N-body dynamics on closed surfaces: the axioms of mechanics

Abstract

A major challenge for our understanding of the mathematical basis of particle dynamics is the formulation of N-body and N-vortex dynamics on Riemann surfaces. In this paper, we show how the two problems are, in fact, closely related when considering the role played by the intrinsic geometry of the surface. This enables a straightforward deduction of the dynamics of point masses, using recently derived results for point vortices on general closed differentiable surfaces M endowed with a metric g. We find, generally, that Kepler's Laws do not hold. What is more, even Newton's First Law (the law of inertia) fails on closed surfaces with variable curvature (e.g. the ellipsoid).

1. Introduction

Every theoretical model relies on what mathematicians call axioms and physicists call working hypotheses, the foundations of theory. In geometry, this is what characterizes, for example, Euclidean geometry, where parallel geodesics—straight lines—never cross, and spherical geometry, where parallel geodesics—great circles—always cross. In Newtonian mechanics in Euclidean spaces, a so-called mechanical system is one that verifies the three Newtonian laws, which we could consider as the axioms of mechanics (F Acker 2013, personal communication).¹

In this article, we show that Newton's First Law, the law of inertia, is not universal. In particular, on compact surfaces without boundaries that are conformal to the sphere, we show that the law of inertia is only valid in special geometries, namely surfaces with constant Gaussian curvature. We can therefore view Newton's First Law as playing a similar role for a mechanical system, on a sphere or on a surface conformal to the sphere, as Euclid's fifth postulate distinguishing planar geometry from spherical geometry.

The crucial insight is to regard a surface in its intrinsic geometry, not as one embedded in ${\mathbb{R}}^3$—i.e. not with the gravitational potential inherited by ${\mathbb{R}}^3$, U(r)∼1/r, as done, for example, in [1]. This leads to significant differences in the formulation of the dynamics, as is already known in the case of point vortices [2]. Here, particular attention is given to closed surfaces of revolution, for example the ellipsoid of revolution and the bean surface, shown in figure 1. Such surfaces permit a straightforward, explicit formulation of the gravitational potential of point masses in analogy to what is done for the point-vortex Hamiltonian in [2].

Figure 1.

(a) A sphere, (b) an oblate ellipsoid of revolution and (c) a bean-shaped surface with parameters a=0.6 and b=0.4 [2].

The structure of this paper is as follows. In the next section, we explain why Newton's law of inertia does not generally hold for masses restricted to moving on a closed surface with variable Gaussian curvature. We briefly review surfaces of revolution in §3, then formulate the dynamics of a ‘test particle’ in §4. This is generalized to N point masses in §§5 and 6, where we provide the explicit equations of motion. Section 7 shows how symmetries of the Hamiltonian and of the equations of motion allow one to reduce the original system to one of lower dimension. Section 8 focuses on the dynamics of a single mass to emphasize how the interaction with the surface geometry induces acceleration. In §9, we revisit the planar two-body problem from its intrinsic geometry point of view, we discuss the existence of an extra integral and we show that Kepler's Laws do not hold in this case.

2. The first axiom of Newtonian mechanics revisited

In 1687, Isaac Newton published in his Principia [3] three famous laws which have become widely regarded as the ‘axioms of mechanics’. The first law, the law of inertia, states:

The vis insita, or innate force of matter, is a power of resisting by which every body, as much as in it lies, endeavours to preserve its present state, whether it be of rest or of moving uniformly forward in a straight line.

On a general surface, the analogue of a straight line is a geodesic [4]. It is well known that a particle on a plane either remains at rest, if its initial velocity is zero, or continues in a straight line with uniform velocity equal to its initial velocity. On a sphere, we have a similar scenario to that on the plane. Either a particle remains at rest or it travels at constant speed around a great circle [5,6]. Both the plane and the sphere are surfaces with constant Gaussian curvature [4]. What happens if, for example, we deform the sphere into an ellipsoid of revolution? In this case, as shown in figure 2 and as explained in §8, a particle initially at rest generally begins to move along a meridian! It then stops and reverses its direction of motion. The particle oscillates about the equator on an oblate ellipsoid (shown), and about the closest pole on a prolate ellipsoid.

Figure 2.

