generalization of Euler's rotation theorem. It reduces many questions about SO(4) symmetries to questions about symmetries with respect to the action of the double planar rotation group.

We choose to focus on the double planar rotation group rather than the full symmetry group SO(4) since it simplifies our study (in particular, reduction simplifies greatly) and we still obtain a rich variety of dynamical features. In particular, any relative equilibrium of the full SO(4) action is conjugate to a relative equilibrium with respect to the action of the double planar rotation group (see remark 4.2).

We start by taking a close look at the central force problem. We find that on each of the principal planes, the projection of the motions obeys the area laws. We also identify various invariant manifolds, in particular that of proportional motions for which the polar radii of the projections of onto the principal planes have a constant ratio.

We briefly consider the Kepler problem for the ${\mathbb{R}}^3$-Newtonian potential V (r) =  − k/r, k > 0 (r being the distance between two mass points). Note that this potential is not the solution of the Laplace equation in ${\mathbb{R}}^4$, and so is not the standard model in mathematical physics. However, the induced dynamics is interesting and is considered for theoretical significance (for instance [11,13,15–18]). We observe that the dynamics under the ${\mathbb{R}}^4$ gravitational potential, that is V (r) =  − k/r², k > 0, also known as the Jacobi potential (see [19] and references within), presents intriguing degeneracies and we defer its detailed study for future projects.

In the Newtonian potential V (r) = − 1/r, we prove that in the Kepler problem collisions are possible if and only if the angular momentum is zero. We also note that this conclusion is true for any homogeneous law, except the aforementioned attractive inverse square Jacobi potential.

Recall that in the classical (${\mathbb{R}}^2$ or ${\mathbb{R}}^3$) n-body problem, a homographic solution is one such that the configurations formed by the masses are similar modulo, a rotation and a scalar multiplication. Two particular cases are usually highlighted: the relative equilibria (RE), for which the scalar multiplication is the identity; and homothetic solutions, which display no rotation. Along a homographic solution the bodies form a central configuration. In the classical Newtonian case, this is equivalent to having the configuration as a critical point of the potential among configurations with a given moment of inertia. Perhaps the most important property of central configuration concerns total collision: when the bodies are released from the central configuration with zero initial velocity, they end in total collapse.

The Newtonian n-body problem in higher-dimensional Euclidean spaces was previously studied by Palmore [16–18] and Albouy & Chenciner [11,13], but see also [14]. Palmore focuses on homographic motions and RE, finding, among other results, that there are RE solutions which are not central configurations, and that the configurations of the homothetic solutions must always be central. Albouy & Chenciner present an extensive study in Euclidean spaces of any dimension. They prove that a necessary condition for the existence of non-homothetic homographic motion is that the motion takes place in an even-dimensional space. Further, two cases are possible: either the configuration is central (the latter being defined as a critical point of the potential among configurations with a given moment of inertia) and the space where the motion takes place is endowed with a hermitian structure; or it is balanced, that is we have a critical point of the potential among configurations with a given inertia spectrum, and the motion is a new type, quasi-periodic, of relative equilibrium.

In ${\mathbb{R}}^4$, we define RE via the general definition for Lie-symmetric ODE systems: a solution is an RE if it is also a one-parameter group orbit. We show that balanced configurations are exactly the configurations of RE, and that these are central configurations if and only if the components of the angular velocity on the two principal planes are equal. We further investigate collinear RE and find that either they have configurations in one of the principal planes, or their angular velocity components are equal.

When the masses are equal, due to the finite symmetries, we are able to detect low-dimensional invariant manifolds using the method of discrete reduction [20] and Palais' principle of symmetric criticality [21]. These invariant manifolds consist of equilibria and relative equilibria with configurations that are ‘regular’ n-gons in ${\mathbb{R}}^4$. By a ‘regular’ n-gon, we do not simply mean that the side lengths are equal (which would include e.g. all rhombuses), nor do we wish to consider only planar shapes. Instead, we follow Coxeter [22]: ‘A polygon (which may be skew) is said to be regular if it possesses a symmetry which cyclically permutes the vertices (and therefore also the sides) of the polygon’.

We classify regular n-gons in ${\mathbb{R}}^4$ in proposition 5.1. A planar n-gon lies in a two-dimensional plane in ${\mathbb{R}}^4$; either it lies in a principal plane, or it projects to similar n-gons in each of the principal planes. These two n-gons are synchronized, meaning that there exists a labelling of the points such that each of the projected n-gons is convex. For a non-planar (skew) n-gon, the projections onto the two principal planes are not similar.

The non-planar regular n-gons are of two types. In type (I), both projections onto the principal planes are regular n-gons, but they are not synchronized in the sense defined above. In type (II), at least one of the two projections onto a principal plane has fewer than n sides; denoting by b₁ and b₂ the number of sides of the two projections, n is the lowest common multiplier (l cm) of b₁ and b₂. Regular planar polygons may have any number of vertices. The smallest non-planar polygon of type (I) has n = 5 vertices, with one projection convex and the other a pentagram. The smallest non-planar polygon of type (II) has n = 4 vertices, with one projection a square and the other a digon (i.e. having two vertices). The next smallest non-planar polygon of type II has n = 6 vertices, with one projection a triangle and the other a digon. The three examples are the only non-planar polygons with fewer than seven vertices.

We also specifically address the three-body problem. Taking advantage of the Abelian structure of symmetry group SO(2)_xy × SO(2)_zw, we are able to reduce the system from 12 to 6 d.f. using directly the integrals of motion. Then we write the RE conditions and deduce that for the ${\mathbb{R}}^3$-Newtonian V (r) = − 1/r law, any non-collinear RE must be isosceles; this condition was observed previously in [11].

When the three masses are equal we study the stability of the equilateral triangle RE (in the case of attractive interactions). We find that such RE exist for any attractive potential. We determine that equilateral RE generically are unstable. Specifically, provided a certain sub-block of the Hessian of the amended potential has non-zero determinant, the matrix linearization at the RE displays a nilpotent component responsible for a ‘drift’ component in the dynamics.

The paper is organized as follows: in §2, we review the geometry of the SO(4) action on ${\mathbb{R}}^4$, including the orbit types and the calculation of the momentum map. In §3, we study the central force problem, detect certain invariant manifolds and discuss briefly the Kepler problem and collisions. In the next section, we introduce the general n-body problem. We examine RE, balanced and central configurations and collinear RE. In §5, we explore regular n-gons in ${\mathbb{R}}^4$, and apply the discrete reduction method to detect the invariant manifolds of regular n-gons. In §6, we set up the three-body problem and reduce the dynamics to a system of 6 d.f. In the case of equal masses, we study the stability of equilateral configurations REs and find that with the class of attractive potentials these are generically unstable. We conclude with some remarks and open questions.

2. Geometric set-up

The Lie group SO(4) has a standard action on ${\mathbb{R}}^4$ and further acts on $T^{\ast }{\mathbb{R}}^4\simeq {\mathbb{R}}^4\times {\mathbb{R}}^4$ via the co-tangent lift (R, (q, p))) = (Rq, Rp) for any R∈SO(4) and $(\mathbf{q},\mathbf{p})\in T^{\ast }{\mathbb{R}}^4$. The diagonal action of SO(4) on configurations of n bodies in ${\mathbb{R}}^{4n}$ is R(q₁, …, qⁿ) = (Rq₁, …, Rqⁿ); this too has a co-tangent lifted action on $T^{\ast }{\mathbb{R}}^{4n}$. We consider n-body problems with SO(4) symmetry, arising from SO(4)-invariant potentials $U:{\mathbb{R}}^4\to \mathbb{R}$. Note that any U that depends only on pairwise distances has this property.

Given any R∈SO(4), it is always possible to find an orthogonal change of coordinates that transforms R to

$$S:=\left[\begin{array}{cccc}\cos {\theta }_1& -\sin {\theta }_1& 0& 0\\ \sin {\theta }_1& \cos {\theta }_1& 0& 0\\ 0& 0& \cos {\theta }_2& -\sin {\theta }_2\\ 0& 0& \sin {\theta }_2& \cos {\theta }_2\end{array}\right],$$ 2.1

for some ${\theta }_1,{\theta }_2\in \mathbb{R}$. In other words, there exists a Q∈SO(4) such that R = QSQ^t, with S in the above form [23, Ex. 1.15]. The form of S in (2.1) is a normal form for R. Note that S and R have the same eigenvalues. The normal form is uniquely determined by these eigenvalues, cosθ_j ± isinθ_j for j = 1, 2, except for the possible exchange of the two diagonal blocks (if θ₁ and θ₂ are unequal). Exchange of the two blocks in S can be accomplished by the special orthogonal transformation τ defined by τ(x, y, z, w) = (z, w, x, y), with matrix

$$\left[\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ 1& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right].$$ 2.2

From the normal form, it is clear that R has two mutually orthogonal invariant planes, and R consists of a rotation in each of these planes. This is analogous to Euler's rotation theorem: in ${\mathbb{R}}^3$, any rotation is equivalent to a single rotation about some axis that runs through the origin.

Note that in the three-dimensional case, the axis of rotation is uniquely defined if and only if R is not the identity. In the four-dimensional case, we have a similar situation. Indeed, each invariant plane corresponds to a pair of complex conjugate eigenvalues of S, cosθ_j ± isinθ_j, for j = 1, 2. Generically, these are four distinct eigenvalues in two distinct pairs, each pair corresponding to an invariant subspace. It is easily checked that the only exceptions are (i) θ₁ = θ₂ = 0, i.e. R = S = Id; and (ii) θ₁ =  ± θ₂ =  ± π/2, in which case there are two double eigenvalues ± i. In both cases, there exist orthogonal changes of coordinates that move the invariant subspaces while leaving the form of S unchanged, for example, if θ₁ = θ₂ =  ± π/2 then, for any $\alpha \in \mathbb{R}$,

$$P=\left[\begin{array}{cccc}\cos \alpha & 0& -\sin \alpha & 0\\ 0& \cos \alpha & 0& -\sin \alpha \\ \sin \alpha & 0& \cos \alpha & 0\\ 0& \sin \alpha & 0& \cos \alpha \end{array}\right]\hspace{0.17em}\hspace{0.17em}⟹\hspace{0.17em}PS{P}^t=S.$$ 2.3

In summary, we have shown the following.

Proposition 2.1. —

• (i)Any R∈SO(4) is orthogonally similar to the normal form given in (2.1), which is unique except for the possible exchange of the two diagonal blocks. Two matrices R₁, R₂∈SO(4) have the same normal form (up to a possible exchange of diagonal blocks) if and only if they are orthogonally similar, i.e. R₂ = QR₁Q^t for some Q∈SO(4).
• (ii)The invariant planes of an R∈SO(4) are uniquely defined unless R = Id or each of the two blocks in (2.1) equals either $\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$ or $\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$, i.e. θ₁ =  ± θ₂ =  ± π/2, in which cases there are an infinite number of invariant planes.

The Lie algebra of SO(4) is so(4), the set of infinitesimal rotations in ${\mathbb{R}}^4$. It consists of all skew-symmetric matrices. There is a normal form for so(4) closely related to the one above for SO(4): for any ξ∈so(4), there exists a Q∈SO(4) such that $\xi =Q\hat{\mathit{\omega }}{Q}^t$, with

$$\hat{\mathit{\omega }}=\left[\begin{array}{cccc}0& -{\omega }_1& 0& 0\\ {\omega }_1& 0& 0& 0\\ 0& 0& 0& -{\omega }_2\\ 0& 0& {\omega }_2& 0\end{array}\right],$$ 2.4

for some ${\omega }_1,{\omega }_2\in \mathbb{R}$. This equation also defines the ‘hat’ notation $\hat{\mathit{\omega }}$ for a vector $\mathit{\omega }=({\omega }_1,{\omega }_2)\in {\mathbb{R}}^2$. This normal form is well known, however, since we are unaware of a reference for it, we give a brief proof in (i) below that it is a consequence of proposition 2.1.

Proposition 2.2. —

• (i)Any ξ∈so(4) is orthogonally similar to the normal form given in (2.4), which is unique except for the possible exchange of the two diagonal blocks. Two matrices ξ₁, ξ₂∈so(4) have the same normal form (up to a possible exchange of diagonal blocks) if and only if they are orthogonally similar, i.e. ξ₂ = Qξ₁Q^t for some Q∈SO(4).
• (ii)The invariant planes of $\hat{\mathit{\omega }}$ are uniquely defined unless ω = (ω, ω), i.e. ω₁ = ω₂, in which case there is an infinite family of invariant planes.

