On the n-body problem on surfaces of revolution

TL;DR

This paper investigates the n-body problem on surfaces of revolution, revealing new dynamical properties, symmetry results, and bifurcation phenomena, extending classical mechanics to curved geometries.

Contribution

It introduces a geometric framework for the n-body problem on surfaces of revolution, proving new results about invariant manifolds, symmetry, and bifurcations not previously known.

Findings

Saari's conjecture fails on certain surfaces of revolution.

Homographic motions with equal masses form an invariant manifold.

Regular n-gon relative equilibria undergo pitchfork bifurcations.

Abstract

We explore the $n$-body problem, $n\geq 3,$ on a surface of revolution with a general interaction depending on the pairwise geodesic distance. Using the geometric methods of classical mechanics we determine a large set of properties. In particular, we show that Saari's conjecture fails on surfaces of revolution admitting a geodesic circle. We define homographic motions and, using the discrete symmetries, prove that when the masses are equal, they form an invariant manifold. On this manifold the dynamics are reducible to a one-degree of freedom system. We also find that for attractive interactions, regular $n$-gon shaped relative equilibria with trajectories located on geodesic circles typically experience a pitchfork bifurcation. Some applications are included.

Equations

$$G(x)=-m_i{m}_j\cot x\text{on}{\mathbb{S}}^2\text{and}G(x)=-m_i{m}_j\coth x\text{on}{\mathbb{H}}_{\text{one}}^2$$

$$(z,\phi )\to \mathbf{x}(z,\phi ):=\left(f(z)\cos \phi ,f(z)\sin \phi ,z\right)$$

$$\frac{d\phi }{dz}=\frac{c}{f(z)}\frac{\sqrt{f^{\prime 2}(z)+1}}{\sqrt{f^2(z)-c^2}}$$

$$f^2(z)\dot{\phi }(z)=f^2(z_0)\dot{\phi }(z_0)=const.=c.$$

$$f(z)\cos \theta =c$$

$$d:(-a,b)\times (-a,b)\times {\mathcal{S}}^1\times {\mathcal{S}}^1$$

$$\left(z_1,z_2,{\phi }_1,{\phi }_2\right)$$

$$d\left(z_1,z_2,{\phi }_1,{\phi }_2\right)=d\left(z_2,z_1,({\phi }_2-{\phi }_1)\right).$$

$$L(\mathbf{q},\dot{\mathbf{q}})=T_{\mathbf{q}}(\dot{\mathbf{q}})-V(\mathbf{q})$$

$$T_{\mathbf{q}}(\dot{\mathbf{q}})=\sum\limits_{i=1}^{n}m_{i}\left<\dot{\mathbf{q}}_{i}\,,\dot{\mathbf{q}}_{i}\right>:=\sum\limits_{i=1}^{N}\frac{1}{2}m_{i}\left[\dot{z}_{i}\,\,\dot{\varphi}_{i}\right]\left[\begin{array}[]{cc}1+f^{\prime 2}(z_{i})&0\\ 0&f^{2}(z_{i})\end{array}\right]\left[\begin{array}[]{c}\dot{z}_{i}\\ \dot{\varphi}_{i}\end{array}\right]$$

$$V(\mathbf{q}):=\sum _{1\le i<j\le n}G(d({\mathbf{q}}_i,{\mathbf{q}}_j))=\sum _{1\le i<j\le n}G(d(z_i,z_j,{\phi }_i,{\phi }_j))$$

$$m_i\frac{d}{dt}\left[(1+f^{\prime 2}(z_i)){\dot{z}}_i\right]=m_i{\dot{z}}_i^2{f}^{\prime }(z_i)f^{\prime \prime }(z_i)+m_i{\dot{\phi }}^{2}f(z_i)f^{\prime }(z_i)-\frac{\partial V}{\partial z_i}$$

$$m_i\frac{d}{dt}\left[f^2(z_i){\dot{\phi }}_i\right]=-\frac{\partial V}{\partial {\phi }_i}i=1,2,\dots ,n.$$

$${\dot{z}}_i=v_{z_i}$$

$${\dot{v}}_{z_i}=-\frac{f^{\prime \prime }(z_i)f^{\prime }(z_i)}{1+f^{\prime 2}(z_i)}{v}_{z_i}^2+\frac{f(z_i)f^{\prime }(z_i)}{1+f^{\prime 2}(z_i)}{v}_{{\phi }_i}^2-\frac{2f(z_i)f^{\prime }(z_i)}{1+f^{\prime 2}(z_i)}{v}_{z_i}^2-\frac{1}{m_i(1+f^{\prime 2}(z_i))}\frac{\partial V}{\partial z_i},$$

$${\dot{\phi }}_i=v_{{\phi }_i}$$

$${\dot{v}}_{{\phi }_i}=-\frac{2f^{\prime }(z_i)}{f(z_i)}{v}_{z_i}{v}_{{\phi }_i}-\frac{1}{m_i{f}^2(z_i)}\frac{\partial V}{\partial {\phi }_i}i=1,2,\dots ,n.$$

$$v_{z_i}(t_0)=0\text{for all}i.$$

$${\dot{z}}_i(t_0)=0$$

$$\displaystyle\dot{v}_{z_{i}}(t_{0})=-\frac{1}{m_{i}}\frac{\partial V}{\partial z_{i}}\Big{|}_{{\mathbf{q}}_{i}={\mathbf{q}}_{i0}}\,.$$

$$\frac{\partial d_{ij}({\mathbf{q}}_{i}\,{\mathbf{q}}_{j})}{\partial z_{i}}\Big{|}_{t=t_{0}}=0.$$

$$\displaystyle\frac{\partial V({\mathbf{q}})}{\partial z_{i}}\Big{|}_{{\mathbf{q}}_{i}={\mathbf{q}}_{i0}}=\sum\limits_{j\neq i\,,j=1}^{n}\,g^{\prime}\,\left(d({\mathbf{q}}_{i0},{\mathbf{q}}_{j0})\right)\frac{\partial d({\mathbf{q}}_{i}\,{\mathbf{q}}_{j})}{\partial z_{i}}\Big{|}_{t=t_{0}}=0$$

$$H(\mathbf{q},\mathbf{p})=K_{\mathbf{q}}(\dot{\mathbf{q}})+V(\mathbf{q})$$

$$K_{\mathbf{q}}({\mathbf{p}})=\frac{1}{2}\sum\limits_{i=1}^{n}\frac{1}{2m_{i}}\left[p_{z_{i}}\,p_{\varphi_{i}}\right]\left[\begin{array}[]{cc}\frac{1}{1+f^{\prime 2}(z_{i})}&0\\ 0&\frac{1}{f^{2}(z_{i})}\end{array}\right]\left[\begin{array}[]{c}p_{z_{i}}\\ p_{\varphi_{i}}\end{array}\right]$$

$${\dot{z}}_i=\frac{p_{z_i}}{m_i(1+f^{\prime 2}(z_i))},{\dot{p}}_{z_i}=\frac{f^{\prime }(z_i)f^{\prime \prime }(z_i)}{m_i(1+f^{\prime 2}(z_i){)}^2}{p}_{z_i}^2+\frac{f^{\prime }(z_i)}{m_i{f}^3(z_i)}{p}_{{\phi }_i}^2-\frac{\partial V}{\partial z_i},$$

$${\dot{\phi }}_i=\frac{p_{{\phi }_i}}{m_i{f}^2(z_i)},{\dot{p}}_{{\phi }_i}=-\frac{\partial V}{\partial {\phi }_i}.$$

$$(R_{\alpha },(z,\phi ,v_z,v_{\phi }))\to (z,\phi +\alpha ,v_z,v_{\phi }),$$

$$(R_{\alpha },(z,\phi ,p_z,p_{\phi }))\to (z,\phi +\alpha ,p_z,p_{\phi }).$$

$$\omega\cdot(z,\varphi)=\frac{d}{dt}\Big{|}_{t=0}\exp(t\omega)(z,\varphi)=(0\,,\omega)\,,\quad\quad\omega\in so(2)\,,\,\,(z,\varphi)\in\mathbb{M}\,.$$

$$\omega \cdot \mathbf{q}$$

$$=\omega \cdot ((z_1,{\phi }_1),(z_2,{\phi }_2),\dots ,(z_n,{\phi }_n))$$

$$=(\omega \cdot (z_1,{\phi }_1),\omega \cdot (z_2,{\phi }_2),\dots ,\omega \cdot (z_n,{\phi }_n))=((0,\omega ),(0,\omega )\dots ,(0,\omega )).$$

$$J(\mathbf{q},\mathbf{p})=J\left((z_1,{\phi }_1,p_{z_1},p_{{\phi }_1}),(z_2,{\phi }_2,p_{z_2},p_{{\phi }_2}),\dots ,(z_n,{\phi }_n,p_{z_n},p_{{\phi }_n})\right)=p_{{\phi }_1}+p_{{\phi }_2}+\dots +p_{{\phi }_n}.$$

$$J(\mathbf{q}(t),\mathbf{p}(t))=p_{{\phi }_1}(t)+p_{{\phi }_2}(t)+\dots +p_{{\phi }_n}(t)=const.:=\mu .$$

$${⟨\mathbb{I}(\mathbf{q})(\omega ),\eta ⟩}_{so(2)}:=\ll \omega \cdot \mathbf{q},\eta \cdot \mathbf{q}\gg$$