Dynamicsof a single particle, initially at rest, on an oblate ellipsoid of revolution. (a) Time evolution of the co-latitude θ for two cases starting at θ=θ_o=10° (solid line), and at θ=θ_o=60° (dashed line). (b) Corresponding time evolution of the meridional momentum, p_θ.

This oscillatory motion can be understood as follows. The particle is initially at a co-latitude θ_o, which divides the surface into two, generally unequal, regions. Because gravity is a central force, the uniform background mass (which is required on a closed surface as discussed later) acts as if it were concentrated at two points along the axis of revolution between the poles. The net force is generally unbalanced and sets the particle in motion. On an oblate ellipsoid, the particle moves towards the equator where the net force changes sign. The momentum of the particle then carries it to θ=π−θ_o before it reverses direction and returns to its initial position. On a prolate ellipsoid, the particle instead oscillates about the nearest pole.

3. Surfaces of revolution

As in [2] we consider a surface of revolution M (about the vertical z-axis) that is a deformation of a sphere of radius R. The Cartesian coordinates (x,y,z) in ${\mathbb{R}}^3$ of any point on M may be expressed as functions of two surface coordinates θ and ϕ, co-latitude and longitude, respectively. For surfaces of revolution, it is sufficient to take

$$x=\rho (\theta )\cos \phi ,y=\rho (\theta )\sin \phi \mathrm{and}z=\zeta (\theta ),$$

where ρ(θ) and ξ(θ) are specified functions of θ—which in the plane yz describe the curve generating the surface. Without loss of generality, we may take 0≤φ≤2π and 0≤θ≤π over S. Note that:

• — for the sphere:

  $$\rho (\theta )=R\sin \theta \mathrm{and}\zeta (\theta )=R\cos \theta ;$$ 3.1
• — for the ellipsoid of revolution:

  $$\rho (\theta )=R\sin \theta \mathrm{and}\zeta (\theta )=bR\cos \theta ,$$ 3.2

  where b is the height-to-width aspect ratio; and
• — for the bean surface:

  $$\rho (\theta )=R\sin \theta \mathrm{and}\zeta (\theta )=R(a{\sin }^2\theta +b\cos \theta ),$$ 3.3

  where a is an asymmetry parameter.

(a). Metric and metric tensor of a surface of revolution

The differential distance ds—also called the metric of M—between two points on S is

$$\mathrm{d}{s}^2=|\mathrm{d}x{|}^2=\mathrm{d}{x}^2+\mathrm{d}{y}^2+\mathrm{d}{z}^2=[{({\rho }^{\prime })}^2+{({\zeta }^{\prime })}^2]\hspace{0.17em}\mathrm{d}{\theta }^2+{\rho }^2\hspace{0.17em}\mathrm{d}{\phi }^2,$$ 3.4

where primes denote differentiation with respect to θ. From equations (3.1)–(3.3) it follows that:

• — for the sphere of radius R:

  $$\mathrm{d}{s}^2=R^2(\mathrm{d}{\theta }^2+{\sin }^2\theta \hspace{0.17em}\mathrm{d}{\phi }^2);$$ 3.5
• — for the ellipsoid of revolution:

  $$\mathrm{d}{s}^2=R^2\{({\cos }^2\theta +b^2{\sin }^2\theta )\mathrm{d}{\theta }^2+{\sin }^2\theta \hspace{0.17em}\mathrm{d}{\phi }^2\};\text{ and}$$ 3.6
• — for the bean surface:

  $$\mathrm{d}{s}^2=R^2\{[{\cos }^2\theta +{\sin }^2\theta {(2a\cos \theta -b)}^2]\hspace{0.17em}\mathrm{d}{\theta }^2+{\sin }^2\theta \hspace{0.17em}\mathrm{d}{\varphi }^2\}.$$ 3.7

4. The motion of a test particle in a gravitational field

By analogy with the fluid dynamics problem [2], we start by considering the motion of a test particle on a surface M with a metric g. We suppose that the density of matter ρ(r,t) is given. To put things into context, we may think of a satellite of mass m₀ in the gravitational field of the planets in our Solar System, with masses m₁,…,m_N. We then assume m₀≪m_j, j=1,…,N. This is what Poincaré called the restricted N-body problem [7]. Following Poincaré, we assume that the presence of the satellite does not affect the motion of the planets. Moreover, for the moment, we also assume that the planets' trajectories, r₁(t),…,r_N(t), are known. The question then is: how do we deduce the motion of the satellite on a general surface M?