Proof. —

(i) Since $\exp (\xi )\in \mathrm{SO}(4)$, proposition 2.1 implies that $\exp (\xi )=QS{Q}^t$ for some orthogonal Q and some S in normal form (2.1) for some θ₁, θ₂. Let ω = (ω₁, ω₂) = (θ₁, θ₂) and $\hat{\mathit{\omega }}$ as in (2.4). Then since $S=\exp (\hat{\mathit{\omega }})$ and

$$\exp (\xi )=QS{Q}^t=Q\exp (\hat{\mathit{\omega }})Q^t=\exp (Q\hat{\mathit{\omega }}{Q}^t).$$

It follows that $\exp (t\xi )=\exp (Q\widehat{t\mathit{\omega }}{Q}^t)$ for all t. Since $\exp$ is locally invertible in a neighbourhood of zero [23], the above calculation, applied to a small enough t, implies that $\xi =Q\hat{\mathit{\omega }}{Q}^t,$ so ξ has the normal form (2.4). The normal form is completely determined by the eigenvalues, which are ± iω₁ and ± iω₂. Thus, two unequal matrices $\hat{\mathit{\omega }}$ and ${\hat{\mathit{\omega }}}^{\prime }$ are orthogonally similar if and only if they differ by an exchange of diagonal blocks, i.e. $\hat{\mathit{\omega }}=\tau \hat{\mathit{\omega }}{\tau }^t$ for the orthogonal matrix τ in (2.3).

(ii) The eigenvalues of $\hat{\mathit{\omega }}$ are ± iω₁,  ± iω₂. If these are distinct, then each pair uniquely defines an invariant plane. If ω₁ = ω₂, then (2.3) gives a change of coordinates leaving the form of $\hat{\mathit{\omega }}$ unchanged, showing that there is an infinite family of invariant planes. ▪

Remark 2.1. —

The preceding two propositions concern single elements of SO(4) or so(4). There is a related result for subgroups of SO(4): any compact abelian subgroup is conjugate to a subgroup of SO_xy × SO_zw. This is a consequence of the Torus theorem for Lie groups [23], and the fact that SO_xy × SO_zw is a maximal torus in SO(4) [24].

Because of the properties just stated, many questions about SO(4) symmetries can be reduced to questions about symmetries with respect to SO(2)_xy × SO(2)_zw. With the exception of §4b, throughout the paper we consider the symmetries associated with this group, which we call the double planar rotation group. The principal planes are Oxy and Ozw. Note that whenever an element R∈SO(2)_xy × SO(2)_zw has uniquely defined invariant planes, they equal the principal planes; this is also true for any ξ∈so(2)_xy × so(2)_zw.

Any element ξ of the Lie algebra so(4) determines a unique one parameter group $R(t):=\exp t\xi$, and the derivative of this path at t = 0 is ξ. Given a ‘base point’ $\mathbf{q}\in {\mathbb{R}}^4$, it follows that ξ determines a path in ${\mathbb{R}}^4$ given by $(\exp t\xi )\mathbf{q}$, and the derivative of this path at t = 0 is ξq (matrix product), which is called the infinitesimal action of ξ on q. Note that if $\xi =\hat{\mathit{\omega }}$ as defined in (2.4), and $\mathbf{q}=(x,y,z,w)\in {\mathbb{R}}^4,$ then

$$\hat{\mathit{\omega }}\mathbf{q}=\left[\begin{array}{c}-{\omega }_{1}y\\ {\omega }_{1}x\\ -{\omega }_{2}w\\ {\omega }_{2}z\end{array}\right].$$ 2.5

If SO(4), or its subgroup T = SO(2)_xy × SO(2)_zw, is acting diagonally on ${\mathbb{R}}^{4n}$, then the infinitesimal action of ξ on q = (q₁, …, q_n) is ξq = (ξq₁, …, ξq_n).

For any group G acting on a space X, the isotropy subgroup of any x∈X is the set of all group elements that leave x fixed, i.e.

$$G_x:=\{g\in G\hspace{0.17em}|\hspace{0.17em}gx=x\}.$$ 2.6

It is easily seen that if y = gx then G_y = gG_xg⁻¹, that is the isotropy subgroups of two points in the same group orbit are conjugate. Thus to each orbit is associated a conjugacy class of subgroups of G, called the orbit type of the orbit.

In SO(4), conjugacy is the same as orthogonal similarity. In any abelian group, such as the double planar rotation group, conjugacy leaves every subgroup invariant, i.e. gG_xg⁻¹ = G_x, so each orbit type contains only one subgroup.

Proposition 2.3. —

The orbit types for the standard action of G: = SO(2)_xy × SO(2)_zw on ${\mathbb{R}}^4$ are

• (i)q = (0, 0, 0, 0) has T_q = T;
• (ii)q = (x, y, 0, 0) with x, y≠0 has G_q = SO(2)_zw (rotations of the Ozw plane);
• (iii)q = (0, 0, z, w) with z, w≠0 then G_q = SO(2)_xy =  rotations of the Oxy plane;
• (iv)if q = (x, y, z, w) with (x, y)≠(0, 0) and (z, w)≠(0, 0) then $G_{\mathbf{q}}=\{{\mathbb{I}}_4\}$ (where ${\mathbb{I}}_4$ is the identity matrix).

Remark 2.2. —

In SO(4), the two isotropy subgroups SO(2)_xy and SO(2)_zw are conjugate by the matrix τ in (2.2) that exchanges blocks, so these are in the same SO(4) orbit type. In fact, by remark 2.1, all compact abelian subgroups of SO(4) of a given dimension are conjugate to each other, i.e. all copies of SO(2) are conjugate to each other, and all copies of SO(2) × SO(2) are conjugate to each other. In SO(4), there are also non-abelian isotropy subgroups isomorphic to SO(3).

Associated with the cotangent-lifted action of SO(2)_xy × SO(2)_zw on $T^{\ast }{\mathbb{R}}^4$ is a momentum map $J:T^{\ast }{\mathbb{R}}^4\to \mathrm{so}{(2)}^{\ast }\times \mathrm{so}{(2)}^{\ast }$ defined by $⟨J(\mathbf{q},\mathbf{p}),\mathit{\xi }⟩=⟨\mathbf{p},\mathit{\xi }\cdot \mathbf{q}⟩$. A straightforward calculation shows

$$J(\mathbf{q},\mathbf{p})=\left[\begin{array}{cccc}0& -(p_y{q}_x-p_x{q}_y)& 0& 0\\ (p_y{q}_x-p_x{q}_y)& 0& 0& 0\\ 0& 0& 0& -(p_w{q}_z-p_z{q}_w)\\ 0& 0& (p_w{q}_z-p_z{q}_w)& 0\end{array}\right]$$

or, using the identification $\mathrm{so}{(2)}^{\ast }\times \mathrm{so}{(2)}^{\ast }\simeq \mathbb{R}\times \mathbb{R}$,

$$J(\mathbf{q},\mathbf{p})=J(q_x,q_y,q_z,q_w,p_x,p_y,p_z,p_w)=(p_y{q}_x-p_x{q}_y,p_w{q}_z-p_z{q}_w).$$ 2.7

If we pass to the double-polar coordinates (r₁, θ₁, r₂, θ₂) defined by

$$x=r_1\cos {\theta }_1,y=r_1\sin {\theta }_1$$ 2.8

and

$$z=r_2\cos {\theta }_2,w=r_2\sin {\theta }_2$$ 2.9

then the momentum map reads

$$J(\mathbf{q},\mathbf{p})=J(r_1,{\theta }_1,r_2,{\theta }_2,p_{r_1},p_{{\theta }_1},p_{r_2},p_{{\theta }_2})=(p_{{\theta }_1},p_{{\theta }_2}).$$ 2.10

Let μ = (μ₁, μ₂)∈so(2)* × so(2)* be fixed momentum value. For future reference, we note that the isotropy group of μ in the double planar group is given by

$$\begin{array}{rl}\{R\in \mathrm{SO}{(2)}_{xy}\times \mathrm{SO}{(2)}_{zw}\hspace{0.17em}|\hspace{0.17em}{\text{Ad}}_{R^{-1}}^{\ast }\hat{\mathit{\mu }}=\hat{\mathit{\mu }}\}& =\{\hat{\mathit{\mu }}\in \mathrm{SO}{(2)}_{xy}\times \mathrm{SO}{(2)}_{zw}\hspace{0.17em}|\hspace{0.17em}R\hat{\mathit{\mu }}{R}^{-1}=\hat{\mathit{\mu }}\}\\ & =\mathrm{SO}{(2)}_{xy}\times \mathrm{SO}{(2)}_{zw}.\end{array}$$

3. Central force problem

Consider the motion of a unit mass point in ${\mathbb{R}}^4$ under the influence of a source field located in the origin. The Hamiltonian is

$$H(\mathbf{q},\mathbf{p})=\frac{1}{2}{\mathbf{p}}^2+V(\parallel \mathbf{q}\parallel ),$$ 3.1

where $V:D\subseteq \mathbb{R}\to \mathbb{R}$ is some smooth potential depending only on the distance to the origin, and D some subset of $\mathbb{R}$. In double-polar coordinates, we have

$$H(r_1,{\theta }_1,r_2,{\theta }_2,p_{r_1},p_{{\theta }_1},p_{r_2},p_{{\theta }_2})=\frac{1}{2}(p_{r_1}^2+\frac{p_{{\theta }_1}^2}{r_1^2}+p_{r_2}^2+\frac{p_{{\theta }_2}^2}{r_2^2})+V(\sqrt{r_1^2+r_2^2}),$$ 3.2

where D₁ × D₂⊆(0, ∞) × (0, ∞) denotes the domain where V is well defined. Recall that we consider as symmetry group the subgroup SO(2)_xy × SO(2)_zw. Since the Hamiltonian is invariant under its action, by Noether theorem, we obtain the conservation of the angular momentum, that is

$$\mathbf{p}(t)=(p_{{\theta }_1}(t),p_{{\theta }_2}(t))=\mathrm{const}.=\mathit{\mu }=({\mu }_1,{\mu }_2),$$

along any solution. We obtain the reduced 2 d.f. system with the Hamiltonian

$$H_{\mathrm{red}}(r_1,p_{r_1},r_2,p_{r_2})=\frac{1}{2}(p_{r_1}^2+p_{r_2}^2)+\frac{{\mu }_1^2}{2r_1^2}+\frac{{\mu }_2^2}{2r_2^2}+V\left(\sqrt{r_1^2+r_2^2}\right),$$ 3.3

coupled with the reconstruction equations

$${\dot{\theta }}_1=\frac{{\mu }_1^2}{r_1^2(t)}\mathrm{and}{\dot{\theta }}_2=\frac{{\mu }_2^2}{r_2^2(t)}.$$ 3.4

Remark 3.1. —

For μ = (μ₁, μ₂)≠0, from the conservation of angular momentum it follows that the projections of the motion on each of the principal planes obey the area laws, and the ratio is these areas is constant. Specifically, let $A_{xy}(t)=\frac{1}{2}{r}_1^2(t){\dot{\theta }}_1$ and $A_{zw}(t)=\frac{1}{2}{r}_2^2(t){\dot{\theta }}_2(t)$ be the projected areas. Then on each plane, we have equal areas in equal times and

$$\frac{A_{xy}(t)}{A_{zw}(t)}={\left(\frac{{\mu }_1}{{\mu }_2}\right)}^2\text{for all}\hspace{0.17em}t.$$ 3.5

Remark 3.2 (The harmonic oscillator). —

For potentials of the form $V(\sqrt{r_1^2+r_2^2})=k(r_1^2+r_2^2)$, $k\in \mathbb{R},$ the system decouples and it is integrable.

Since the Hamiltonian is time-independent, the energy is conserved, and so along any solution H_red(r₁(t), p_r1(t), r₂(t), p_r2(t)) = const. The equations of motion read as

$${\dot{r}}_1=p_{r_1},{\dot{p}}_{r_1}=\frac{{\mu }_1^2}{r_1^3}-V^{\prime }\left(\sqrt{r_1^2+r_2^2}\right)\frac{r_1}{\sqrt{r_1^2+r_2^2}}$$ 3.6

and

$${\dot{r}}_2=p_{r_2},{\dot{p}}_{r_2}=\frac{{\mu }_2^2}{r_2^3}-V^{\prime }\left(\sqrt{r_1^2+r_2^2}\right)\frac{r_2}{\sqrt{r_1^2+r_2^2}}.$$ 3.7

We define the effective (or amended) potential

$${\tilde{V}}_{{\mu }_1,{\mu }_2}(r_1,r_2):=\frac{{\mu }_1^2}{2r_1^2}+\frac{{\mu }_2^2}{2r_2^2}+V(\sqrt{r_1^2+r_2^2}).$$

Given the conservation of energy and since the kinetic energy $\frac{1}{2}(p_{r_1}^2+p_{r_2}^2)$ is positive, for a fixed level of energy h we retrieve the allowed (Hill) regions of motion

$${\mathcal{R}}_h({\mu }_1,{\mu }_2):=\{(r_1,{\theta }_1,r_2,{\theta }_2)\hspace{0.17em}|\hspace{0.17em}{\tilde{V}}_{{\mu }_1,{\mu }_2}(r_1,r_2)\le h\}.$$

For example, if for some h₀ fixed, the set $\{(r_1,r_2)\hspace{0.17em}|\hspace{0.17em}{\tilde{V}}_{{\mu }_1,{\mu }_2}(r_1,r_2)\le h_0\}=[a_1,b_1]\times [a_2,b_2]$, $a_j,b_j\in \mathbb{R}$, j = 1, 2, then all trajectories are bounded and belong to ([a₁, b₁] × S¹) × ([a₂, b₂] × S¹), i.e. the product of two annular regions.