Full text

1. 1 Introduction

2. 2 Motion on a surface of revolution

3. 2.1 Surfaces of revolution

4. 2.2 Lagrangian formulation and geodesics as invariant manifolds

5. 2.3 Hamiltonian formulation and momentum conservation

6. 2.4 Relative equilibria

7. 2.5 Motion on a cylinder and the “vertical linear momentum” integral

8. 3 Saari’s conjecture on surfaces of revolution

9. 4 Equal masses and homographic motions

10. 4.1 Homographic motions as a symplectic invariant manifold

11. 4.2 Discrete reduction and the equal mass homographic invariant manifold

12. 4.3 Dynamics. The generalized Clairaut relation

13. 4.4 Homographic dynamics on a cylinder

14. 4.5 Hill’s regions and topology of the phase space

15. 5 Lagrangian homographic relative equilibria

16. 5.1 Some general existence criteria

17. 5.2 Bifurcations of geodesic Lagrangian homographic RE and curvature

18. 6 Some examples

19. 6.1 Homographic dynamics for the “quasi-harmonic” interaction

20. 6.1.1 Quasi-harmonic homographic motion on $\mathbb{S}^{2}$

21. 6.1.2 Quasi-harmonic homographic motion on a peanut-like surface

22. 6.2 Gravitational homographic motion on $\mathbb{S}^{2}$ and $\mathbb{H}_{\text{one}}^{2}$

23. 7 Acknowledgements

1 Introduction

There are many generalizations of the classical $n$-body problem. For instance, one may modify or generalize the interaction potential ([4]), or enlarge the configuration space to a higher dimensional Euclidean space ([1]), or endow the configuration space with a non-Euclidean structure ([28, 37]). In particular, in the last decade a good body of work was dedicated to the study of the curved $n$-body problem, that is the generalization of the classical $n$-body problem to surfaces of constant curvature ([5, 9, 10, 12, 15, 16, 23, 32, 39, 36, 40]). Also recently, a unified formulation of $n$-body and $n$-vortex dynamics on Riemann surfaces was advanced, bringing together two related but quite distinct systems ([3]).

In this paper we report findings on the generalization of the $n$-body problem, $n\geq 3$, to a system confined to a surface of revolution. The mass points are interacting pairwise via some potential depending on the shortest geodesic path. The potential remains unspecified until the final section of the paper when we present some examples.

Using the geometric methods of classical mechanics, we find a large set of properties. We retrieve the integrals of motion, find which geodesics are configurations of invariant manifolds, and give a negative answer to the generalization of Saari’s conjecture on surfaces of revolution admitting at least one geodesic circle (e.g. a 2-sphere). Using discrete symmetries, we also prove that when all masses are equal, homographic motions (defined below) form an invariant manifold, and recognize a generalization of the Clairaut relation on surfaces of revolution. One of the most interesting properties is that for motions with (unspecified) attractive interaction, regular $n$-gon shaped relative equilibria with trajectories located on a geodesic circles normal to the rotation axis typically undergo a pitchfork bifurcation. To our knowledge, this property was not remarked upon even in particular cases such as the gravitational equal mass $3$-body problem on a 2-sphere (see, for instance, [12, 23]).

The generalized $n$-body problem on a surface of revolution is a mechanical system with $2n$ degrees of freedom symmetric with respect to rotations about the axis of revolution. We model the dynamics in Lagrangian and Hamiltonian formulations. Considering the axis of revolution of the configuration surface as vertical, we remind the reader that the geodesics of a surface of revolution are either meridians, parallel circles at “heights” at which the generatrix has a critical point, or Clairaut curves. (see Section 2.1 or [30]). We prove that motions with initial positions on, and velocities tangent to, parallel geodesic circles or meridians are invariant manifolds, a result natural from a physics standpoint. While one might expect that motions aligned to Clairaut curves also form invariant manifolds, this is true only locally. We retrieve the conservation of energy and angular momentum, the latter due to the invariance of the dynamics to rotations about the axis of revolution. We also calculate the moment of inertia associated to the rotational symmetry of the surface, and give some criteria for the existence of relative equilibria (RE). We observe that the dynamics on a cylinder exhibits of an extra integral, namely the linear momentum along the vertical; moreover, an associated “vertical centre of mass” integral is also present.

Recall the so-called Saari’s conjecture: proposed in 1970 by Don Saari in the context of the planar $n$-body problem, it claims that solutions with constant moment of inertia are RE. This conjecture has been proven for the planar 3-body problem with equal masses in [26] and for the general planar $3$-body problem in [27]. Several researchers have worked on various restrictions or generalizations of this conjecture ([8, 33, 20, 22, 35]). In recent years, it was tackled by Diacu et al. ([10]) in the context of the $3$-body problem on surfaces of constant curvature. In this paper we consider the natural generalization of the Saari conjecture to $n$-body problems on surfaces of revolution. The definition of moment of inertia is generalized in this context using using the more general geometric mechanics notion of locked inertia tensor, which in this case is a scalar quantity. The history of the moment of inertia as understood in celestial versus classical mechanics may be found in [13]. We show that the Saari’s conjecture on surfaces of revolution which admit at least one geodesic parallel circle is not true. The result is independent of the potential or the curvature of the surface and, in particular, it settles the generalization of Saari’s Conjecture on a 2-sphere.

By definition, in the classical $n$-body problem, homographic solutions are those for which the configuration formed by the bodies at a given time moves in the inertial barycentric coordinates system in such a way as to remain similar to itself when $t$ varies - see [41]. Without the conservation of linear momentum and consequently, in the absence of a barycentric system, homographic solutions are defined as those with trajectories forming a self-similar shape in the ambient $\mathbb{R}^{3}$ space (see [11]). Here we define homographic motions as the set of homographic solutions which, besides maintaing a self-similar shape in $\mathbb{R}^{3}$, have the points located on a plane orthogonal to the axis of revolution at all times. Homographic motions include as special cases homothetic motions (the bodies move without rotations along the surface meridians), and RE.

Recall that for an ODE system with discrete symmetries, an effective method to obtain invariant manifolds of solutions consists in restricting the dynamics to the associated fixed point spaces (see, for instance, [17]). This method, known as discrete reduction, was specialized to the symplectic and the cotangent bundle categories ([18, 24, 29]), with the outcome that the fixed point spaces are symplectic and cotangent bundle systems, respectively. Also, by Palais’ Principle of Symmetric Criticality ([31]), any RE in a fixed point space is a RE in the full phase space. Moreover, one can demonstrate the lack of stability of certain RE, by proving that an RE is unstable on a fixed point space (and thus is unstable in the full phase-space).

When the masses are equal, we apply Discrete Reduction to show the presence of homographic motions. In our context, these form a symplectic invariant manifold with solutions for which the configuration of the bodies maintains a regular polygonal shape at all times. We recognize the homographic dynamics as given by a two degrees of freedom system of the form “kinetic + potential” with rotational symmetry. We then show that equal mass homographic solutions obey a generalized Clairaut relation, which is essentially a reformulation of the angular momentum conservation. Further, using the conservation of angular momentum, we reduce the dynamics to a one degree of freedom, and thus integrable, system parametrized by the energy and the angular momentum. As known, for such systems a sketch of the amended potential is sufficient to provide a complete qualitative picture of the dynamics ([2, 38]). Consequently, if the generatrix of the surface of revolution and the binary potential are specified, then one is able to completely describe the dynamics on the equal mass homographic invariant manifold, both quantitatively (i.e. the solutions up to some quadrature) and qualitatively (i.e. the topological portrait of the phase space and all orbit types).

We remark that the same problem was recently considered in a purely mathematical context by Fomenko et al. [14]: the geodesic flow on a surface of revolution augmented by a rotationally symmetric potential. This flow can be recognized as the homographic flow for the equal mass $n$-body problem (modulo a straight-forward scaling) on surface of revolution. In this interesting study, the authors provide the topological picture of the phase space in the terms of the so-called Fomenko-Zieschang invariants.