By analogy with Maxwell's Laws [8], we start with the fundamental equations of a mechanical system in the presence of a central force. Let a(r,t) be the acceleration field surrounding the satellite located at r at time t. Because the force (per unit mass) is irrotational (curl free) and attractive, a must satisfy essentially the same laws satisfied by an electric field E, namely:

$$\left.\begin{array}{rl}& (\mathrm{i})\mathrm{curl}(\mathit{\text{a}})=0,\\ & (\mathrm{ii})\mathrm{div}(\mathit{\text{a}})=-\frac{\rho (\mathit{\text{r}},t)}{ϵ}\\ \text{and}& (\mathrm{iii})\frac{\mathrm{\partial }\mathit{\text{a}}}{\mathrm{\partial }t}+\frac{1}{ϵ}\mathit{\text{J}}=0,\end{array}\right\}$$ 4.1

where the minus sign in the second equation reflects the fact that gravity is a force of attraction [5]. Above, ρ(r,t) is the mass density, ϵ is a suitable constant analogous to permittivity in electrostatics and J is the mass current. The last equation is nothing more than the statement of mass conservation. The second equation can be re-expressed as

$$\mathrm{div}(\mathit{\text{a}})=-\frac{1}{ϵ}\rho (\mathit{\text{r}},t)=-\gamma \rho (\mathit{\text{r}},t),$$ 4.2

where the gravitational constant γ satisfies γ=1/ϵ.

Considering surfaces that are smooth deformations of the sphere ${\mathbb{S}}^2$ (figure 1), equation (4.1) allows us to express the acceleration field as the gradient of an unknown function Φ,

$$\mathit{\text{a}}=-\mathrm{grad}(\mathrm{\Phi }),$$

where Φ plays the role of the satellite's gravitational potential. It is determined by substituting the above into equation (4.2), giving

$${\mathrm{\Delta }}_g\mathrm{\Phi }=\gamma \rho (\mathit{\text{r}},t),$$ 4.3

where Δ_g is the Laplace–Beltrami operator, generalizing the Laplacian for a surface with metric g [9].

Remark 4.1 —

Given the density of matter ρ(r,t)—i.e. given the dynamics of the planets—and the metric g of the surface under consideration, the gravitational potential of the satellite, Φ, is the solution of the Poisson equation (4.3). A similar equation arises for the streamfunction Ψ of an incompressible fluid [2] for a given vorticity field ω(r,t), i.e. Δ_gΨ=ω; see appendix A.

Remark 4.2 —

The Poisson equation (4.3) is linear. Hence, there exists a fundamental solution G(r,r′), called the Green function [10], in terms of which the solution of equation (4.3) is

$$\mathrm{\Phi }(\mathit{\text{r}},t)=\gamma {\iint }_{M}G(\mathit{\text{r}},{\mathit{\text{r}}}^{\prime })\rho ({\mathit{\text{r}}}^{\prime },t)\hspace{0.17em}\mathrm{d}{\mathit{\text{r}}}^{\prime }.$$ 4.4

For ${\mathbb{R}}^3$, and for surfaces conformal to the plane, G(r,r′) is the solution of

$${\mathrm{\Delta }}_{g}G(\mathit{\text{r}},{\mathit{\text{r}}}^{\prime })=\delta (\mathit{\text{r}}-{\mathit{\text{r}}}^{\prime }),$$ 4.5

where δ(r−r′) is the usual Dirac delta distribution. More specifically [9,11,12], for ${\mathbb{R}}^3$, the punctured sphere ${\mathbb{S}}_{\mathrm{p}}^2$, the plane ${\mathbb{R}}^2$ and the hyperbolic surface ${\mathbb{H}}^2$ (of constant negative Gaussian curvature K) we have that