(a). Some invariant manifolds

In cartesian coordinates, we observe the sets

$${\mathcal{P}}_{xy}:=\{(x,y,z,w,p_x,p_y,p_z,p_w)\hspace{0.17em}|\hspace{0.17em}z=w=p_z=p_w=0\}$$ 3.8

and

$${\mathcal{P}}_{zw}:=\{(x,y,z,w,p_x,p_y,p_z,p_w)\hspace{0.17em}|\hspace{0.17em}x=y=p_x=p_y=0\}$$ 3.9

are invariant under the dynamics induced by (3.1) and that on each of these the dynamics is given by the standard planar central force problem.

Let us write the reduced Hamiltonian (3.3) using polar coordinates r₁ = Rcosφ, r₂ = Rsinφ:

$$H_{\mathrm{red}}(R,\phi ,P_R,P_{\phi })=\frac{P_R^2}{2}+\frac{P_{\phi }^2}{2R^2}+\frac{{\mu }_1^2}{2R^2{\cos }^2\phi }+\frac{{\mu }_2^2}{2R^2{\sin }^2\phi }+V(R),$$ 3.10

which holds for all φ≠0, π/2, π, 3π/2. The equations of motion read

$$\dot{R}=P_R,{\dot{P}}_R=\frac{P_{\phi }^2}{R^3}+(\frac{{\mu }_1^2}{{\cos }^2\phi }+\frac{{\mu }_2^2}{{\sin }^2\phi })\frac{1}{R^3}-V^{\prime }(R),$$ 3.11

and

$$\dot{\phi }=\frac{P_{\phi }^2}{R^2},{\dot{P}}_{\phi }=(\frac{{\mu }_1^2\sin \phi }{{\cos }^3\phi }-\frac{{\mu }_2^2\cos \phi }{{\sin }^3\phi })\frac{1}{R^2}.$$ 3.12

For all non-zero momenta μ = (μ₁, μ₂)≠0, we find that the sets

$${\mathcal{M}}_{\pm \phi }:=\{\begin{array}{l}\{(R,\phi ,P_R,P_{\phi })\hspace{0.17em}|\hspace{0.17em}\tan \phi =\pm \sqrt{\frac{{\mu }_2}{{\mu }_1}},\hspace{0.17em}{P}_{\phi }=0\},\text{if}\hspace{0.17em}{\mu }_1\ne 0,\\ \{(R,\phi ,P_R,P_{\phi })\hspace{0.17em}|\hspace{0.17em}\tan \phi =\pm \sqrt{\frac{{\mu }_1}{{\mu }_2}},\hspace{0.17em}{P}_{\phi }=0\},\text{if}\hspace{0.17em}{\mu }_2\ne 0\end{array}$$ 3.13

are invariant manifolds on which the dynamics is given by the one-degree of freedom system with the Hamiltonian

$$\tilde{H}(R,P_R)=\frac{P_R^2}{2}+\frac{{({\mu }_1+{\mu }_2)}^2}{2R^2}+V(R).$$ 3.14

Given that these are motions with constant ratio r₁(t)/r₂(t) of the polar radii, we introduce

Definition 3.1. —

Motions with a constant ratio of the polar radii, that is r₁(t) = λr₂(t) for some λ > 0 and all t, are called proportional motions.

Thus, we can state that proportional motions form an invariant manifold. Since the system associated with $\tilde{H}$ had one-degree of freedom, it is integrable. For future reference, we introduce

Definition 3.2. —

A potential $V:D\subset \mathbb{R}\to \mathbb{R}$ is attractive if V ′(∥q∥)≥0 for all q.

We also note that for any attractive potential V , V (R)≠ − 1/R², the equilibria of (3.14) are given by

$$(R,\phi ,P_R,P_{\phi })=(R_0,\pm \sqrt{\frac{{\mu }_2}{{\mu }_1}},0,0),$$

where R₀ is the root of R³V ′(R) = (μ₁ + μ₂)². (As the reader can easily verify, the case of the Jacobi potential V (R) =  − 1/R² is degenerate in the sense that either all R are equilibria or there are no equilibria alt all, and will be discussed elsewhere.) Using the reconstruction equation (3.4), every equilibrium on ${\mathcal{M}}_{\pm \phi }$ corresponds to orbits that either are quasi-periodic, or densely fill in a torus.

Various choices for V lead to different problems. Below we recall the Kepler problem and find the necessary and sufficient conditions for collisional motion.

(b). The Kepler problem

In this subsection, we consider the classical Newtonian potential in ${\mathbb{R}}^3$, that is V (q) =  − k/∥q∥, k > 0. As mentioned in the Introduction, while this is not the solution of the Laplace equation in ${\mathbb{R}}^4$ and so from a physical standpoint this is not the ${\mathbb{R}}^4$-gravitational potential, the induced dynamics is interesting from a theoretical standpoints.

$$H(\mathbf{q},\mathbf{p})=\frac{1}{2}{\mathbf{p}}^2-\frac{k}{\parallel \mathbf{q}\parallel },k>0.$$ 3.15

Following [15] in this case, the Laplace–Runge–Lenz vector

$$\mathbf{A}:=({\mathbf{p}}^2-\frac{1}{\sqrt{r_1^2+r_2^2}})\mathbf{q}-(\mathbf{q}\cdot \mathbf{p})\mathbf{p}$$

is a conserved quantity. Since it provides four integrals of motion, the dynamics drops to an integrable system. The dynamics resembles the Kepler problem in ${\mathbb{R}}^3$; for instance, for allowed negative energies h < 0, all orbits are ellipses Θ → p/(1 + ϵcosΘ), where p: = (μ²₁ + μ²₂)/k, $ϵ:=\sqrt{1+(2h({\mu }_1^2+{\mu }_2^2))/k^2}$ and Θ is the angle between q and A; for details, see [15].

Proposition 3.1. —

In the Kepler problem in ${\mathbb{R}}^4$, the motion is collisional if and only if μ = (μ₁, μ₂) = 0.

Proof. —

For the classical Newtonian potential, the energy conservation reads

$$\frac{1}{2}(p_{r_1}^2(t)+p_{r_2}^2(t))+\frac{{\mu }_1^2}{2r_1^2(t)}+\frac{{\mu }_2^2}{2r_2^2(t)}-\frac{k}{\sqrt{r_1^2(t)+r_2^2(t)}}=\mathrm{const}.=h.$$ 3.16

Let μ²₁ + μ²₂ > 0 and let us assume that lim_t → t*(r²₁(t) + r²₂(t)) = 0 for some t* ≤ ∞. Then at least one of μ₁ or μ₂ is non-zero; without loosing generality, say μ²₁≠0. As t → t*, the left-hand side of (3.16) tends to ∞, whereas the right-hand side is the finite energy, which is a contradiction.

If μ₁ = μ₂ = 0, then the reduced Hamiltonian (3.3) (with Newtonian interaction) becomes

$$H_{\mathrm{red}}(r_1,p_{r_1},r_2,p_{r_2})=\frac{1}{2}(p_{r_1}^2+p_{r_2}^2)-\frac{k}{\sqrt{r_1^2+r_2^2}}$$ 3.17

from where we have the equations of motion

$${\dot{r}}_i=p_{r_i}$$ 3.18

and

$${\dot{p}}_{r_i}=-\frac{r_1}{{(r_1^2+r_2^2)}^{3/2}},i=1,2.$$ 3.19

It is immediate that r_i(t) eventually becomes decreasing for all t greater than some t*. ▪

Remark 3.3. —

The same result is valid if V (r) = − 1/r^α with α < 2.

Proposition 3.2. —

In the Kepler problem in ${\mathbb{R}}^4$, collisional orbits are proportional motions.

Proof. —

For μ = 0, in polar coordinates r₁ = Rcosφ, r₂ = Rsinφ, the Hamiltonian (3.17) reads

$$H_{\mathrm{red}}(R,\phi ,P_R,P_{\phi })=\frac{P_R^2}{2}+\frac{P_{\phi }^2}{2R^2}-\frac{k}{R}$$ 3.20

and so it is identical to the classical planar Kepler problem in polar coordinates. For the latter, it is known that collision is attained only by motions on a straight line, that is those for φ(t) = constant = :φ₀. Thus along a collision path in ${\mathbb{R}}^4$, since tanφ₀ = r₁(t)/r₂(t), the conclusion follows. ▪

Remark 3.4. —

Along a collisional paths, the coordinates of the mass point is given by

$$x(t)=aR(t),y(t)=bR(t),z(t)=cR(t),w(t)=dR(t),$$ 3.21

where R(t) solves (3.20) and the constants a, b, c, d are determined by the initial conditions. All collisional paths are collinear.

4. The n-body problem

(a). Generalities

Consider n points with masses m₁, m₂, …, m_n in ${\mathbb{R}}^4$ with mutual interaction via some potential. Their positions is given by $\mathbf{q}=({\mathbf{q}}_1,{\mathbf{q}}_2,\dots {\mathbf{q}}_n)\in Q\subseteq {\mathbb{R}}^{4n}$ on which the symmetry group SO(2) × SO(2) acts diagonally on the principal planes. Further, the group acts on TQ and T*Q by tangent and co-tangent lift, respectively. The masses (or weights) of the points induce the mass metric on Q

$$⟨⟨\dot{\mathbf{q}},\dot{\mathbf{r}}⟩⟩:={\dot{\mathbf{q}}}^t\hspace{0.17em}\mathbb{M}\hspace{0.17em}\dot{\mathbf{r}},\dot{\mathbf{q}},\dot{\mathbf{r}}\in T{\mathbb{R}}^{4n},$$ 4.1

where $\mathbb{M}=\text{diag}(m_1{\mathbb{I}}_4,m_2{\mathbb{I}}_4,\dots ,m_n{\mathbb{I}}_4)i=1,2,\dots ,n$, where ${\mathbb{I}}_4$ is the 4 × 4 identity matrix, is the mass matrix.

The dynamics is given by the Lagrangian

$$\begin{array}{r}L(\mathbf{q},\dot{\mathbf{q}})=\frac{1}{2}{\dot{\mathbf{q}}}^t\mathbb{M}\hspace{0.17em}\dot{\mathbf{q}}-U(\mathbf{q}),\end{array}$$ 4.2

$$\begin{array}{rl}\text{where}U(\mathbf{q})& :=\sum _{1\le j<k\le n}V(\parallel {\mathbf{q}}_j-{\mathbf{q}}_k\parallel ).\end{array}$$ 4.3

Given the invariance of L to translations, one may prove that the linear momentum is conserved.

Thus, since the centre of mass has a rectilinear and uniform motion, we choose without loosing generality the location of the centre of mass to be the origin.

Remark 4.1 (Collinear, planar and spatial motions). —

Any line in ${\mathbb{R}}^4$ is an invariant manifold for the dynamics. More precisely if initially all points are on a line with velocities tangent to that line, then the points will remain on that line at all times. Similar statements are valid for motions in a plane (${\mathbb{R}}^2$) or hyperplane (${\mathbb{R}}^3$).

In Hamiltonian formulation, the dynamics is given by

$$H(\mathbf{q},\mathbf{p})=\frac{1}{2}{\mathbf{p}}^t{\mathbb{M}}^{-1}\mathbf{p}+U(\mathbf{q}),$$ 4.4

i.e.

$$H(\mathbf{q},\mathbf{p})=\frac{1}{2}{\mathbf{p}}^t{\mathbb{M}}^{-1}\mathbf{p}+\sum _{1\le j<k\le n}V(\parallel {\mathbf{q}}_j-{\mathbf{q}}_k\parallel )$$ 4.5

for some smooth $v:D\subseteq \mathbb{R}\to \mathbb{R}$ and where the momenta are denoted p = (p₁, p₂, …, p_n) with p_j = (p_jx, p_jy, p_jz, p_jw), j = 1, 2, …, n. The energy is given by the Hamiltonian H(q, p) and it is conserved along any solution. The (angular) momentum map is

$$J(\mathbf{q},\mathbf{p}):=(\sum _{j=1}^n(p_{jy}{q}_{jx}-p_{jx}{q}_{jy}),\sum _{j=1}^n(p_{jw}{q}_{jz}-p_{jz}{q}_{jw}))$$ 4.6

and by Noether theorem, since H is invariant with respect to the SO(2)_xy × SO(2)_zw action, J is conserved as well along any solution. In double-polar coordinates, we have

$$\begin{array}{rl}H(\mathbf{q},\mathbf{p})& =\frac{1}{2}\sum _{j=1}^n\hspace{0.17em}(p_{r_{j1}}^2+\frac{p_{{\theta }_1}^2}{m_j{r}_{j1}^2}+p_{r_{j2}}^2+\frac{p_{{\theta }_2}^2}{m_j{r}_{j1}^2})\\ & +\sum _{1\le j<k\le n}V(\sqrt{r_{j1}^2+r_{k1}^2-2r_{j1}{r}_{k1}\cos ({\theta }_{j1}-{\theta }_{k1})+r_{j2}^2+r_{k2}^2-2r_{j2}{r}_{k2}\cos ({\theta }_{j2}-{\theta }_{k2}}\hspace{0.17em})\end{array}$$ 4.7

and

$$J(\mathbf{q},\mathbf{p}):=(\sum _{j=1}^n{p}_{{\theta }_{j1}},\sum _{j=1}^n{p}_{{\theta }_{j2}}).$$ 4.8

(b). Relative equilibria are the Albouy–Chenciner balanced configurations

In the following two subsections, we are considering the full symmetry group SO(4).