We next study Lagrangian homographic RE, that is RE which are solutions on the equal mass homographic invariant manifold with their configuration on a plane orthogonal to the symmetry axis. These RE maintain a regular $n$-gon configuration and rotate with (an appropriate) constant angular velocity on a parallel circle. After some existence criteria, we state and prove one of the main results in this paper (Proposition 5.7 and Remark 5.8): if the binary interaction is attractive then a Lagrangian homographic RE with its trajectory on a geodesic circle typically undergoes a pitchfork bifurcation. The bifurcation parameter is the angular momentum.

We choose to not specify any potentials until the last section, where we demonstrate our theoretical findings on some examples. First we choose an attractive interaction, which we call quasi-harmonic, given by a pairwise potential of the form $G(x)=x^{2}/2$, where $x$ is the (geodesic) distance between two unit mass points. This potential has the exceptional feature that the sum of the potential terms can be collapsed to a single term. (A $n$-body potential is a sum of $n(n-1)/2$ terms; in general, this sum does not collapse.) Thus, in our examples we obtain the bifurcation momenta and their location as expressions (depending on the generatrix) for general $n$. Using this potential, we describe the homographic dynamics for motions on the unit sphere $\mathbb{S}^{2}$, a symmetric peanut-like surface and a one-sheet hyperboloid $\mathbb{H}^{2}_{\text{one}}$ with a geodesic circle of unit radius .

Next, we consider homographic motions on $\mathbb{S}^{2}$ and $\mathbb{H}^{2}_{\text{one}}$ with 3-d gravitational potentials. Recall that the later were found as solutions of the Laplace equation on 3-d surfaces of constant curvature and then restricted to $\mathbb{S}^{2}$ and $\mathbb{H}^{2}_{\text{one}}$ (for more on gravitational potentials and historical notes, see for instance, [9, 40]). These potentials are

[TABLE]

where $x$ is the (geodesic) distance between two points of mass $m_{i}$ and $m_{j}$. Note that for the potential on $\mathbb{S}^{2}$, configurations with diametrically opposite points are ill-defined; in particular, for motions on $\mathbb{S}^{2}$, we thus consider Lagrangian homographic RE for $n$ odd only. Using Proposition 5.7 we find the condition which guarantees that a Lagrangian homographic RE with its trajectory on the Equator will experience a subcritical pitchfork bifurcation. (To simplify exposition, we also call Equator the geodesic circle of $\mathbb{H}^{2}_{\text{one}}$.) We verify this condition for $n=3$ and conjecture that it will be fulfilled for any $n\geq 5$ odd.

A physically relevant harmonic or gravitational potential on a general surface of revolution varies with the curvature. The gravitational case was discussed by Santoprete [34], whereas the harmonic case awaits investigation. (As a remark, it would be interesting to find the proper harmonic potential on surfaces of revolution using a symmetry Cayley-Klein-type approach such as in the paper of Cari$\tilde{\text{n}}$ena et al [6]). The work we present here focuses on finding generic properties for the $n$-body problem of revolution, leaving for the future more involved studies for specific potentials.

The paper is organized as follows: in Section 2 we set up the problem in Lagrangian and Hamiltonian formulations. We discuss geodesics as invariant manifolds, deduce the conservation laws and the RE existence conditions, and observe on the existence of an additional conservation law for motions on a cylinder. In Section 3 we prove that the generalization of Saari’s conjecture fails on surfaces of revolution with at least one geodesic parallel circle. In Section 4 we define homographic motions, show that they form an invariant manifold, and state and prove a generalization of the Clairaut relation. Further, since homographic motions are given by a two degree of freedom Hamiltonian system of the form “kinetic +potential” with rotational symmetry, we reduce the dynamics to a one degree of freedom system and remind the reader that a full qualitative picture of the dynamics is provided by the analysis of the amended potential. In Section 5 we retrieve some RE existence criteria, and state and prove Proposition 5.7 on the bifurcations of RE with trajectories on a geodesic circle. In Section 6 we discuss apply our findings on some examples for the quasi-harmonic and gravitational interactions.

2 Motion on a surface of revolution

2.1 Surfaces of revolution

Consider a surface of revolution $\mathbb{M}$ generated by rotations about the vertical $Oz$ axis of a profile smooth curve $f=f(\cdot)$ defined on some open $(-a,b)$ interval with $a$ and $b$ finite or $\infty$. We chose to parametrize $\mathbb{M}$ by

[TABLE]

with $z\in(-a,b)\,,\,\varphi\in{\mathbb{S}}^{1}$, where $z\to f(z)$ is the given smooth profile curve with $f(z)>0$. Recall that the geodesics of a surface of revolution are:

• geodesic circles, that is parallel circles $z=z_{c}=$constant, where $z_{c}$ is a critical point of the generatrix $f(z)$, and

• Clairaut curves, that is curves $\varphi=\varphi(z)$ that are solutions of ODEs

[TABLE]

where

[TABLE]

The latter category includes meridians $\varphi=$ constant obtained for $c=0.$ The relation (4) is the expression of the angular momentum for the dynamics conservation associated to the geodesic equations. For future reference, we the same relation written as

[TABLE]

where $\theta\in(0\,,\pi/2)$ is the angle between the geodesic and a parallel circle is known as the Clairaut relation (see [7], pp. 257).

On $\mathbb{M}$ we define the distance function $d:\mathbb{M}\times\mathbb{M}\longrightarrow\mathbb{R}$ , $d\,({\mathbf{q}}_{1},{\mathbf{q}}_{2}):=$ the shortest distance from ${\mathbf{q}}_{1}$ to ${\mathbf{q}}_{2}$ along a geodesic arc . We observe that in the parametrization above

[TABLE]

and that $d$ is rotationally invariant, that is

[TABLE]

2.2 Lagrangian formulation and geodesics as invariant manifolds

Consider $n$ mass points $P_{i}$ of mass $m_{i}$ on $\mathbb{M}$ that are interacting mutually via a potential depending on the (shortest) geodesic distance between the points. Denote the coordinates of $P_{i}$ by ${\mathbf{q}}_{i}=(z_{i},\varphi_{i})$, $i=1,2\ldots,n$, and let ${\mathbf{q}}:=({\mathbf{q}}_{1},{\mathbf{q}}_{2},\ldots,{\mathbf{q}}_{n}).$ The configurations space is ${\mathbb{Q}}:={\mathbb{M}}^{n}\setminus\left\{\right.$collisions and configurations where the vector field is undefined (see Remark 2.1 below)$\left.\right\}$. The dynamics is given by the Lagrangian $L:T{\mathbb{Q}}\to\mathbb{R}$

[TABLE]

where $T$ is a mass-weighted metric on ${\mathbb{M}}^{n}$ defined by

[TABLE]

and $V({\mathbf{q}})$ is the interaction potential. We assume that any two points, say $P_{i}$ and $P_{j}$ of coordinates ${\mathbf{q}}_{i}$ and ${\mathbf{q}}_{j},$ respectively, interact via a law depending on $d({\mathbf{q}}_{i},{\mathbf{q}}_{j})$. Further, we assume that the potential $V$ is either independent of the masses, and so

[TABLE]

where $G:D\to\mathbb{R}$, $D\subseteq[0,\infty)$, is some given smooth function. Throughout the paper, unless otherwise stated, $G$ is assumed to be non-constant.

The Euler-Lagrange equations of motion associated to $L$ are

[TABLE]

or

[TABLE]

Remark 2.1

Throughout the paper, we assume that all the vector fields are defined on domains on which they are smooth enough for the purpose in question. In particular, we assume that configurations for which the Euler-Lagrange vector field (13)-(16) is not at least ${\cal C}^{1}$ are excluded.

For instance, observe that for a pair of bodies the distance function $d$ defined by (6) is smooth at all points on its domain, except at those for which the bodies are on the same parallel circle and diametrically opposite. On such a circle, the distance between the bodies $\tilde{d}(\varphi):=d(z,z,\varphi,\varphi-\pi/2)$ behaves as the absolute value function $f(\varphi)=|\varphi-\pi/2|$ at $\varphi=\pi/2$ and so it is Lipschitz. Since we take $G$ smooth, the composition function $G\circ d$ is at least Lipschitz on its domain $\tilde{D}:=\{(z_{i},z_{j},\varphi_{i},\varphi_{j})\in(-a,b)\times(-a,b)\times\mathcal{S}^{1}\times\mathcal{S}^{1}\,|\,d((z_{i},z_{j},\varphi_{i},\varphi_{j}))\in D\}$ and so the local existence and uniqueness of the ODE solutions of the Euler-Lagrange equations is guaranteed everywhere on the $\tilde{D}$. However, part of our study requires better smoothness of the vector field (in particular the analysis of the relative equilibria bifurcations).Thus, without further notice, we consider that all the vector fields appearing in the paper are sufficiently smooth.