Definition 4.1. —

A solution $(\mathbf{q}(t),\dot{\mathbf{q}}(t))$ of the n-body problem in ${\mathbb{R}}^4$ as given by the Lagrangian (4.2) is a relative equilibrium solution if there is group velocity $\hat{\mathit{\omega }}\in \mathrm{so}(4)$ so that

$$\mathbf{q}(t)=\exp (t\hat{\mathit{\omega }}){\mathbf{q}}_0\mathrm{and}\dot{\mathbf{q}}(t)=\exp (t\hat{\mathit{\omega }}){\dot{\mathbf{q}}}_0$$ 4.9

for some base point q₀∈Q. Alternatively, given $\hat{\mathit{\omega }}$, an element q₀∈Q such that (4.9) is a solution is called a relative equilibrium (RE) (with group velocity $\hat{\mathit{\omega }}$). If instead we use the Hamiltonian formulation (4.5), a solution (q(t), p(t)) is a relative equilibrium solution if there is a $\hat{\mathit{\omega }}\in \mathrm{so}(4)$ so that

$$\mathbf{q}(t)=\exp (t\hat{\mathit{\omega }}){\mathbf{q}}_0\mathrm{and}\mathbf{p}(t)=\text{FL}(\exp (t\hat{\mathit{\omega }}){\dot{\mathbf{q}}}_0)$$ 4.10

for some and q₀∈Q, where FL:TQ → T*Q is the Legendre transform.

Remark 4.2. —

Any relative equilibrium with respect to the SO(4) action is conjugate to a relative equilibrium with respect to the action of the double planar rotation group. Indeed, by proposition 2.2, for any relative equilibrium q₀ with group velocity $\hat{\mathit{\omega }}\in \mathrm{so}(4)$, there exists an orthogonal matrix P such that $P\hat{\mathit{\omega }}{P}^t\in \mathrm{SO}{(2)}_{xy}\times \mathrm{SO}{(2)}_{zw}$. Then the trajectory $\mathbf{r}(t):=P\exp (t\hat{\mathit{\omega }}){\mathbf{q}}_0$ satisfies $(\mathbf{r}(t),\dot{\mathbf{r}}(t))=(P\mathbf{q}(t),P\dot{\mathbf{q}}(t))$, so it is also a solution of the n-body problem, and

$$\mathbf{r}(t)=P\exp (t\hat{\mathit{\omega }})P^t\mathbf{r}(0),=\exp (tP\hat{\mathit{\omega }}{P}^t){\mathbf{r}}_0=\exp (t\widehat{P\mathit{\omega }}){\mathbf{r}}_0,$$

where r₀: = r(0). So r₀ is a relative equilibrium with group velocity $\widehat{P\mathit{\omega }}$.

The following proposition is standard, see e.g. [20].

Proposition 4.1. —

There is an RE with base point q₀ and group velocity $\hat{\mathit{\omega }}\in \mathrm{so}(4)$ if and only if q₀ is a critical point of the augmented potential

$$U_{\hat{\mathit{\omega }}}(\mathbf{q}):=U(\mathbf{q})-\frac{1}{2}{(\hat{\mathit{\omega }}\cdot \mathbf{q})}^t\mathbb{M}(\hat{\mathit{\omega }}\mathbf{q}),$$

$\mathbb{M}=\text{diag}(m_1{\mathbb{I}}_4,m_2{\mathbb{I}}_4,\dots ,m_n{\mathbb{I}}_4)$.

Note that in [20], the definition of the augmented potential is more general, involving the so-called locked inertia tensor. However for the n-body problem, the definition reduces to the form shown in the above proposition.

In our case, we are considering the diagonal action of SO(4) on ${\mathbb{R}}^{4n}$, so $\mathit{\omega }\cdot \mathbf{q}=(\hat{\mathit{\omega }}{\mathbf{q}}_1,\dots ,\hat{\mathit{\omega }}{\mathbf{q}}_n)$, and hence the augmented potential takes the form

$$\begin{array}{rl}{U}_{\mathit{\omega }}(\mathbf{q})& =U(\mathbf{q})-\frac{1}{2}\sum _j{(\hat{\mathit{\omega }}{\mathbf{q}}_j)}^t(m_j{\mathbb{I}}_4)(\hat{\mathit{\omega }}\mathbf{q})\\ & =U(\mathbf{q})-\frac{1}{2}\sum _j{m}_j{\mathbf{q}}_j^t{\hat{\mathit{\omega }}}^t\hat{\mathit{\omega }}{\mathbf{q}}_j.\end{array}$$

Thus, for all $j=1\dots n$,

$$\begin{array}{rl}{\mathrm{\nabla }}_j{U}_{\mathit{\omega }}(\mathbf{q})& ={\mathrm{\nabla }}_{j}U(\mathbf{q})-m_j{\hat{\mathit{\omega }}}^t\hat{\mathit{\omega }}{\mathbf{q}}_j\\ & ={\mathrm{\nabla }}_{j}U(\mathbf{q})+{\hat{\mathit{\omega }}}^2(m_j{\mathbf{q}}_j).\end{array}$$

It follows that q is a relative equilibrium with group velocity $\hat{\mathit{\omega }}$ if and only if

$${\mathrm{\nabla }}_{j}U(\mathbf{q})=-m_j{\hat{\mathit{\omega }}}^2{\mathbf{q}}_j\mathrm{for}\hspace{0.17em}\mathrm{all}\hspace{0.17em}j=1\dots n.$$ 4.11

Note that this criterion, which determines a relative equilibrium in the present context, is equivalent to the following definition introduced by Albouy & Chenciner [11] (also [14]):

Definition 4.2. —

A configuration is 4-balanced in ${\mathbb{R}}^4$ if there is a 4 × 4 antisymmetric matrix A such that, for all $j=1\dots n$,

$${\mathrm{\nabla }}_{j}U(\mathbf{q})=-m_j{A}^2{\mathbf{q}}_j.$$

Specifically, a configuration is 4-balanced if and only if is the base point of a relative equilibrium solution with group velocity given by A in the above definition, i.e. $\hat{\mathit{\omega }}=A$.

Another definition introduced in [11] and reproduced in [14] is:

Definition 4.3. —

A central configuration is an arrangement of the n-point masses whose configuration vector satisfies $\mathrm{\nabla }U(\mathbf{q})+\mathrm{\lambda }\mathbb{M}\mathbf{q}=0$ for some real constant λ.

This is clearly a special case of the relative equilibrium condition (4.11), with ${\hat{\mathit{\omega }}}^2=\mathrm{\lambda }{\mathbb{I}}_4$. In fact, as we note below, for ω = (ω₁, ω₂)∈SO(2)_xy × SO(2)_zw, we have ${\hat{\mathit{\omega }}}^2=\mathrm{diag}(-{\omega }_1^2,-{\omega }_1^2,-{\omega }_2^2,-{\omega }_2^2)$. Thus, q₀ is a central configuration if and only if q₀ is a relative equilibrium with group velocity ω = (ω, ω) with −ω² = λ.

(c). Collinear relative equilibria

An n-point collinear configuration, for some n > 1, is one satisfying

$${\mathbf{q}}_j={\mathrm{\lambda }}_j{\mathbf{q}}_0,$$ 4.12

for some λ_j≠0 for every j = 1, 2, …, n, all distinct, and ${\mathbf{q}}_0\in {\mathbb{R}}^4,{\mathbf{q}}_0\ne 0$.

Recall the relative equilibrium condition (4.11):

$${\mathrm{\nabla }}_{j}U(\mathbf{q})=-m_j{\hat{\mathit{\omega }}}^2{\mathbf{q}}_j\text{for all }j=1\dots n.$$

Given that U has the form $U(\mathbf{q})=\sum _{1\le i<j\le n}V(\parallel {\mathbf{q}}_i-{\mathbf{q}}_j\parallel )$, for a some function V , it follows that for every j,

$${\mathrm{\nabla }}_{j}U(\mathbf{q})=\sum _{i\ne j}{V}^{\prime }(\parallel {\mathbf{q}}_i-{\mathbf{q}}_j\parallel )\frac{{\mathbf{q}}_j-{\mathbf{q}}_i}{\parallel {\mathbf{q}}_i-{\mathbf{q}}_j\parallel }.$$

If q is collinear, i.e. satisfies (4.12), then

$$\begin{array}{rl}{\mathrm{\nabla }}_{j}U(\mathbf{q})& =\sum _{i\ne j}{V}^{\prime }(|{\mathrm{\lambda }}_i-{\mathrm{\lambda }}_j|\parallel {\mathbf{q}}_0\parallel )\frac{({\mathrm{\lambda }}_j-{\mathrm{\lambda }}_i){\mathbf{q}}_0}{|{\mathrm{\lambda }}_i-{\mathrm{\lambda }}_j|\parallel {\mathbf{q}}_0\parallel }\\ & =\sum _{i\ne j}{V}^{\prime }(|{\mathrm{\lambda }}_i-{\mathrm{\lambda }}_j|\parallel {\mathbf{q}}_0\parallel )\hspace{0.17em}\mathrm{sign}({\mathrm{\lambda }}_j-{\mathrm{\lambda }}_i)\frac{{\mathbf{q}}_0}{\parallel {\mathbf{q}}_0\parallel }.\end{array}$$

So q is a relative equilibrium if and only if, for every j,

$$\sum _i{V}^{\prime }(|{\mathrm{\lambda }}_i-{\mathrm{\lambda }}_j|\parallel {\mathbf{q}}_0\parallel )\hspace{0.17em}\mathrm{sign}({\mathrm{\lambda }}_j-{\mathrm{\lambda }}_i)\frac{{\mathbf{q}}_0}{\parallel {\mathbf{q}}_0\parallel }=-m_j{\mathrm{\lambda }}_j{\hat{\mathit{\omega }}}^2{\mathbf{q}}_0.$$ 4.13

In particular, any collinear relative equilibrium must satisfy: ${\hat{\mathit{\omega }}}^2{\mathbf{q}}_0$ is a scalar multiple of q₀. If ω = (ω₁, ω₂)∈so(2) × so(2) and q₀ = (q^xy₀, q^zw₀), then we have

$${\hat{\mathit{\omega }}}^2={\left[\begin{array}{cccc}0& -{\omega }_1& 0& 0\\ {\omega }_1& 0& 0& 0\\ 0& 0& 0& -{\omega }_2\\ 0& 0& {\omega }_2& 0\end{array}\right]}^2=\left[\begin{array}{cccc}-{\omega }_1^2& 0& 0& 0\\ 0& -{\omega }_1^2& 0& 0\\ 0& 0& -{\omega }_2^2& 0\\ 0& 0& 0& -{\omega }_2^2\end{array}\right]$$ 4.14

and

$${\hat{\mathit{\omega }}}^2{\mathbf{q}}_0=\left[\begin{array}{c}-{\omega }_1^2{\mathbf{q}}_0^{xy}\\ -{\omega }_2^2{\mathbf{q}}_0^{zw}\end{array}\right],$$ 4.15

which leads us to the following proposition.

Proposition 4.2. —

Let q be a collinear configuration, written as above, i.e.

$$\mathbf{q}=({\mathbf{q}}_1,\dots ,{\mathbf{q}}_n)=({\mathrm{\lambda }}_1{\mathbf{q}}_0,\dots ,{\mathrm{\lambda }}_n{\mathbf{q}}_0),$$

for some distinct non-zero λ_j's and some non-zero ${\mathbf{q}}_0\in {\mathbb{R}}^4$, and let ω = (ω₁, ω₂)∈so(2) × so(2). Then q is the base point of an RE with group velocity $\hat{\mathit{\omega }}$ if and only if the following scalar condition is satisfied

$$\sum _i{V}^{\prime }(|{\mathrm{\lambda }}_i-{\mathrm{\lambda }}_j|\parallel {\mathbf{q}}_0\parallel )\hspace{0.17em}\mathrm{sign}({\mathrm{\lambda }}_j-{\mathrm{\lambda }}_i)={\omega }^2{m}_j{\mathrm{\lambda }}_j\parallel {\mathbf{q}}_0\parallel$$ 4.16

and one of the following holds:

• (i)q₀ lies in the principal plane Oxy, and ω = ω₁;
• (ii)q₀ lies in the principal plane Ozw, and ω = ω₂;
• (iii)q₀ lies in neither of the principal planes, and ω = ω₁ = ω₂.

In all cases, there exists a two-dimensional subspace in which the RE remains, i.e. q_j(t) remains in the subspace for all j and all t. In cases (i) and (ii), this is the principal plane containing q₀. In all cases, the RE has underdetermined group velocity due to isotropy. In case (i), the group velocity is $\widehat{({\omega }_1,\ast )}$ for * an arbitrary element of so(2), while in case (ii) it is $\widehat{(\ast ,{\omega }_2)}$.

In case (iii), there are scaled projections of the RE into each of the principal planes that are themselves REs. Specifically, let q₀ = (q^xy₀, q^zw₀), and similarly for each q_j, and define

$${\mathrm{proj}}_{xy}\mathbf{q}=(({\mathbf{q}}_1^{xy},0),\dots ({\mathbf{q}}_n^{xy},0))=({\mathrm{\lambda }}_1({\mathbf{q}}_0^{xy},0),\dots {\mathrm{\lambda }}_n({\mathbf{q}}_0^{xy},0)),$$

and similarly proj_zwq. Then both (∥q₀∥/∥q^xy₀∥) proj_xyq and (∥q₀∥/∥q^zw₀∥) proj_zwq are relative equilibria with group velocity $\widehat{(\omega ,\ast )}$ or $\widehat{(\ast ,\omega )}$, respectively.

In general, for ω∈so(4), there exists an orthogonal change of coordinates that brings ω into so(2) × so(2), so the above analysis applies. In particular, for any collinear RE, there exists a two-dimensional subspace in which the RE remains.

Proof. —

As observed above, the RE condition (4.13) implies that ${\hat{\mathit{\omega }}}^2{\mathbf{q}}_0=k{\mathbf{q}}_0$ for some $k\in \mathbb{R}$, and from (4.15), this occurs if and only if one of the following is true: (i) q^xy₀ = 0 and k =  − ω²₂; (ii) q^zw₀ = 0 and k =  − ω²₁; (iii) q^xy₀≠0, q^zw₀≠0 and ω₁ = ω₂ = :ω and k =  − ω². In all three cases, (4.13) reduces to the scalar condition (4.16), with ω defined as in the proposition.

If q₀ is in a principal plane, then each q_j is also in that plane, and since the principal planes are invariant under the SO(2) × SO(2) action, any RE remains in that plane. Note that if q₀∈Oxy, then it has isotropy group SO(2)_zw, so the second component of the angular velocity ω₂ is undetermined; similarly, if q₀∈Ozw, then ω₁ is undetermined.

In case (iii), the RE still remains in a fixed two-dimensional subspace. This follows from the special form of the group velocity ω = (ω, ω), which says that both projections of q onto the principal planes rotate at the same speed. Indeed, let ${\mathbf{v}}_0^{xy}=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]{\mathbf{q}}_0^{xy}$ and ${\mathbf{v}}_0^{zw}=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]{\mathbf{q}}_0^{zw}$ and v₀ = (v^xy₀, v^zw₀). Then for any t and j,

$$\begin{array}{rl}\exp (t\hat{\mathit{\omega }}){\mathbf{q}}_j& ={\mathrm{\lambda }}_j\exp (t\hat{\mathit{\omega }}){\mathbf{q}}_0={\mathrm{\lambda }}_j(\exp (t\omega ){\mathbf{q}}_0^{xy},\exp (t\omega ){\mathbf{q}}_0^{zw})\\ & ={\mathrm{\lambda }}_j(\cos (t\omega ){\mathbf{q}}_0^{xy}+\sin (t\omega ){\mathbf{v}}_0^{xy},\cos (t\omega ){\mathbf{q}}_0^{zw}+\sin (t\omega ){\mathbf{v}}_0^{zw})\\ & ={\mathrm{\lambda }}_j(\cos (t\omega ){\mathbf{q}}_0+\sin (t\omega ){\mathbf{v}}_0).\end{array}$$

Hence

$$\begin{array}{rl}\exp (t\hat{\mathit{\omega }})\cdot \mathbf{q}& =({\mathrm{\lambda }}_1(\cos (t\omega ){\mathbf{q}}_0+\sin (t\omega ){\mathbf{v}}_0),\dots ,{\mathrm{\lambda }}_n(\cos (t\omega ){\mathbf{q}}_0+\sin (t\omega ){\mathbf{v}}_0))\\ & =\cos (t\omega )\mathbf{q}+\sin (t\omega )\mathbf{v},\mathrm{where}\mathbf{v}:=({\mathrm{\lambda }}_1{\mathbf{v}}_0,\dots ,{\mathrm{\lambda }}_n{\mathbf{v}}_0).\end{array}$$

Thus, the RE remains in the two-dimensional subspace spanned by q and v.

Finally, in case (iii), let r₀ = (∥q₀∥/∥q^xy₀∥)(q^xy₀, 0), and note that ∥r₀∥ = ∥q₀∥ and (∥q₀∥/∥q^xy₀∥)proj_xyq = (λ₁r₀, …, λ_nr₀). Therefore, the scalar condition (4.16) for q and $\hat{\mathit{\omega }}$ is the same as one that arises from (∥q₀∥/∥q^xy₀∥) proj_xyq and $\widehat{(\omega ,\ast )}$ (where the asterisk indicates an arbitrary value). A similar argument applies to the projections into the zw-plane. Hence both (∥q₀∥/∥q^xy₀∥) proj_xyq and (∥q₀∥/∥q^zw₀∥) proj_zwq are relative equilibria with group velocity $\widehat{(\omega ,\ast )}$ or $\widehat{(\ast ,\omega )}$, respectively. ▪

Remark 4.3. —

Since all collinear REs in ${\mathbb{R}}^4$ remain in a two-dimensional subspace, they have the same configurations as in the corresponding n-body problem in ${\mathbb{R}}^2$.

In ${\mathbb{R}}^3$, each RE of the n-body problem remains in a plane perpendicular to the angular velocity vector. The situation is almost the same for the n-body problem in ${\mathbb{R}}^4$, where ‘angular velocity’ is replaced by ‘group velocity’. Each RE still moves in a fixed plane, however due to isotropy, it is not always possible to determine this plane from the group velocity $\hat{\mathit{\omega }}$ alone. (Consider the case (iii) above.)

5. Regular n-gons and discrete reduction

When the masses are equal, due to the finite symmetries, we are able to detect low-dimensional invariant manifolds using the method of discrete reduction (reviewed below). These invariant manifolds will consist of equilibria and relative equilibria with configurations that are regular n-gons, so we begin with a general discussion of these in ${\mathbb{R}}^4$. We consider an n-gon to be an n-tuple of points, which are the vertices. By ‘regular’, we do not simply mean that the side lengths are equal (which would include e.g. all rhombuses), nor do we wish to consider only planar shapes. Instead, we follow Coxeter [22]: ‘A polygon (which may be skew) is said to be regular if it possesses a symmetry which cyclically permutes the vertices (and therefore also the sides) of the polygon’. Here, ‘skew’ means non-planar. The symmetry transformation is required to belong to a predefined group; for example, SO(2) gives rise to the usual planar regular polygons, centred at the origin. The requirement of a cyclic permutation means that the symmetry transformation, which we call the symmetry generator, must have order equal to the number of points n. (Recall that the order of a group element g is the smallest k such that g^k = e, the identity.) Since we are defining polygons by n-tuples (q₁, …, q_n), the order in which the points are listed matters; but by relabelling them if necessary, we may assume that there exists a symmetry transformation g such that gq_i = q_i+1(modn) for all i. In summary, we arrive at:

Definition 5.1. —

A regular n-gon in ${\mathbb{R}}^d$, with respect to a group of transformations G, is a configuration q = (q₁, …, q_n), with distinct points q_i, such that there exists an element g∈G of order n, called the symmetry generator, such that gq_i = q_i+1(modn) for all i.

In our application G = SO(4), and we have seen that every element of SO(4) is conjugate to an element of the double planar rotation group SO(2)_xy × SO(2)_zw. Thus, it suffices to consider G = SO(2)_xy × SO(2)_zw. The finite-order elements of this group are those of the form R = (R_2πa1/b1, R_2πa2/b2), for positive integers a₁, b₁, a₂, b₂, where R_θ is counterclockwise rotation by angle θ. Without loss of generality, we assume that a_j and b_j are relatively prime, for j = 1, 2, so that the order of R_2πaj/bj is b_j. It follows that the order of R is lcm(b₁, b₂), the least common multiple of b₁ and b₂. Note that if b₁ = b₂ = n, then the projections of the polygon onto each principal plane are also regular n-gons. Otherwise the projections are not injective, so the distinctness requirement in the above definition is not satisfied. However in general, (q₁₁, …, q_bj) is a regular b_j-gon. The projection of the original n-gon onto the xy-plane covers the b_j-gon n/b_j times.

Since a_j and b_j are relatively prime, the group generated by R_2πaj/bj is also generated by R_2π/bj. A b_j-gon in ${\mathbb{R}}^2$ with symmetry generator R_2π/bj is convex, meaning that the vertices are numbered in order around the polygon, and so joining the points in order gives a curve that bounds a convex set. A b_j-gon with symmetry generator R_2πaj/bj, for a_j∈2, …, b_j − 2 is non-convex, i.e. ‘star-shaped’, e.g. a pentagram. In this case, it is always possible to relabel the points of the b_j-gon so that it has symmetry generator R_2π/bj and is convex. Given an n-gon in ${\mathbb{R}}^4$ that projects to n-gons in both principal planes, by relabelling points if needed, it is possible to make either one of the two n-gons convex, but not necessarily both at once. We call an n-gon synchronized if it is possible to make both projections convex at once.

Proposition 5.1. —

Let n≥2. There are three types of regular n-gon in ${\mathbb{R}}^4$ with respect to SO(2)_xy × SO(2)_zw (and hence w.r.t. SO(4) as well):

• (i)Planar, lying in a two-dimensional plane in ${\mathbb{R}}^4$. Either the n-gon lies in a principal plane, or it projects to similar n-gons in each of the principal planes. These two n-gons are synchronized, meaning that there exists a labelling of the points such that each of the projected n-gons is convex.
• (ii)Non-planar, i.e. skew. The projections of the n-gon onto the two principal planes are not similar.
  • (a)Type I (unsynchronized n-gons): both projections are n-gons, but there does not exist a labelling of the points that makes both projections convex simultaneously.
  • (b)Type II (lower-order projections): the projection of the n-gon onto at least one principal plane is a regular b_j-gon for some b_j < n. In this case, n = lcm(b₁, b₂).

Proof. —

Let q = (q₁, …, q_n) be a regular n-gon in ${\mathbb{R}}^4$. If the n-gon is planar, then for every i, q_i is a linear combination of q₁ and q₂, which implies that q_ij is the same linear combination of q_1j and q_2j for j = 1, 2, so the two projected polygons are similar to each other, meaning that there exists an invertible linear transformation taking one to the other; since both projections are regular polygons, this just means that the order of the points on the polygon is the same in both projections. If the two projections of the n-gon are similar b-gons, with b₁ = b₂ = b, then b = n, since we must have n = lcm(b, b). There exists a (re)labelling of the original points (q₁, …, q_n) such that each of the projected n-gons is convex, since this is always true of a single n-gon, and the two n-gons are congruent. Conversely, suppose that both projections of the n-gons are also n-gons, i.e. for each j = 1, 2 the projected points (q_1j, …, q_nj) are distinct. Suppose there exists a (re)labelling of the original points (q₁, …, q_n) such that both of the projected n-gons are convex. Then they are congruent. For each j = 1, 2, each projected point q_ij is a linear combination of q_1j and q_2j. Since the two projected n-gons are congruent, the coefficients in this combination are independent of j, so each q_i is a linear combination of q₁ and q₂. This shows that the n-gon is planar.

The above argument also shows that if the two projections are unsynchronized n-gons, then the original n-gon q is non-planar. Finally, we consider the case (Type II) in which at least one of the projections of q is a b-gon for some b < n. In this case, it is not possible for the two projections to be congruent. ▪

Remark 5.1. —

Regular planar polygons may have any number of vertices. The smallest non-planar polygon of type I has n = 5 vertices, with one projection convex and the other a pentagram. The smallest non-planar polygon of type II has n = 4 vertices, with one projection a square and the other a digon (i.e. having two vertices). The next smallest non-planar polygon of type II has n = 6 vertices, with one projection a triangle and the other a digon. The three examples are the only non-planar polygons with fewer than seven vertices.