Definition 2.2

The generalized n-body problem on a surface of revolution consists in the dynamics induced by the Lagrangian (8) with a potential of the form (10).

Remark 2.3

If the potential is constant the Euler-Lagrange (13)-(16) become the geodesic equations on $\mathbb{M}$ for $n$ free mass points.

Proposition 2.4** (Parallel geodesic circles are invariant manifolds)**

Assume that $z=z_{c}$ is an isolated critical point of $f$ and so $z=z_{c}$ is a parallel geodesic circle. If the bodies have their initial positions on the parallel geodesic circle $z=z_{c}$ and their initial velocities tangent to that circle, then their motion, on its domain of existence, will remain on that geodesic circle for all times.

Proof: At the initial time $t_{0}$ we have the initial positions ${\mathbf{q}}_{i0}=\left(z_{i}(t_{0})\,,\varphi_{i}(t_{0}))\right)=(z_{c}\,,\varphi_{i0})$ and null components of the velocities along the parallel circle $z=z_{c}$, i.e.

[TABLE]

Thus at the initial time the ODE system (13)-(14) reads:

[TABLE]

Thus it is sufficient to show that \displaystyle{\frac{\partial V}{\partial z_{i}}\Big{|}_{{\mathbf{q}}_{i}={\mathbf{q}}_{i0}}=0\,.} The key observation is that since at the initial time all distances $d\left({\mathbf{q}}_{i0}\,,{\mathbf{q}}_{j0}\right)$ are arcs of the (same) parallel geodesic circle, the tangent vectors to $d\left({\mathbf{q}}_{i0}\,,{\mathbf{q}}_{j0}\right)$ are tangent to that parallel geodesic circle. Equivalently, we have that at the initial time, the component along the meridian $\varphi=\varphi_{i0}$ of the tangent vector to the arc $d\left({\mathbf{q}}_{i0}\,,{\mathbf{q}}_{j0}\right)$ is null, i.e.

[TABLE]

Since

[TABLE]

the conclusion follows.$\,\square$

Analogously, we have

Proposition 2.5** (Meridians are invariant manifolds)**

If the bodies have their initial positions on a meridian with their initial velocities tangent to that meridian, then their motion, on its domain of existence, will remain on that meridian circle for all times.

2.3 Hamiltonian formulation and momentum conservation

By applying the Legendre transform to (8) we find the corresponding Hamiltonian $H:T^{*}\mathbb{Q}\to\mathbb{R}$

[TABLE]

with the kinetic energy

[TABLE]

where ${\mathbf{p}}_{i}:=(p_{z_{i}}\,p_{\varphi_{i}})\in\mathbb{R}^{2}$ and ${\mathbf{p}}:=({\mathbf{p}}_{1},{\mathbf{p}}_{2},\ldots{\mathbf{p}}_{n})$. The equations of motion are

[TABLE]

Remark 2.6

If the potential is constant we retrieve the Hamiltonian formulation of the dynamics of $n$ free mass points on $\mathbb{M}$. The trajectories $\left(z_{i}(t)\,,\varphi_{i}(t)\right)$ describe geodesics on the surface.

The rotation group $SO(2)$ acts isometrically on $\mathbb{M}$ by $(R_{\alpha},(z,\varphi))\to(z,\varphi+\alpha)\,,$ where $R_{\alpha}$ represents a rotation of angle $\alpha$ and $(z,\varphi)\in\mathbb{M}\,.$ Further, $SO(2)$ acts on $T\mathbb{M}$ by

[TABLE]

and on $T^{*}\mathbb{M}$ by

[TABLE]

The associated infinitesimal generator vector field is

[TABLE]

The rotational actions on $\mathbb{M},$ $T\mathbb{M}$ and $T^{*}\mathbb{M}$ extend naturally to diagonal acting on $\mathbb{M}^{n}$, $T\mathbb{M}^{n}$ and $T^{*}\mathbb{M}^{n}$. For instance, the infinitesimal generator corresponding to the rotation group action on $T\mathbb{M}^{n}$ is given by

[TABLE]

Associated to the rotational group action on $T^{*}\mathbb{M}^{n}$ is the (angular) momentum map $J:T^{*}\mathbb{M}^{n}\to so^{*}(2)$ which may be calculated using the co-tangent bundle formula $\left<J({\mathbf{q}},{\mathbf{p}})\,,\omega\right>=\left<{\mathbf{p}}\,,\omega\cdot{\mathbf{q}}\right>$ for all $\omega\in so(2)$ (see [21]). We obtain

[TABLE]

Since the Hamiltonian (19) is rotationally invariant under the diagonal action of the rotation group on $T^{*}\mathbb{Q},$ by Noether theorem, the momentum $J({\mathbf{q}},{\mathbf{p}})$ is constant along any solution $\displaystyle{\left({\mathbf{q}}(t),{\mathbf{p}}(t)\right)}$, and so

[TABLE]

The effect of rotations on the bodies is given by the (locked) inertia tensor, denoted $\mathbb{I},$ which maps each point ${\mathbf{q}}$ to a linear application $\mathbb{I}({\mathbf{q}}):so(2)\to so^{*}(2)$

[TABLE]

where $\left<\cdot\,,\cdot\right>_{so(2)}$ is the pairing between $so^{(}2)$ and its dual $so^{*}(2)$ (in our case, just number multiplication), and $\ll\cdot\,,\cdot\gg$ is the metric on $T\mathbb{M}^{n}.$ Since $\text{dim}\,so(2)=1,$ the inertia tensor $\mathbb{I}$ is a scalar and is in fact the analogue of the usual moment of inertia associated to the action of the rotation group on the plane. Using (26) and (9) we obtain that the moment of inertia for the motion of $n$ mass points on a surface of revolution is

[TABLE]

2.4 Relative equilibria

For a Hamiltonian system with continuous (Lie) symmetry, relative equilibria (RE) are solutions which are also (one-parameter) orbits of the symmetry group ([24]). In our case, taking into account the action (24), its diagonal extension to $\mathbb{M}^{n}$ and the lift to $T^{*}\mathbb{M}^{n}\supset T^{*}\mathbb{Q}$ , these solutions must be of the form

[TABLE]

for some group velocity $\omega\in so(2)\simeq\mathbb{R}$ and some suitable base point $({\mathbf{q}}_{0}\,,{\mathbf{p}}_{0})\in T^{*}\mathbb{Q}$, ${\mathbf{q}}_{0}=({\mathbf{q}}_{10}\,,{\mathbf{q}}_{20}\,,\ldots\,,{\mathbf{q}}_{n0})$ , ${\mathbf{p}}_{0}=({\mathbf{p}}_{10}\,,{\mathbf{p}}_{20}\,,\ldots\,,{\mathbf{p}}_{n0})$ , ${\mathbf{q}}_{i0}=(z_{i0}\,,\varphi_{i0})$ , ${\mathbf{p}}_{i0}=(p_{z_{i0}}\,,p_{\varphi_{i0}})\,.$ The base points ${\mathbf{q}}_{i0}=(z_{i0}\,,\varphi_{i0})$ are found as critical points of the augmented potential ([24])

[TABLE]

that is, solutions of

[TABLE]

whereas

[TABLE]

Alternatively, one may determine the base points as critical points of the amended potential

[TABLE]

where $\mu$ is (a fixed value of) the angular momentum. Thus one needs to solve

[TABLE]

The relationship between the group velocity $\omega$ and the angular momentum $\mu$ is

[TABLE]

and can be retrieved following a general geometric mechanics context (see [24]) . We also note that a RE with zero $\omega=0$, or equivalently, with $\mu=0,$ is in fact an equilibrium.

Definition 2.7

A solution is called Lagrangian if, at every time t, the masses form a polyhedron that is orthogonal to the z axis.

Proposition 2.8

If there is an equilibrium so that the points lie on a parallel geodesic circle then for every nonzero group (angular) velocity $\omega\neq 0$ there is a RE with the same configuration that rotates along the parallel geodesic circle.

Proof: Let $z=z_{c}$ be a parallel geodesic circle. Assume there is an equilibrium $(z_{i}=z_{c}\,,\varphi_{i}=\varphi_{i0})$, $i=1,2\ldots n$ and so

[TABLE]

Since $f^{\prime}(z_{c})=0$ and RE are given by (32)-(33), the conclusion follows.