We now briefly recall the theory of discrete reduction as presented in [20] (but also see [25, pp. 203]). Let Σ be a finite group acting on a cotangent bundle T*Q, and consider its fixed point set:

$$\text{Fix}(\mathit{\Sigma },T^{\ast }Q):=\{(q,p)\in T^{\ast }Q\hspace{0.17em}|\hspace{0.17em}g(q,p)=(q,p)\hspace{0.17em}\mathrm{\forall }\hspace{0.17em}\sigma \in \mathit{\Sigma }\}.$$ 5.1

If a Hamiltonian $H:T^{\ast }Q\to \mathbb{R}$ is Σ-invariant and the symplectic structure is preserved under the Σ-action, then Fix (Σ, T*Q) is a symplectic submanifold of T*Q and it is an invariant manifold for the dynamics of H. Moreover, if the symplectic structure and H are also invariant under the action of a Lie group G giving rise to an equivariant momentum map, and the actions of G and Σ commute (or satisfy a more general compatibility condition) then $H{|}_{\mathrm{Fix}(\mathit{\Sigma },T^{\ast }Q)}\to \mathbb{R}$ is a G symmetric Hamiltonian system. By Palais' principle of criticality [21] any equilibrium or RE in Fix (Σ, T*Q) is also an equilibrium or RE, respectively, in the full T*Q phase space, where the RE are with respect to the same commuting group G mentioned above, if such a group exists.

Returning to the dynamics of n mass points in ${\mathbb{R}}^4$, let us assume all masses to be unity, i.e. m₁ = m₂ = … = m_n = 1. We consider regular n-gon configurations in ${\mathbb{R}}^4$, centred at the origin. As discussed above, these configurations have associated symmetry generators R∈SO(4), R = (R_(2πa1/b1), R_(2πa2/b2)), for positive integers a₁, b₁, a₂, b₂, with a_j and b_j relatively prime, for j = 1, 2, and n = lcm(b₁, b₂). Once the symmetry generator R is specified, the set of all n-gon configurations with this symmetry is

$$\{({\mathbf{q}}_1,\dots ,{\mathbf{q}}_n)\hspace{0.17em}:\hspace{0.17em}R{\mathbf{q}}_i={\mathbf{q}}_{i+1(\mathrm{mod}n)}\mathrm{for}\hspace{0.17em}\mathrm{all}\hspace{0.17em}i\}.$$ 5.2

We can express this set in terms of a group action as follows. For every k∈{0, …, n − 1}, let σ_k be the cyclic permutation of {1, …, n} given by σ_k(i) = (i − k)(mod n), i.e. a ‘backwards shift’ by k positions. Let $C_n={\mathbb{Z}}_n=\{0,\dots ,n-1\}$ be the finite cyclic group of order n, and let C_n act on $Q={({\mathbb{R}}^4)}^n\simeq {({\mathbb{R}}^2\times {\mathbb{R}}^2)}^n$ as follows:

$$k\cdot ({\mathbf{q}}_1,\dots ,{\mathbf{q}}_n)=({\mathcal{R}}^k{\mathbf{q}}_{{\sigma }_k(1)},\dots ,{\mathcal{R}}^k{\mathbf{q}}_{{\sigma }_k(n)}).$$

By abuse of notation, we denote this action by $C_n^{\mathcal{R}}$. Note that in particular, $1\cdot ({\mathbf{q}}_1,\dots ,{\mathbf{q}}_n)=(\mathcal{R}{\mathbf{q}}_{{\sigma }_1(1)},\dots ,\mathcal{R}{\mathbf{q}}_{{\sigma }_1(n)})$, and this generates the whole action. So

$$\begin{array}{rl}\mathrm{Fix}(C_n^{\mathcal{R}},Q)& =\{({\mathbf{q}}_1,\dots ,{\mathbf{q}}_n)\hspace{0.17em}:\hspace{0.17em}({\mathbf{q}}_1,\dots ,{\mathbf{q}}_n)=(\mathcal{R}{\mathbf{q}}_{{\sigma }_1(1)},\dots ,\mathcal{R}{\mathbf{q}}_{{\sigma }_1(n)})\}\\ & =\{({\mathbf{q}}_1,\dots ,{\mathbf{q}}_n)\hspace{0.17em}:\hspace{0.17em}R{\mathbf{q}}_i={\mathbf{q}}_{{\sigma }_1^{-1}(i)}\mathrm{for}\hspace{0.17em}\mathrm{all}\hspace{0.17em}i\}\\ & =\{({\mathbf{q}}_1,\dots ,{\mathbf{q}}_n)\hspace{0.17em}:\hspace{0.17em}R{\mathbf{q}}_i={\mathbf{q}}_{i+1(\mathrm{mod}n)}\mathrm{for}\hspace{0.17em}\mathrm{all}\hspace{0.17em}i\},\end{array}$$

which is the same expression as above in (5.2). Thus, $\text{Fix }(C_n^{\mathcal{R}},Q)$ consists of all regular n-gon configurations with symmetry generator R. Let ${\hat{Q}}^R=\text{Fix}(C_n^R,Q)$. In polar coordinates, the action C^R_n is expressed as

$$\begin{array}{rl}& k\cdot (r_{11},{\theta }_{11},r_{12},{\theta }_{12},\dots ,r_{n1},{\theta }_{n1},r_{n2},{\theta }_{n2})\\ & =(r_{{\sigma }_k(1)1},{\theta }_{{\sigma }_k(1)1}+\frac{2\pi a_{1}k}{b_1},r_{{\sigma }_k(1)2},{\theta }_{{\sigma }_k(1)2}+\frac{2\pi a_{2}k}{b_2},\\ & \dots ,r_{{\sigma }_k(n)1},{\theta }_{{\sigma }_k(n)1}+\frac{2\pi a_{1}k}{b_1},r_{{\sigma }_l(n)2},{\theta }_{{\sigma }_l(n)2}+\frac{2\pi a_{2}k}{b_2}).\end{array}$$

When R = (R_(2π/n), R_(2π/n)), which corresponds to planar n-gons, we have

$$(\mathrm{planarcase}){\hat{Q}}^R=\{\mathbf{q}\in Q\hspace{0.17em}|\hspace{0.17em}{r}_{ij}=r_{kj},\hspace{0.17em}{\theta }_{ij}-{\theta }_{{\sigma }_1(i)j}=\frac{2\pi }{n},\hspace{0.17em}\mathrm{for}\hspace{0.17em}\mathrm{all}\hspace{0.17em}i,k=1,2,\dots n,\hspace{0.17em}j=1,2\},$$

i.e. all configurations such that both projections are regular n-gons. Note that there is no restriction on the relationship between these two n-gons: they may have different radii and/or different phases θ_ij.

In all cases (planar or non-planar), every element of ${\hat{Q}}^R$ is determined by the two common radii r₁ and r₂, defined by r_j: = r_kj for any k; and the phases of the first point, θ₁: = θ₁₁ and θ₂: = θ₁₂. This observation defines double-polar coordinates (r₁, θ₁, r₂, θ₂) on ${\hat{Q}}^R$. The action of C^R_n lifts to an action on T*Q, expressed in polar coordinates as

$$\begin{array}{rl}& (k,l)\cdot (r_{11},{\theta }_{11},p_{r_{11}},p_{{\theta }_{11}},r_{12},{\theta }_{12},p_{r_{12}},p_{{\theta }_{12}},\dots ,r_{n1},{\theta }_{n1},p_{r_{n1}},p_{{\theta }_{n1}},r_{n2},{\theta }_{n2},p_{r_{n2}},p_{{\theta }_{n2}})\\ & =(r_{{\sigma }_k(1)1},{\theta }_{{\sigma }_k(1)1}+\frac{2\pi a_{1}k}{b_1},p_{r_{{\sigma }_k(1)1}},p_{{\theta }_{{\sigma }_k(1)1}},r_{{\sigma }_k(1)2},{\theta }_{{\sigma }_k(1)2}+\frac{2\pi a_{2}k}{b_2},p_{r_{{\sigma }_k(1)2}},p_{{\theta }_{{\sigma }_k(1)2}},\\ & \dots ,r_{{\sigma }_k(n)1},{\theta }_{{\sigma }_k(n)1}+\frac{2\pi a_{1}k}{b_1},p_{r_{{\sigma }_k(n)1}},p_{{\theta }_{{\sigma }_k(n)1}},r_{{\sigma }_k(n)2},{\theta }_{{\sigma }_k(n)2}+\frac{2\pi a_2}{b_2},p_{r_{{\sigma }_k(n)2}},p_{{\theta }_{{\sigma }_k(n)2}},).\end{array}$$

The fixed point set in T*Q is

$$\{(\mathbf{q},\mathbf{p})\in T^{\ast }Q\hspace{0.17em}|\hspace{0.17em}\mathbf{q}\in \hat{Q}\text{ and }{p}_{r_{ij}}=p_{r_{kj}}\text{ and }{p}_{{\theta }_{ij}}=p_{{\theta }_{kj}},\hspace{0.17em}\text{for all }i,k=1,2,\dots n,\hspace{0.17em}j=1,2\},$$

i.e. the momenta of all points are required to be equal when expressed in polar coordinates. Note that this equals $T^{\ast }\hat{Q}$, which can be checked directly, and also follows from a general result [20].

By the discrete reduction method, $T^{\ast }\hat{Q}$ is a symplectic invariant manifold and the dynamics on it is given by the restriction $\hat{H}$ of the Hamiltonian (4.5) to $T^{\ast }\hat{Q}$. Also, any RE of $\hat{H}$ is a RE of the full system.

Any planar configuration in $\hat{Q}$ projects to two regular n-gons with side lengths 2r_jsin(π/n), where r_j is the common radius of the jth n-gon, for j = 1, 2. So the original n-gon has side length

$$l=2\sqrt{r_1^2+r_2^2}\sin \frac{\pi }{n}.$$ 5.3

The distance between any two points on the n-gon, with angle kπ/n between them, is

$$(\mathrm{planarcase})2\sqrt{r_1^2+r_2^2}|\sin \frac{k\pi }{n}|.$$ 5.4

The general formula, for R = (R_(2πa1/b1), R_(2πa2/b2)), for the distance between q and R^kq, is

$$2\sqrt{r_1^2{\sin }^2\frac{k{a}_1\pi }{b_1}+r_2^2{\sin }^2\frac{k{a}_2\pi }{b_2}}.$$ 5.5

As a consequence, we have the following formula for the restricted Hamiltonian, where for j = 1, 2, the momentum of the jth n-gon in polar coordinates is (p_rj, p_θj)

$$\begin{array}{rl}& \hat{H}(r_1,{\theta }_1,p_{r_1},p_{{\theta }_1},r_2,{\theta }_2,p_{r_2},p_{{\theta }_2})\\ & =\frac{n}{2}(p_{r_1}^2+\frac{p_{{\theta }_1}^2}{r_1^2}+p_{r_2}^2+\frac{p_{{\theta }_2}^2}{r_2^2})+\sum _{1\le k<l\le n}V\left(2\sqrt{r_1^2{\sin }^2\frac{(l-k)a_1\pi }{b_1}+r_2^2{\sin }^2\frac{(l-k)a_2\pi }{b_2}}\right)\end{array}$$ 5.6

$$\begin{array}{rl}& =\frac{n}{2}(p_{r_1}^2+\frac{p_{{\theta }_1}^2}{r_1^2}+p_{r_2}^2+\frac{p_{{\theta }_2}^2}{r_2^2})+\frac{n}{2}\sum _{1\le k<n}V\left(2\sqrt{r_1^2{\sin }^2\frac{k{a}_1\pi }{b_1}+r_2^2{\sin }^2\frac{k{a}_2\pi }{b_2}}\right).\end{array}$$ 5.7

In the planar case, this simplifies to

$$\begin{array}{rl}& \hat{H}(r_1,{\theta }_1,p_{r_1},p_{{\theta }_1},r_2,{\theta }_2,p_{r_2},p_{{\theta }_2})\\ & =\frac{n}{2}(p_{r_1}^2+\frac{p_{{\theta }_1}^2}{r_1^2}+p_{r_2}^2+\frac{p_{{\theta }_2}^2}{r_2^2})+\frac{n}{2}\sum _{1\le k<n}V\left(2\sqrt{r_1^2+r_2^2}|\sin \frac{k\pi }{n}|\right).\end{array}$$ 5.8

Remark 5.2. —

If the angular velocity is zero, the dynamics on $T^{\ast }\hat{Q}$ consist of homothetic solutions.

It is immediate that due to the SO(2)_xy × SO(2)_zw-invariance of $\hat{H},$ the angular momentum is conserved and so (p_θ1(t), p_θ2(t)) = (c₁, c₂) for all t. Note that nc_j = μ_j, j = 1, 2, where μ = (μ₁, μ₂) is the total angular momentum of the (equal mass) n-body problem. The dynamics on $\hat{H}$ is reducible to a 2 d.f. system and the reduced Hamiltonian is

$$\begin{array}{rl}{\hat{H}}_{c_1,c_2}(r_1,p_{r_1},r_2,p_{r_2})& =\frac{n}{2}(p_{r_1}^2+\frac{c_1^2}{r_1^2}+p_{r_2}^2+\frac{c_2^2}{r_2^2})\\ & +\frac{n}{2}\sum _{1\le k<n}V\left(2\sqrt{r_1^2{\sin }^2\frac{k{a}_1\pi }{b_1}+r_2^2{\sin }^2\frac{k{a}_2\pi }{b_2}}\right).\end{array}$$ 5.9

If V is attractive (and so V ′(r) > 0 for all r > 0), then regular n-gon RE configurations arise as solutions of a system of two equations in r₁, r₂ of the following form, for j = 1 and j = 2:

$$\frac{c_j^2}{r_j^4}=\sum _{1\le k<n}\frac{V^{\prime }\left(2\sqrt{r_1^2{\sin }^2(k{a}_1\pi /b_1)+r_2^2{\sin }^2(k{a}_2\pi /b_2)}\right)}{\sqrt{r_1^2{\sin }^2(k{a}_1\pi /b_1)+r_2^2{\sin }^2(k{a}_2\pi /b_2)}}{\sin }^2\frac{k{a}_j\pi }{b_j}.$$ 5.10

In the planar case, the condition is

$$\frac{c_j^2}{r_j^4}=\sum _{1\le k<n}\frac{V^{\prime }\left(2\sqrt{r_1^2{\sin }^2(k\pi /n)+r_2^2{\sin }^2(k\pi /n)}\right)}{\sqrt{r_1^2{\sin }^2(k\pi /n)+r_2^2{\sin }^2(k\pi /n)}}{\sin }^2\frac{k\pi }{n}.$$ 5.11

In any case (planar or non-), we observe that for at least some (c₁, c₂) with c₁≠0 and c₂≠0, the systems above admits solutions r₁₀ = r₁₀(c₁, c₂), r₂₀ = r₂₀(c₁, c₂) and these satisfy

$$r_{20}=\gamma r_{10}\text{where}\hspace{0.17em}\gamma :=\sqrt{\left|\frac{c_2}{c_1}\right|}.$$ 5.12

The dynamics on the invariant manifolds of synchronized homographic motions coincide to the central force problem on ${\mathbb{R}}^4$ as studied in §3, with the reduced Hamiltonian given in (5.9).

6. The three-body problem

(a). Reduction

In this section, we reduce the 12 d.f. the three-body problem in ${\mathbb{R}}^4$ to a 6 d.f. system. This is possible due to translational and rotational SO(2)_xy × SO(2)_zw-symmetries of the dynamics.

Rather than applying the general symplectic reduction theory, we chose to deduce the reduced space and dynamics working specifically on our system.

Consider the three-body problem in ${\mathbb{R}}^4$ with the configuration given by $({\mathbf{q}}_1,{\mathbf{q}}_2,{\mathbf{q}}_3)\in {({\mathbb{R}}^4)}^3\setminus \{\text{possible collisions}\}$. We start by introducing the Jacobi coordinates

$$\mathbf{u}:={\mathbf{q}}_2-{\mathbf{q}}_1\mathrm{and}\mathbf{v}:={\mathbf{q}}_3-\frac{m_1{\mathbf{q}}_1+m_2{\mathbf{q}}_2}{m_1+m_2}$$ 6.1

that we write in double-polar coordinates

$$\mathbf{u}:=(R_1,{\mathit{\Theta }}_{R1},R_2,{\mathit{\Theta }}_{R2})\mathrm{and}\mathbf{v}:=(S_1,{\mathit{\Theta }}_{S1},S_2,{\mathit{\Theta }}_{S2}).$$ 6.2

The Hamiltonian (4.5) reads

$$\begin{array}{rl}H& =\frac{1}{2M_1}(P_{R_1}^2+\frac{P_{{\mathit{\Theta }}_{R1}}^2}{R_1^2}+P_{R_2}^2+\frac{P_{{\mathit{\Theta }}_{R2}}^2}{R_2^2})+\frac{1}{2M_2}(P_{S_1}^2+\frac{P_{{\mathit{\Theta }}_{S1}}^2}{S_1^2}+P_{S_2}^2+\frac{P_{{\mathit{\Theta }}_{S2}}^2}{S_2^2})\\ & -V(d_{12})-V(d_{13})-V(d_{23}),\end{array}$$ 6.3

where M₁: = m₁m₂/(m₁ + m₂) and M₂ = m₃(m₁ + m₂)/(m₁ + m₂ + m₃), and

$$\begin{array}{rl}& d_{12}=\sqrt{R_1^2+R_2^2},\\ & d_{13}=\sqrt{S_1^2+{\alpha }_1^2{R}_1^2-2{\alpha }_1{R}_1{S}_1\cos ({\mathit{\Theta }}_{S1}-{\mathit{\Theta }}_{R1})+S_2^2+{\alpha }_1^2{R}_2^2-2{\alpha }_1{R}_2{S}_2\cos ({\mathit{\Theta }}_{S2}-{\mathit{\Theta }}_{R2})}\\ \text{and}& d_{23}=\sqrt{S_1^2+{\alpha }_2^2{R}_1^2+2{\alpha }_2{R}_1{S}_1\cos ({\mathit{\Theta }}_{S1}-{\mathit{\Theta }}_{R1})+S_2^2+{\alpha }_2^2{R}_2^2+2{\alpha }_2{R}_2{S}_2\cos ({\mathit{\Theta }}_{S2}-{\mathit{\Theta }}_{R2})}\end{array}$$

with α₁ = m₂/(m₁ + m₂) and α₂ = m₁/(m₁ + m₂). The symmetry becomes obvious after performing the (symplectic) change of variables

$$({\mathit{\Theta }}_{Rj},{\mathit{\Theta }}_{Sj})\to ({\mathit{\Phi }}_j,{\mathit{\Psi }}_j):=({\mathit{\Theta }}_{Sj}-{\mathit{\Theta }}_{Rj},{\mathit{\Theta }}_{Sj})$$ 6.4

and

$$(P_{{\mathit{\Theta }}_{Rj}},P_{{\mathit{\Theta }}_{Sj}})\to (P_{{\mathit{\Phi }}_j},P_{{\mathit{\Psi }}_j}):=(P_{{\mathit{\Theta }}_{Rj}},P_{{\mathit{\Theta }}_{Rj}}+P_{{\mathit{\Theta }}_{Sj}}),j=1,2.$$ 6.5

The Hamiltonian (6.3) becomes

$$\begin{array}{rl}H& =\frac{1}{2M_1}(P_{R_1}^2+\frac{P_{{\mathit{\Phi }}_1}^2}{R_1^2}+P_{R_2}^2+\frac{P_{{\mathit{\Phi }}_2}^2}{R_2^2})\\ & +\frac{1}{2M_2}(P_{S_1}^2+\frac{{(P_{{\mathit{\Psi }}_1}-P_{{\mathit{\Phi }}_1})}^2}{S_1^2}+P_{S_2}^2+\frac{{(P_{{\mathit{\Psi }}_2}-P_{{\mathit{\Phi }}_2})}^2}{S_2^2})\\ & -V(d_{12})-V(d_{13})-V(d_{23})\end{array}$$ 6.6

with

$$\begin{array}{rl}& d_{12}=\sqrt{R_1^2+R_2^2},\end{array}$$ 6.7

$$\begin{array}{r}{d}_{13}=\sqrt{S_1^2+{\alpha }_1^2{R}_1^2-2{\alpha }_1{R}_1{S}_1\cos {\mathit{\Phi }}_1+S_2^2+{\alpha }_1^2{R}_2^2-2{\alpha }_1{R}_2{S}_2\cos {\mathit{\Phi }}_2}\end{array}$$ 6.8

$$\begin{array}{r}\text{and}{d}_{23}=\sqrt{S_1^2+{\alpha }_2^2{R}_1^2+2{\alpha }_2{R}_1{S}_1\cos {\mathit{\Phi }}_1+S_2^2+{\alpha }_2^2{R}_2^2+2{\alpha }_2{R}_2{S}_2\cos {\mathit{\Phi }}_2}.\end{array}$$ 6.9

It is immediate that along any solution the angular momentum is conserved and so

$$(P_{{\mathit{\Psi }}_1}(t),P_{{\mathit{\Psi }}_2}(t))=\mathrm{const}.=({\mu }_1,{\mu }_2).$$ 6.10

Thus, denoting the reduced configurations space M: = [0, ∞) × (0, ∞) × (0, ∞) × (0, ∞) × S¹ × S¹\ {possible collisions}, we obtain the reduced Hamiltonian

$$\begin{array}{rl}{H}_{\mathrm{equil}}& :T^{\ast }M\to \mathbb{R},\\ H_{\mathrm{equil}}& =\frac{1}{2M_1}(P_{R_1}^2+\frac{P_{{\mathit{\Phi }}_1}^2}{R_1^2}+P_{R_2}^2+\frac{P_{{\mathit{\Phi }}_2}^2}{R_2^2})\\ & +\frac{1}{2M_2}(P_{S_1}^2+\frac{{({\mu }_1-P_{{\mathit{\Phi }}_1})}^2}{S_1^2}+P_{S_2}^2+\frac{{({\mu }_2-P_{{\mathit{\Phi }}_2})}^2}{S_2^2})\\ & -V(d_{12})-V(d_{13})-V(d_{23}).\end{array}$$ 6.11

The equations of motion are

$$\begin{array}{rl}{\dot{R}}_j& =\frac{P_{R_j}}{M_1},\end{array}$$ 6.12

$$\begin{array}{rl}{\dot{P}}_{R_j}& =\frac{P_{{\mathit{\Phi }}_j}^2}{M_1{R}_j^3}-\frac{V^{\prime }(d_{12})}{d_{12}}{R}_j-\frac{V^{\prime }(d_{13})}{d_{13}}({\alpha }_1^2{R}_j-{\alpha }_1{S}_j\cos {\mathit{\Phi }}_j)\\ & -\frac{V^{\prime }(d_{23})}{d_{13}}({\alpha }_2^2{R}_j+{\alpha }_2{S}_j\cos {\mathit{\Phi }}_j),\end{array}$$ 6.13

$$\begin{array}{rl}{\dot{S}}_j& =\frac{P_{S_j}}{M_2},\end{array}$$ 6.14

$$\begin{array}{rl}{\dot{P}}_{S_j}& =\frac{{({\mu }_j-P_{{\mathit{\Phi }}_j})}^2}{M_2{S}_j^3}-\frac{V^{\prime }(d_{13})}{d_{13}}(S_j-{\alpha }_1{R}_j\cos {\mathit{\Phi }}_j)-\frac{V^{\prime }(d_{23})}{d_{13}}(S_j+{\alpha }_2{R}_j\cos {\mathit{\Phi }}_j),\end{array}$$ 6.15

$$\begin{array}{rl}{\dot{\mathit{\Phi }}}_j& =\frac{P_{{\mathit{\Phi }}_j}}{M_1{R}_j^2}-\frac{{\mu }_j-P_{{\mathit{\Phi }}_j}}{M_2{S}_j^2}\end{array}$$ 6.16

$$\begin{array}{rl}\text{and}{\dot{P}}_{{\mathit{\Phi }}_j}& =-\frac{V^{\prime }(d_{13})}{d_{13}}({\alpha }_1{R}_j{S}_j\sin {\mathit{\Phi }}_j)+\frac{V^{\prime }(d_{23})}{d_{23}}({\alpha }_2{R}_j{S}_j\sin {\mathit{\Phi }}_j).\end{array}$$ 6.17

(b). Relative equilibria

Recall that in a Lie-symmetric mechanical system, RE solutions project to equilibria in a reduced space, and vice versa, any equilibrium in a reduced space lifts to an RE solution in the unreduced space (for details, see [20]).

In the case of the three-body problem previously presented, for every μ = (μ₁, μ₂)\(0, 0), the RE (of momentum μ) are found as the equilibria of the system (6.12)–(6.17). We observe that equating to zero the r.h.s. of the (6.17) we obtain

$$R_j{S}_j\sin {\mathit{\Phi }}_j({\alpha }_1\frac{V^{\prime }(d_{13})}{d_{13}}-{\alpha }_2\frac{V^{\prime }(d_{23})}{d_{23}})=0,j=1,2$$ 6.18

and so

• —
  either sinΦ_j = 0 i.e. the projections of u and v are parallel on the principal planes, i.e.

  $${\text{Proj}}_{xy}(\mathbf{u})\parallel {\text{Proj}}_{xy}(\mathbf{v})\text{and}{\text{Proj}}_{zw}(\mathbf{u})\parallel {\text{Proj}}_{zw}(\mathbf{v});$$

  these RE correspond to the collinear Euler configurations in the classical three-body problem;
• —
  or

  $$m_2\frac{V^{\prime }(d_{13})}{d_{13}}=m_1\frac{V^{\prime }(d_{23})}{d_{23}},$$ 6.19

  where we use that α₁ = m₂/(m₁ + m₂) and α₂ = m₁/(m₁ + m₂).