$\square$

2.5 Motion on a cylinder and the “vertical linear momentum” integral

If $\mathbb{M}$ is a cylinder, i.e. $f^{\prime}(z)=0$ and $f(z)=const.=:f_{0}$ for all $z$, then the Hamiltonian (19) becomes

[TABLE]

and the equations of motion are

[TABLE]

On a cylinder the geodesic distance between two points $(z_{i},\varphi_{i})$ and $(z_{j},\varphi_{j})$ is invariant to translations along the $z$ direction:

[TABLE]

It follows that the Hamiltonian is invariant to translations along the $z-$axis, that is it is invariant under the diagonal action of $\mathbb{R}$ on $T^{*}\mathbb{M}^{n}$, where the action on each component is

[TABLE]

This symmetry leads to the conservation of the “vertical linear momentum”

[TABLE]

which can be found by either calculating the corresponding momentum map and using Noether’s theorem, or by a direct guess and verification. Further, summing the equations for $z_{i}$ from (39), integrating, and taking into account (41), we obtain the vertical centre of mass integral

[TABLE]

where $k_{0}=M(t_{0})-kt_{0}\,.$

3 Saari’s conjecture on surfaces of revolution

From formula (29) we deduce that the moment of inertia on surfaces of revolution $\mathbb{I}$ is constant when all points are on the same parallel circle. In particular, since by Proposition 2.4 parallel geodesic circles are invariant manifolds, we obtain the following:

Proposition 3.1

Any motion of the $n$-bodies on a geodesic parallel circle has constant moment of inertia.

Corollary 3.2

The generalization of Saari’s Conjecture to $n$-body problems on surfaces of revolution with at least one geodesic parallel circle is not true.

In particular we settle the the conjecture posed on surfaces of constant positive curvature as presented by Diacu & al. in [10]:

Corollary 3.3

The generalization of Saari’s Conjecture to $n$-body problems on a 2-spheres is not true.

Also, we have:

Corollary 3.4

The generalization of Saari’s Conjecture to $n$-body problems on a cylinder is not true.

4 Equal masses and homographic motions

4.1 Homographic motions as a symplectic invariant manifold

Definition 4.1

A solution of the generalized $n$-body problem on a surface of revolution is homographic if it is of the form

[TABLE]

for some functions $z(t)$ and $\varphi(t)$ and some constant angles $\theta_{k}\in[0,2\pi)$ .

In other words, the particles are all in a plane orthogonal to the $z$ axis and move simultaneously along the given surface of revolution while keeping a self-similar shape in the ambient $\mathbb{R}^{3}$ space at all times. When the dynamics is given in Hamiltonian formulation, a solution of the Hamiltonian system (21)-(22) is called homographic if it is of the form

[TABLE]

If they exist, homographic solutions are parametrized by the their (constant) angular momentum $p_{\varphi}.$ Specifically, let $z(t)\,,\varphi(t)\,,p_{z}(t)\,,p_{\varphi}(t)$ be a homographic solution. Using the total angular momentum conservation (27) and since all homographic momenta (44) are equal we have

[TABLE]

Thus to every homographic solution corresponds a value of the angular momentum $c.$ To zero momentum $c=0$ correspond homothetic motions, that is solutions for which the particles move synchronously along meridians. For values $c\neq 0$ we obtain purely homographic solutions; the particles move rotate while ”sliding” at an unison on the surface. Homographic motions such that $z(t)=const.=:z_{0}$ for all $t$ are RE; this can be seen from the RE equations (30).

4.2 Discrete reduction and the equal mass homographic invariant manifold

When the masses are equal, homographic motions form an invariant manifold for which the points remain at the vertices of a regular $n$-gon and move simultaneously on paths along $\mathbb{Q}$. We retrieve this manifold by applying the method of Discrete reduction (see [24, 29]) that we briefly recall below.

Let $\Sigma$ be a discrete group act on a cotangent bundle $T^{*}Q$. Its fixed point set $\text{Fix}\,(\Sigma,T^{*}Q)$ is defined by:

[TABLE]

If a Hamiltonian $H:T^{*}Q\to\mathbb{R}$ is $\Sigma$-invariant and the symplectic structure is preserved under the $\Sigma$-action, then $\text{Fix}\,(\Sigma,T^{*}Q)$ is an invariant manifold for the dynamics of $H$. If in addition, $\Sigma$ is compact, then $\text{Fix}\,(\Sigma,T^{*}Q)$ is a symplectic invariant submanifold, and the restrictions of the Hamiltonian and the symplectic form to $\text{Fix}\,(\Sigma,T^{*}Q)$ is a co-tangent bundle Hamiltonian system which coincides with the restriction of the given Hamiltonian system. Moreover, if the symplectic structure and $H$ are invariant under the action of a Lie group $G$ giving rise to an equivariant momentum map, and $\Sigma$ acts on $G$ in such a way that the actions of $G$ an $\Sigma$ are compatible then $\text{Fix}\,(\Sigma,T^{*}Q)$ is a co-tangent bundle Hamiltonian system with $G$ symmetry. Also, by Palais’ Principle of Criticality ([31]) any equilibrium or RE in $\text{Fix}\,(\Sigma,T^{*}Q)$ is also an equilibrium or a RE, respectively, in the full $T^{*}Q$ phase space.

Returning to the $n$-body problem on surface of revolution, assume that all masses are equal, i.e., $m_{1}=m_{2}=\ldots=m_{n}$. Now consider the cyclic group $C_{n}$ generated by counterclockwise rotations $R_{2\pi/n}$ of angle $2\pi/n$ about the vertical; this group is isomorphic to $(\mathbb{Z}_{n},+)$. Also, consider the subgroup of permutations $S_{n}$ of the set $\{1,2,\ldots,n\}$ generated by a shift to the right by one unit of the elements, which we denote $\sigma$.

The product group $S_{n}\times C_{n}$ acts on $\mathbb{M}^{n}$ by first rotating (all) the points by a multiple of $2\pi/n$, and then relabelling the masses. A generator for this action is

[TABLE]

Further, $S_{n}\times C_{n}$ acts on $T^{*}\mathbb{M}^{n}$ by co-tangent lift

[TABLE]

It follows that the fixed point set $\text{Fix}(S_{n}\times C_{n})$ os

[TABLE]

which coincides to the set of homographic solutions (44) with constant angles $\theta_{k}=2\pi/n$. By the Discrete reduction method, we have:

Proposition 4.2

The generalized n-body problem on a surface of revolution with equal masses admits an invariant manifold that is homographic with the particles forming a regular $n$-gon at al times. If the problem is modeled in Hamiltonian formulation, this invariant manifold is symplectic and on it the dynamics is given by a co-tangent bundle Hamiltonian system.

Remark 4.3

To ease the reading of the paper, we avoid checking the compatibility condition of the actions of $S_{n}\times C_{n}$ and $SO(2)$ on $T^{*}\mathbb{M}$ as given by Assumption $1_{Q}$ in Marsden’s book [24], pp.155, which would guarantee that the homographic Hamiltonian system has $SO(2)$ symmetry. Instead we will just write the homographic Hamiltonian and observe that it is rotationally symmetric.

Thus, when the masses are equal, homographic solutions are of the form

[TABLE]

Equivalently, if the masses are equal, a solution of the Hamiltonian system (21)-(22) is homographic if it is of the form

[TABLE]

Definition 4.4

An equal mass homographic trajectory is a curve on $\mathbb{Q}$ given by the configuration $\left(z(t),\phi(t)\right)$ of a homographic solution (52) defined above.

From now on we assume that all bodies have equal mass, which without loosing generality, we take to be unity. Also, to simplify denominations, in what follows we’ll refer to equal mass homographic dynamics, solutions, trajectories, etc. as homographic dynamics, solutions, trajectories, etc. However, given appropriate specifications for the masses, there are other situations in which homographic dynamics is present. For instance, it is easy to see that for a three body problem, if two masses are equal, solutions for which the bodies are all in a plane orthogonal to the axis of revolution and form isosceles triangles at all times are homographic; in this case, the discrete symmetry group is $S_{2}\times\mathbb{Z}_{2},$ (acting as a reversal of the equal masses followed by their relabelling).

4.3 Dynamics. The generalized Clairaut relation

The dynamics on the homographic invariant manifold is given by the Hamiltonian

[TABLE]

where

[TABLE]

The equation of motion are:

[TABLE]

The motion of body $k$, $k=1,2,\ldots,n$ is a solution of the initial value problem given by the system above and initial conditions of the form

[TABLE]

Remark 4.5

For a constant potential (i.e., $G\equiv const.$ ) we retrieve the Hamiltonian formulation for the homographic dynamics of $n$ free mass points on $\mathbb{M}$. The bodies are situated on a plane perpendicular to the axis of revolution at all times with each trajectory $\left(z_{k}(t)\,,\varphi_{k}(t)\right)=\left(z_{0}\,,\phi_{(}t)+\frac{2k\pi}{n}\right)$ describing a geodesics on the surface.

The conservation of energy reads

[TABLE]

The angular momentum conservation

[TABLE]

is retrieved immediately from equation (63). (Alternatively, one can use the total angular momentum conservation given by (27) specified to the context of homographic motions). Note that

[TABLE]

where $\mu$ is the total angular momentum given by (27).

Proposition 4.6

(Clairaut relation for homographic motions) Let $\left(z(t),\varphi(t)\right)$ be a homographic trajectory, and denote by $\theta(t)\in(0,\pi/2)$ the angle of the trajectory with the parallel circle $z(t).$ Then the product of the radious’ trajectory (i.e. $f(z(t))$), speed, and $\cos$ of the angle $\theta(t)\in(0,\pi/2)$ is constant, that is

[TABLE]

Proof. Let $\left(z(t)\,,\varphi(t)\right)$ be a solutions’ trajectory on $\mathbb{Q}$. Substituting $p_{\varphi}(t)=c$ into (60) we write

[TABLE]

Given the $(z,\varphi)\to{\bf x}(z,\varphi)$ parametrization in equation (2), we have

[TABLE]

Thus

[TABLE]

and so

[TABLE]

Multiplying the relation above by $f\left(z(t)\right)$ and using (68) the conclusion follows. $\square$

Remark 4.7

If the potential is constant (i.e. $G\equiv const.$ ), the system (60) - (63) describe geodesic motion (in Hamiltonian formulation) and the homographic trajectories are the surfaces’ geodesics. In particular, since the speed along any geodesic is constant, in this case the generalized Clairaut relation (67) above becomes the well-known Clairaut relation on a surface of revolution.

Remark 4.8

The generalized Clairaut relation (67) expresses the fact that when the potential is non-zero, one cannot find a parametrization for which all homographic motions have unit speed.

Remark 4.9

The generalized Clairaut relation (67) is a consequence of the angular momentum conservation associated to any rotational-invariant two-degrees of freedom system defined on a surface of revolution.

Substituting $p_{\varphi}(t)=c$ into the Hamiltonian $\tilde{H}$ we reduce the dynamics to a one-degree of freedom (and thus integrable) system given by

[TABLE]

where $W_{c}(z)$ is the amended potential

[TABLE]

The time evolution of the angle $\varphi$ is given by (62) and thus, once a solution of the reduced system $(z(t),p_{z}(t))$ is found, $\varphi(t)$ is given by the reconstruction equation

[TABLE]

4.4 Homographic dynamics on a cylinder

When $\mathbb{M}$ is a cylinder i.e., $f(z)=const.=:f_{0}$, the dynamics reduces to a linear flow. Indeed, in this case the Hamiltonian (71) of the reduced dynamics on the homographic invariant manifold is

[TABLE]

and the homographic dynamics consists in uniform motions no matter the potential:

[TABLE]

4.5 Hill’s regions and topology of the phase space

As a two-degrees of freedom Hamiltonian system with rotational symmetry, the dynamics on the homographic invariant manifold possesses two independent integrals (the energy $\tilde{H}$ and angular momentum $\tilde{J}$) and so is integrable. Moreover, given the ”kinetic + (amended) potential” structure of the reduced one-degree of freedom system, a sketch of the amended potential is sufficient for extracting a full qualitative picture of the dynamics, including the Hill’s regions of motions, the topology of the phase-space and the orbit types (see [2, 38]). This is the subject of this subsection.

Recall that the Hill’s regions of motions of a mechanical ”kinetic + (amended) potential ” system at a given fixed energy level $h$ and momentum $c$, are defined as the regions of allowed motion in the configuration space; the later are determined by taking into account that the kinetic energy is non-negative. In our case the Hill’s regions are

[TABLE]

The topological characterization of the phase space is obtained by considering the level sets of the energy-momentum map

[TABLE]

For $h\in\mathbb{R}\,,c\in\mathbb{R}$, the level sets

[TABLE]

are invariant submanifolds and provide a foliation of the phase-space. For instance, if we assume that the flow is complete (i.e. collisions are not possible) and $c\neq 0$, we have:

[TABLE]

We will discuss some examples in Section 6.

5 Lagrangian homographic relative equilibria

5.1 Some general existence criteria

The RE of the dynamics on the ( homographic manifold correspond to RE of the full system with the particles form a regular $n$-ring at situated on a plane perpendicular to the axis of rotation; we call these Lagrangian homographic RE. In order to determined these it is sufficient to determine their base points in the $z$-direction. Indeed, once a such a base point $z_{0}$ is determined, the RE solution is

[TABLE]

where we used the RE definition (30), the angular momentum conservation (65) and the reconstruction equation (73).

The base points may be found as the critical points of the augmented potential $W_{c}(z)$ (see [24]). Since

[TABLE]

we can write

[TABLE]

and so we have

[TABLE]

We immediately deduce that if $\mathbb{M}$ admits a geodesic circle $z=z_{0}$ (and so $f^{\prime}(z_{0})=0$), then $z_{0}$ is critical points of $W_{c}(z)$ and thus a base point for a RE.

Definition 5.1

The potential (10) is attractive (repulsive) if the function $G$, is such that $G^{\prime}(x)>0$ ($G^{\prime}(x)<0$) for all $x>0$.

Taking into account the formula of $W^{\prime}_{c}(z)$ given by (84), and that $f(z)>0$ for all $z$, the next propositions are immediate.

Proposition 5.2

In the generalized $n$-body problem on surfaces of revolution, if the masses are equal and the potential is repulsive, then the only Lagrangian homographic RE with the shape of a regular $n$-gon are those with trajectories on the geodesic circles.

Corollary 5.3

In the generalized $n$-body problem on surfaces of revolution, if the masses are equal, the potential is repulsive and the profile curve $f(z)$ has no critical points, then there are no Lagrangian homographic RE.

Corollary 5.4

In the generalized $n$-body problem on $\mathbb{S}^{2}$, if the masses are equal and the potential is repulsive, then the only Lagrangian homographic RE are those with their trajectories on the Equator.

Proposition 5.5

*In the generalized $n$-body problem on surfaces of revolution, if the masses are equal and if the potential is attractive, then any parallel circle (not necessarily geodesic) is a trajectory of a Lagrangian homographic RE. *

5.2 Bifurcations of geodesic Lagrangian homographic RE and curvature

In this subsection we show that for attractive potentials, a Lagrangian homographic RE with its trajectory on a geodesic circle generically experiences a pitchfork bifurcation as the angular momentum vary.

To start, recall that in a two-degrees of freedom Hamiltonian mechanical system with rotational symmetry, the stability modulo rotations of a RE may be determined the sign of second derivative of the augmented potential at the base point. In our context, let $z_{0}$ be a base point for a RE. Then:

• •

if $W^{\prime\prime}_{c}(z_{0})<0$ then the RE is unstable (and thus unstable in the full phase-space);

• •

if $W^{\prime\prime}_{c}(z_{0})>0$ then the RE is stable within the dynamics on invariant homographic manifold only;

• •

if $W^{\prime\prime}_{c}(z_{0})=0$ then the RE stability is undecided and generically $z_{0}$ is the landmark of a bifurcation.

Consider a Lagrangian homographic RE with its trajectory on the geodesic circle $z=z_{0}$. Given the formula (84) of $W^{\prime}_{c}(z)$ and that $f^{\prime}(z_{0})=0$ we calculate

[TABLE]

On the other hand, recall that for a given a surface of revolution generated by a function $f(z)$ the Gaussian curvature is

[TABLE]

Thus the curvatures changes its sign as the second derivative of $f$ does. Considering the expressions of $W^{\prime\prime}_{c}(z_{0})$ and $K(z)$ above, we immediately obtain

Proposition 5.6

Consider a Lagrangian homographic RE with its trajectory on the geodesic circle $z=z_{0}$. If the Gaussian curvature at $z_{0}$ is negative (i.e. $K(z_{0})<0$ and so $f^{\prime\prime}(z_{0})>0$) and $G$ is repulsive (and so $G^{\prime}(x)<0$ for all $x$), then the RE is unstable.

For motions with $G$ attractive, we have:

Proposition 5.7** (Geodesic Lagrangian homographic RE bifurcation criterion)**

Consider a Lagrangian homographic RE with its trajectory on a non-zero curvature geodesic circle $z=z_{0}$, and assume that $G$ is attractive. If

[TABLE]

then $(z_{0},c_{0})$ is the landmark of a pitchfork bifurcation.