Remark 6.1. —

If we chose V to be the Newtonian potential, i.e. for any two mass points m_j, m_k we have V (d_ij) =  − m_jm_k/d_jk then the condition above leads to d₁₃ = d₂₃, and so any non-collinear three-body RE triangle is isosceles. This was also proved (with a different method) in the general case of the n-body problem in ${\mathbb{R}}^n$, n even, by Albouy & Chenciner [11].

(c). Stability of equilateral triangles

In this section, we study the stability of the equilateral triangle solutions in the case of three equal masses interacting via a generic attractive potential.

From §5, equations (5.11) and (5.12), given $(c_1,c_2)\in \in {\mathbb{R}}^2\setminus \{(0,0)\}$ the RE configuration polar radii receive the form

$$r_1=:r_0\mathrm{and}{r}_2=\gamma r_0$$ 6.20

where r₀ solves

$$r^3{V}^{\prime }(r\sqrt{3(1+{\gamma }^2)})=\frac{2\sqrt{3(1+{\gamma }^2)}}{9}{c}_1^2$$ 6.21

where

$$\gamma :=\sqrt{\left|\frac{{\mu }_2}{{\mu }_1}\right|}>0.$$

Note that μ_j = 3c_j, j = 1, 2.

To ease notation, we write μ: = μ₁ = 3c₁. We assume that we are in a generic situation in which for any (μ, γ) in some non-void domain, equation (6.21) has at least one solution r₀ = r₀(μ, γ). The RE with the polar radii above project into the equilibria

$$(R_1,R_2,S_1,S_2,{\mathit{\Phi }}_1,{\mathit{\Phi }}_2)=(r_0\sqrt{3},\hspace{0.17em}\gamma r_0\sqrt{3},\hspace{0.17em}\frac{3r_0}{2},\hspace{0.17em}\frac{3\gamma r_0}{2},-\frac{\pi }{2},-\frac{\pi }{2})$$ 6.22

and

$$(P_{R_1},P_{R_2},P_{S_1},P_{S_2},P_{{\mathit{\Phi }}_1},P_{{\mathit{\Phi }}_2})=(0,0,0,0,\frac{\mu }{2},\frac{{\gamma }^2\mu }{2})$$ 6.23

of system (6.12)–(6.17). We also note that the side of the equilateral triangles formed by the three bodies is

$$l=r_0\sqrt{3(1+{\gamma }^2)}.$$ 6.24

We will show, in theorem 6.1, that for generic potentials and generic values of μ and γ, the equilateral configuration RE is unstable.

Recall that an equilibrium z_e of a Hamiltonian system with Hamiltonian H is Lyapunov (or nonlinearly) stable if the Hessian D²H(z_e) is positive definite; linearly stable if the linearization matrix $\mathbb{J}{D}^{2}H({\mathbf{z}}_e)$, where $\mathbb{J}=\left[\begin{array}{cc}{\mathbb{O}}_6& {\mathbb{I}}_6\\ -{\mathbb{I}}_6& {\mathbb{O}}_6\end{array}\right]$, is semi-simple (diagonalizable) and has all eigenvalues purely imaginary; and spectrally stable if none of its eigenvalues has a positive real part [26]. Since the eigenvalues of a Hamiltonian system always appear in quadruples of the form ± Reλ ± i Im λ (or pairs, in the case of real or purely imaginary values) [27], spectral stability is only possible if all eigenvalues are pure imaginary. Even in this case, the equilibrium is unstable if the semi-simple-nilpotent decomposition of the linearization has a non-trivial nilpotent component [28]. For symplectic Hamiltonian systems, since the phase space is even dimensional, if there is a zero eigenvalue, then its algebraic multiplicity must be even (since all other eigenvalues must come in quadruples or pairs).

At an equilibrium z_e = (R^e₁, R^e₂, S^e₁, S^e₂, Φ^e₁, Φ^e₁, 0, 0, 0, 0, P^e_Φ1, P^e_Φ2)∈ T*M of the reduced system (6.12)–(6.17), the Hamiltonian Hessian D²H_equil(z_e) is a 12 × 12 matrix arranged into four 6 × 6 blocks

$$D^2{H}_{\mathrm{equil}}({\mathbf{z}}_e)=\left[\begin{array}{cc}A& B\\ B^t& D\end{array}\right]$$ 6.25

with

$$\begin{array}{rl}A& =\left[\begin{array}{ccc}[a_1]& {\mathbb{O}}_2& {\mathbb{O}}_2\\ {\mathbb{O}}_2& [a_2]& {\mathbb{O}}_2\\ {\mathbb{O}}_2& {\mathbb{O}}_2& {\mathbb{O}}_2\end{array}\right]+[D^{2}V]{|}_{\mathbf{z}={\mathbf{z}}_e},\end{array}$$ 6.26

$$\begin{array}{rl}[a_1]& =\left[\begin{array}{cc}\frac{3P_{{\mathit{\Phi }}_1}^2}{M_1{R}_1^4}& 0\\ 0& \frac{3P_{{\mathit{\Phi }}_2}^2}{M_1{R}_2^4}\end{array}\right],[a_2]=\left[\begin{array}{cc}\frac{{({\mu }_1-P_{{\mathit{\Phi }}_1})}^2}{M_2{S}_1^4}& 0\\ 0& \frac{{({\mu }_2-P_{{\mathit{\Phi }}_2})}^2}{M_2{S}_2^4}\end{array}\right],\end{array}$$ 6.27

$$\begin{array}{rl}B& =\left[\begin{array}{ccc}{\mathbb{O}}_2& {\mathbb{O}}_2& [b_1]\\ {\mathbb{O}}_2& {\mathbb{O}}_2& [b_2]\\ {\mathbb{O}}_2& {\mathbb{O}}_2& {\mathbb{O}}_2\end{array}\right],\end{array}$$ 6.28

$$\begin{array}{rl}[b_1]& =\left[\begin{array}{cc}-\frac{P_{{\mathit{\Phi }}_1}}{M_1{R}_1^3}& 0\\ 0& -\frac{P_{{\mathit{\Phi }}_2}}{M_1{R}_2^3}\end{array}\right],[b_2]=\left[\begin{array}{cc}\frac{({\mu }_1-P_{{\mathit{\Phi }}_1})}{M_2{S}_1^3}& 0\\ 0& \frac{({\mu }_2-P_{{\mathit{\Phi }}_2})}{M_2{S}_2^3}\end{array}\right],\end{array}$$ 6.29

$$\begin{array}{rl}D& =\left[\begin{array}{ccc}[d_1]& {\mathbb{O}}_2& {\mathbb{O}}_2\\ {\mathbb{O}}_2& [d_2]& {\mathbb{O}}_2\\ {\mathbb{O}}_2& {\mathbb{O}}_2& [d_3]\end{array}\right],\end{array}$$ 6.30

$$\begin{array}{rl}[d_1]& =\text{diag}\left(\frac{1}{M_1}\right),[d_2]=\text{diag}\left(\frac{1}{M_2}\right)\end{array}$$ 6.31

$$\begin{array}{rl}\text{and}[d_3]& =d\left[\begin{array}{cc}1& 0\\ 0& \gamma \end{array}\right],\text{where}\hspace{0.17em}d=\frac{1}{M_1}\text{diag}\left(\frac{1}{R_1}\right)+\frac{1}{M_2}\text{diag}\left(\frac{1}{S_1}\right).\end{array}$$ 6.32

We now consider the equilateral triangular RE given by formulae (6.22) and (6.23). In this case, the Hessian matrix D²V |_z=ze takes the form

$$D^{2}V{|}_{\mathbf{z}={\mathbf{z}}_e}\left[\begin{array}{cccccc}\ast & \ast & \ast & \ast & 0& 0\\ \ast & \ast & \ast & \ast & 0& 0\\ \ast & \ast & \ast & \ast & 0& 0\\ \ast & \ast & \ast & \ast & 0& 0\\ 0& 0& 0& 0& a& {\gamma }^{2}a\\ 0& 0& 0& 0& {\gamma }^{2}a& {\gamma }^{4}a\end{array}\right],$$ 6.33

where * are the appropriate partial derivatives evaluated at the RE ∂V_ij|_z=ze, 1 ≤ i ≤ j ≤ 4 entries, and

$$a:=\frac{3\sqrt{3}}{8(1+{\gamma }^2)\sqrt{1+{\gamma }^2}}(l{V}^{″}(l)-V^{\prime }(l)).$$ 6.34

Let us denote

$$[a]:=\left[\begin{array}{cc}a& {\gamma }^{2}a\\ {\gamma }^{2}a& {\gamma }^{4}a\end{array}\right]=a\left[\begin{array}{cc}1& {\gamma }^2\\ {\gamma }^2& {\gamma }^4\end{array}\right].$$

The eigenvalues λ₁, λ₂ of [a], and corresponding eigenvectors u₁ and u₂, are

$$\begin{array}{rll}& {\mathrm{\lambda }}_1=0,& {\mathbf{u}}_1=(-{\gamma }^2,1)\\ \text{and}& {\mathrm{\lambda }}_2=a({\gamma }^4+1),& {\mathbf{u}}_2=(1,{\gamma }^2).\end{array}\}$$ 6.35

Since [a] has a zero eigenvalue, it follows from the structure given above (including equation (6.33)) that D²H_equil(z_e) also has a zero eigenvalue. The corresponding eigenspace is

$$\ker D^2{H}_{\mathrm{equil}}({\mathbf{z}}_e)=\{({\mathbf{v}}_1,\dots ,{\mathbf{v}}_6)\in {\mathbb{R}}^{12}\hspace{0.17em}|\hspace{0.17em}{\mathbf{v}}_3\in \ker [a],{\mathbf{v}}_4={\mathbf{v}}_5=0\hspace{0.17em}\mathrm{and}({\mathbf{v}}_1,{\mathbf{v}}_2,{\mathbf{v}}_6)\in \ker [F]\},$$

where

$$[F]=\left[\begin{array}{ccc}\ast & \ast & [b_1]\\ \ast & \ast & [b_2]\\ [b_1]& [b_2]& [d_3]\end{array}\right],$$

in which the upper left-hand 2 × 2 block equals that of A. Note that for generic potentials V , and generic values of μ and γ, the matrix [F] is nonsingular, and hence $\ker D^2{H}_{\mathrm{equil}}({\mathbf{z}}_e)$ has dimension 1.

Since $\mathbb{J}$ is invertible, it follows that the linearization $\mathbb{J}{D}^2{H}_{\mathrm{equil}}({\mathbf{z}}_e)$ also has a kernel of dimension at least 1 under the condition det[F]≠0, the kernel of $\mathbb{J}{D}^2{H}_{\mathrm{equil}}({\mathbf{z}}_e)$ has dimension exactly 1. The infinitesimally symplectic structure of $\mathbb{J}{D}^2{H}_{\mathrm{equil}}({\mathbf{z}}_e)$ implies that the algebraic multiplicity of the zero eigenvalue is at least two. Thus, if det[F]≠0, the Jordan normal form of $\mathbb{J}{D}^2{H}_{\mathrm{equil}}({\mathbf{z}}_e)$ contains a block of the form

$$\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]$$

or, equivalently, the semi-simple-nilpotent decomposition has a non-trivial nilpotent term.

Finally, in a spectrally stable system, the presence of a non-trivial nilpotent term induces a ‘drift’ in the dynamics, implying (nonlinear) instability (for instance [28]). Thus, we have proven the following.

Theorem 6.1. —

For generic potentials, the equilateral configuration RE is unstable.

For example, for the classical Newtonian potential V (r) = − 1/r, numerical evidence shows that the det[F] is strictly negative for all permitted values of μ and γ and so the ‘Newtonian’ equilateral configuration RE is unstable. The same is valid for the Jacobi potential V (r) = − 1/r², with the observation that in this case any r > 0 is a solution of (6.21) (i.e. there is a continuum of solutions) as long as μ and γ lie along the curves given by μ² = 3/(2(1 + γ²)²).

7. Final remarks

The present study is the tip of the iceberg of new and exciting research. One may ask a multitude of questions: Is there an equivalent of Bertrand's theorem in ${\mathbb{R}}^4$? For what potentials can we regularize collisions? What are the equivalent of the figure 8 solutions [29]? What is the equivalent of Saari's conjecture [30] and is it true in ${\mathbb{R}}^4$? How many REs can be found for a fixed n≥3 and what is their stability? etc. Also, an interesting topic is given by considering the gravitational potential in ${\mathbb{R}}^4$, that is the Jacobi potential V (r) = − k/r², k > 0; note that the latter marks the threshold between weak and strong forces as defined in physics. Last but not least, one may also study the generalization of the n-body problem in ${\mathbb{R}}^n$, n≥4, continuing the research started here and elsewhere. We leave all these for future endeavours.

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The authors have contributed equally.

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The authors declare that they have no competing interests.

Funding

The authors have been supported by NSERC Discovery grants.