Proof: Let us denote $W_{c}(z)=W(z,c)$. Standard bifurcation theory for one-degree of freedom Hamiltonians (see, for instance, [19]) guarantees the existence of a pitchfork bifurcation at a point $(z_{0},c_{0})$ if

[TABLE]

and

[TABLE]

For reader’s convenience we re-write formula (84) of $\displaystyle{W^{\prime}_{c}(z)\equiv\frac{\partial W(z,c)}{\partial z}}$ below:

[TABLE]

Let $c_{0}>0$ be the positive root of the parenthesis on the right hand side of the (90) above, and so

[TABLE]

Taking into account that $z_{0}$ is such that $f^{\prime}(z_{0})=0$ and that $(z_{0},c_{0})$ cancels the right hand side parentheses of (90), one verifies that conditions (88) are fulfilled and that

[TABLE]

and

[TABLE]

Using (91), we substitute $c_{0}^{2}$ into the above and get

[TABLE]

Since the curvature at $z=z_{0}$ is non-zero (and so $f^{\prime\prime}(z_{0})\neq 0$) and using the condition (87), the conclusion follows. $\square$

Remark 5.8

It seems likely the inequality (87) is fulfilled by most attractive potentials and for most values of $n$. In this sense, this condition is typical for most $n$-body problems on a surface or revolution. A conjecture stated for the specific case of the 3-d gravitational $n$-body problem $\mathbb{S}^{2}$ will be stated later (Conjecture 6.5).

Remark 5.9

If in the conditions (89) we have \frac{\partial^{4}W(z,c)}{\partial z^{4}}\Big{|}_{(z_{0},c_{0})}>0, then the pitchfork bifurcation is subcritical, that is the inner branch of RE (in between the outer branches, when they exist) is unstable. Note that \frac{\partial^{4}W(z,c)}{\partial z^{4}}\Big{|}_{(z_{0},c_{0})}>0 is equivalent to

[TABLE]

Remark 5.10

The trajectories of the outer branches RE emanating at the pitchfork bifurcation point are on parallel circles $z=\pm z(c)$ which are found by solving the right hand side parenthesis of (90) equal to zero for $|c|\neq c_{0}.$ Specifically, $(\pm z(c),c)$ are points on the curve

[TABLE]

where $|c|\neq c_{0}.$ From the conditions assuring the existence of the pitchfork bifurcation at $|c|=c_{0}$, we know that the equations above has precisely two roots symmetrically disposed with respect to, and in a neighbourhood of, $z_{0}$.

Taking into account the relation (86) between $f^{\prime\prime}(z)$ and the curvature $K(z)$, and given the equation (85) for $W^{\prime\prime}_{c}(z_{0})$, we also deduce:

Proposition 5.11** (Unstable geodesic Lagrangian homographic RE)**

Consider a Lagrangian homographic RE with its trajectory on geodesic circle $z=z_{0}$ and assume that $G$ is attractive. Let $c_{0}>0$ be the momentum value which cancels the parenthesis in the right hand side of (85).

If the curvature at $z_{0}$ is positive (i.e. $K(z_{0})>0$ and so $f^{\prime\prime}(z_{0})<0$), then

1. (a)

for $|c|<c_{0}$ the RE is unstable; 2. (b)

for $|c|\geq c_{0}$ the stability of the RE is undecided. More precisely, for $|c|>c_{0}$ the RE is stable within the homographic invariant manifold only. 2. 2.

If the curvature at $z_{0}$ is negative (i.e. $K(z_{0})<0$ and so $f^{\prime\prime}(z_{0})>0$), then

1. (a)

for $|c|\leq c_{0}$ the stability of the RE is undecided. More precisely, for $|c|>c_{0}$ the RE is stable within the homographic invariant manifold only. The momentum $|c|=c_{0}$ generically marks a pitchfork bifurcation; 2. (b)

for $|c|>c_{0}$ the RE is unstable.

6 Some examples

6.1 Homographic dynamics for the “quasi-harmonic” interaction

We discuss homographic motion with a “quasi-harmonic” binary interaction given by $\displaystyle{G(x)=x^{2}/2}$, $G:[0,\infty)\to\mathbb{R}$. Note that $G^{\prime}(x)=x>0$ for all $x>0$ so $G$ is attractive. The amended potential is

[TABLE]

After some elementary algebra, given that

[TABLE]

we have

[TABLE]

and so

[TABLE]

Assume that $z=z_{0}$ is a geodesic circle and so $f^{\prime}(z_{0})=0$. The pitchfork bifurcation in Proposition (5.7) is retrieved at

[TABLE]

and we distinguish:

for $c=0$ there is a unique equilibrium on the parallel geodesic circle $z=z_{0}$; 2. 2.

for every $|c|$ such that

1. (a)

$|c|<c_{0}(n)$ if $f^{\prime\prime}(z_{0})<0$ (i.e. $z=z_{0}$ has positive curvature) 2. (b)

$|c|>c_{0}(n)$ if $f^{\prime\prime}(z_{0})>0$ (i.e. $z=z_{0}$ has negative curvature)

there are three RE, one with its trajectory on $z=z_{0}$ and the other two with their trajectories symmetrically disposed (see Remark 5.10) on parallel circles at $\pm z(c)$, where the later $z$ values solve

[TABLE]

in a neighbourhood of $z=z_{0};$ 3. 3.

for $|c|=c_{0}(n)$ there is a RE and a pitchfork bifurcation; 4. 4.

for

1. (a)

$|c|>c_{0}(n)$ if $f^{\prime\prime}(z_{0})<0$ (i.e. $z=z_{0}$ has positive curvature) 2. (b)

$|c|<c_{0}(n)$ if $f^{\prime\prime}(z_{0})>0$ (i.e. $z=z_{0}$ has negative curvature)

there is a unique RE with its trajectory on $z=z_{0}.$

6.1.1 Quasi-harmonic homographic motion on $\mathbb{S}^{2}$

We now focus on motion on the unit sphere $\mathbb{S}^{2}$. The generatrix of $\mathbb{S}^{2}$ is $f(z)=\sqrt{1-z^{2}}$ and for $c\neq 0$, the reduced dynamics is given by

[TABLE]

whereas for $c=0$

[TABLE]

When $c=0,$ the equations of motions are

[TABLE]

and we observe that the boundaries $z=\pm 1$ become fictitious invariant manifolds. On the Equator (i.e., at $z=0$) the momentum bifurcation value (101) becomes

[TABLE]

whereas the values of the RE coordinates as given by (102) are

[TABLE]

The topology of the phase space is easily deduced by applying the definition (81) to the present context and the analysis of Figure 2. Thus the phase space of the homographic invariant manifold is foliated by:

for $c=0$

[TABLE] 2. 2.

for $|c|\in(0\,,{c_{0}(n)})$

[TABLE] 3. 3.

for $|c|\geq{c_{0}(n)}$

[TABLE]

Proposition 6.1

Consider $n$ particles constrained to move on $\mathbb{S}^{2}$ and with pairwise quasi-harmonic interaction. As the energy $h$ and the momentum $c$ vary, the dynamics on the homographic Lagrangian invariant manifold consists of:

for $c=0$

1. (a)

for $h<0$ there is no dynamics; 2. (b)

for $h=0$ we have fictitious motion at the Poles $z=\pm 1;$ these points, while not singularities for the equations of motion, correspond to non-physical configurations (all points clustered in one point); 3. (c)

for $h\in(0,W_{0}(0))$ we have homothetic motion (each particle moving on a meridian) consisting in orbits which are ejected from the North pole ($z=1$), reach a minimum value $z_{min}=z_{min}(h)$ and return to the North pole; also, by symmetry with respect to the Equator, we have identical dynamics in the Southern hemisphere; 4. (d)

for $h=W_{0}(0)$ we have homothetic motion (each particle moving on a meridian) consisting in orbits which are ejected from the North pole ($z=1$) and tend to the Equator; by symmetry we have identical dynamics in the Southern hemisphere; also, there is an equilibrium on the Equator. 5. (e)

for $h>W_{0}(0)$ we have homothetic motion (each particle moving on a meridian) consisting in orbits which are ejected from the North pole ($z=1$) and tend to the South Pole; we have identical dynamics in the Southern hemisphere; 2. 2.

for $|c|\in(0\,,c_{0}(n))$, let $z(c)$ be defined as in Remark 5.10. Then

1. (a)

for $h=W_{c}(z(c))$ there is no dynamics; 2. (b)

for $h=W_{c}(z(c))$ there are two RE with their trajectories located at $z=\pm z(c);$ 3. (c)

for $h\in\left(W_{c}(z(c)),W_{c}(0)\right)$, in the Northern hemisphere the orbits are contained in a parallel annular sector $z\in[z_{min}\,,z_{max}]$, where $z_{min}$ and $z_{min}$ depend on the energy level. These orbits may be periodic or quasi-periodic, filling densely the annular sector. By symmetry, we have identical dynamics in the Southern hemisphere; 4. (d)

for $h=W_{c}(0)$ in the Northern hemisphere, the orbits reach an extreme value of $z$ and spiral asymptotically towards the RE on the Equator; identical dynamics are in the Southern hemisphere. 5. (e)

for $h>W_{c}(0)$ the particles visit, in a symmetric way, both hemispheres crossing the Equator, with $z$ in between maximal and minimum values which depend on the energy level; 3. 3.

for $|c|\geq c_{0}$

1. (a)

for $h<W_{c}(0)$ there is no dynamics; 2. (b)

for $h=W_{c}(0)$ we have a RE with its trajectory on the Equator; 3. (c)

for $h>W_{c}(0)$ we have orbits similar to those in the case 2.(e).

Proof: This follows from a a direct analysis of the one-degree of freedom Hamiltonian (103). See also the sketch of the amended potential in in Figure 2 and the phase curves in Figure 3. $\square$

6.1.2 Quasi-harmonic homographic motion on a peanut-like surface

We consider now homographic dynamics on a surface $\mathbb{M}$ generated by

[TABLE]

The sign of curvature $K=K(z,\gamma)$ of $\mathbb{M}$ is that of

[TABLE]

and so the curvature $K(z,\gamma)$ cancels at

[TABLE]

Further,

[TABLE]

The amended potential is

[TABLE]

and, by formula (101), the momentum bifurcation values are

[TABLE]

Using elementary calculus it can be proven that $c_{0}<c_{\gamma}$ for $\gamma>1/2.$

Given the sketch of the amended potential - see Figure 4 - one may fully describe the topology of the phase-space and the dynamics by performing the analysis in an analogous manner as for motion on $\mathbb{S}^{2}$. Instead, we choose to point out some of the main features. Let us call the regions $z>0$ and $z<0$ the Northern and Southern hemisphere, respectively, and the geodesic parallel $z=0$ the Equator. Note that the trajectories are symmetric with respect to the Equator. For appropriate levels of energy $h$ we have

for $c=0$ (in Figure 4, $W_{c}(z)$ corresponds to the black curve)

• •

there are homothetic trajectories connecting from the North to the South Pole, trajectories connecting a pole to a maximum distance, and trajectories going up and down in a symmetric manner with respect to the Equator (in Figure 4, regions bounded on the top by the black dotted curve, on the bottom the black curve and on the sides by $z\pm 1$);

• •

there are three equilibria: two unstable, one in each hemisphere with their coordinates symmetrically disposed with respect to the Equator, and a third one on the Equator (in Figure 4, the critical points on the black curve); 2. 2.

for $c\neq 0$ (in Figure 4, $W_{c}(z)$ corresponds to the blue, red, light green, and dark green curves)

• •

at each energy level there are three RE with trajectories on each of the geodesic parallel circles, two with trajectories on $z=z_{\gamma}$ and one on $z=0$; they undergo three distinct pitchfork bifurcations as prescribed by Proposition 5.7

• •

the RE with trajectories on $z=z_{\gamma}$ (i.e., on geodesic circles with positive curvature) are unstable at low momenta, whereas the Equatorial RE is unstable at high momenta (i.e., on geodesic circles with negative curvature);

• •

due to symmetry, two of the persistent RE appear always in pairs, one of each hemisphere, symmetrically disposed with respect to the Equator;

• •

there are momenta $c\neq 0$ for which, as $h$ vary, we have seven distinct RE (in Figure 4, the critical points on the light green curve);

• •

there are levels of energy for which there are four distinct Hill regions, two on each hemisphere (in Figure 4, regions bounded on the top by the green dotted curve and at the bottom by the light green curve). In this case, in the phase space we have 4 distinct tori.

6.2 Gravitational homographic motion on $\mathbb{S}^{2}$ and $\mathbb{H}_{\text{one}}^{2}$

In this Subsection we consider homographic for motions on $\mathbb{S}^{2}$ and $\mathbb{H}^{2}_{\text{one}}$ for 3-d gravitational interactions. Recall that that 3-d gravitational potentials are defined as spherical-symmetric solutions of the Laplace equation on $\mathbb{S}^{3}$ ($\mathbb{H}^{3}$) and then restricted to the submanifolds $\mathbb{S}^{2}$ ($\mathbb{H}^{2}_{\text{one}}$). (For more on gravitational potentials on surfaces of constant curvature, see [9, 40].) For any two points ${\mathbf{q}}_{i}$ and ${\mathbf{q}}_{j}$, the binary potential is

[TABLE]

Since $G(x)=-\cot x$, $G^{\prime}(x)=1+\cot^{2}x>0$ and so $G$ is attractive (in agreement with our Definition 5.1). By Proposition 5.5, we have that the 3-d gravitational $n$-body problem on $\mathbb{S}^{2}$ ($\mathbb{H}^{2}_{\text{one}}$) admits Lagrangian homographic RE with Equatorial trajectories. However, since configurations with points diametrically opposite on the Equator are ill-defined, we must consider $n$ odd.

Using the results in Propositions 5.7 and 5.11, we deduce:

Proposition 6.2

Consider the homographic dynamics of the 3-d gravitational $n$-body problem on $\mathbb{S}^{2}$, where $n$ is odd. If

[TABLE]

then the Equatorial Lagrangian homographic RE experiences a subcritical pitchfork bifurcation. Moreover, denoting the bifurcation momentum absolute value $c_{0}$, the Equatorial Lagrangian homographic RE are unstable for $|c|<c_{0}$.

Proof. The sphere $\mathbb{S}^{2}$ is described by a parametrization (2) with a generatrix $f(z)=\sqrt{1-z^{2}}$ and the Equator corresponds to the circle $z=0$. The unit sphere $\mathbb{S}^{2}$ has constant positive curvature $K=1$; in particular, in our parametrization, $f^{\prime\prime}(z)<0$ for all $z.$ Since $G(x)=-\cot(x)$ we have $G^{\prime}(x)=1+\cot^{2}x>0$ and $G^{\prime\prime}(x)=-2(1+\cot^{2}x)\cot x$. By Proposition 5.7 and Remark 5.9, the presence of a subcritical pitchfork is insured by the condition (95) written as a strict inequality, that is

[TABLE]

Recall that $n$ must be odd. Expanding the sum above and since $\cot(2\pi-x)=\cot x$ we have

[TABLE]

and thus condition (114) is obtained. The lack of stability of the inner branch of the RE follows as an application of from Proposition 5.11. $\square$

Remark 6.3

The bifurcation value is the positive root of

[TABLE]

Remark 6.4

One may verify that condition (115) is satisfied for $n=3$. Moreover, one may verify that the sum in (115) is strictly positive and so by Remark 5.9 we have a subcritical pitchfork.

Conjecture 6.5

*In the 3-d gravitational $n$-body problem on $\mathbb{S}^{2}$ the Equatorial Lagrangian homographic RE experiences a subcritical pitchfork bifurcation. *

Remark 6.6

To prove the conjecture above it is sufficient to prove the inequality (114) for any odd $n$.

Likewise, we have

Proposition 6.7

Consider the homographic dynamics of the 3-d gravitational $n$-body problem on $\mathbb{H}^{2}_{\text{one}}$. If

[TABLE]

then the Equatorial Lagrangian homographic RE experiences a subcritical pitchfork bifurcation. Moreover, denoting the bifurcation momentum absolute value $c_{0}$, then Equatorial Lagrangian homographic RE is unstable for $|c|>c_{0}$.

The sketch the amended potential for homographic motion on $\mathbb{S}^{2}$ and $\mathbb{H}^{2}_{\text{one}}$ for $n=3$ is given in Figure 5. The orbits on $\mathbb{S}^{2}$ are qualitatively identical to the the quasi-harmonic homographic orbits, with one modification: since the potential is not defined at the Poles, orbits reaching these points correspond to an ejection/fall from/into a collision configuration. The dynamics are described in detail in [9].

We now comment on the dynamics for the hyperbolic case. The amended potential is

[TABLE]

and further we calculate

[TABLE]

There is one critical point at $z=0$, and the bifurcation momentum is

[TABLE]

Applying Proposition 6.7, or just by direct inspection of the profile of $W_{c}(z)$, we deduce the following:

• •

for $|c|=0$ there is one equilibrium and it is located on the Equator;

• •

for $|c|<c_{0}$ there are three RE, one with its trajectory on the Equator and two with their trajectories on the circles $z=\pm z(c)$, the later satisfying

[TABLE]

• •

at $|c|=c_{0}$ there is a pitchfork bifurcation;

• •

for $|c|>c_{0}$ there is a unique RE with its trajectory on the Equator.

The orbit behaviour and the topology of the phase space may be deduced following an analogous procedure as in the previous subsection. We sketch the phase portrait for representative momentum levels in Figure 6.

7 Acknowledgements

The author was supported by NSERC Discovery grant. She thanks Florin Diacu and Stefanella Boatto for suggesting this problem. She also thanks Tanya Schmah for relevant discussions.