$$\begin{array}{rlr}& G_{{\mathbb{R}}^3}(\mathit{\text{r}},\mathbf{\text{0}})=-\frac{1}{4\pi }\frac{1}{\parallel \mathit{\text{r}}\parallel },& \mathit{\text{r}}=(x,y,z),\\ & G_{{\mathbb{S}}_{\mathrm{p}}^2}(\mathit{\text{r}},\mathbf{\text{0}})=\frac{1}{2\pi }\ln \left[\tan \left(\frac{\sqrt{K}}{2}r\right)\right],& \mathit{\text{r}}=(\phi ,\theta ),r=\frac{\theta }{\sqrt{K}},\\ & G_{{\mathbb{R}}^2}(\mathit{\text{r}},\mathbf{\text{0}})=\frac{1}{2\pi }\ln r,& \mathit{\text{r}}=(x,y),r=\sqrt{x^2+y^2},\\ & G_{{\mathbb{H}}^2}(\mathit{\text{r}},\mathbf{\text{0}})=\frac{1}{2\pi }\ln \left[\tanh \left(\frac{\sqrt{|K|}}{2}r\right)\right],& \mathit{\text{r}}=(\phi ,\theta ),r=\frac{\theta }{\sqrt{|K|}},\end{array}$$

respectively, where r is the geodesic distance. Notably, for the punctured sphere [12], the Green function converges to the planar one in the limit K→0 (or, equivalently, the radius of the sphere $R=1/\sqrt{K}\to \mathrm{\infty }$):

$$2\pi G_{{\mathbb{S}}_{\mathrm{p}}^2}(\mathit{\text{r}},0)=\ln \left[\tan \left(\frac{\sqrt{K}r}{2}\right)\right]+\ln \left(\frac{\sqrt{K}}{2}\right)=\ln r+\frac{1}{12}K{r}^2+O(K^2{r}^4),$$

for small K and fixed r, such that K r²≪1.²

Up to this point, we have considered surfaces with their intrinsic geometry and not as embedded in ${\mathbb{R}}^3$. We have addressed the question: given the surface metric and the distribution of matter, can we deduce the mass dynamics? However, it is also possible to consider the dynamics from an extrinsic geometry point of view, i.e. now regarding M as embedded in ${\mathbb{R}}^3$. One way to do so is to restrict the ${\mathbb{R}}^3$ potential to the plane ${\mathbb{R}}^2$ and pull it back to the sphere, by the inverse of a stereographic projection from the south pole ($r=\tan (\theta /2)$). In this case, the Green function for the unit punctured sphere is

$${\tilde{G}}_{{\mathbb{S}}_{\mathrm{p}}^2}(\mathit{\text{r}})=-\frac{1}{4\pi }\hspace{0.17em}\frac{1}{\parallel \mathit{\text{r}}\parallel }=-\frac{1}{4\pi }\cot \left(\frac{\theta }{2}\right).$$

Such a potential is a potential for the punctured sphere, but it is not the solution of the Poisson equation (4.5), i.e. it is not the potential associated with the intrinsic geometry of the surface.

Moreover, various authors in the literature [1,13,14] use

$${\tilde{G}}_{{\mathbb{S}}^2}(\mathit{\text{r}})=-k\cot \theta$$

for the Green function of a sphere, where k is a constant. In this case, observe that if we hold one mass fixed at the north pole and consider a second mass at a co-latitude θ, then the force on the second mass is

$$\tilde{F}(\theta )=\tilde{\gamma }\frac{m_1{m}_2}{{\sin }^2\theta },$$

which exhibits a minimum at the equator but increases without bound towards each pole (figure 3). This does not enable one to consider the equilibrium configuration consisting of one mass at the north pole and a second mass at the south pole, a configuration for which we recover with our potential (see remark 4.3) the corresponding force

$$F(\theta )=\frac{\gamma }{4\pi }\frac{m_1{m}_2\sin \theta }{1-\cos \theta }.$$

Figure 3.

Consider the unit sphere ${\mathbb{S}}^2$ and a mass m₁ fixed at the north pole. The force on a second mass m₂ at a co-latitude θ is proportional to $1/{\sin }^2\theta$ if derived from the potential $\mathcal{U}(\theta )=-\tilde{\gamma }{m}_1{m}_2\cot \theta$, whereas it is proportional to $\sin \theta /(1-\cos \theta )$ if derived from the potential $\mathcal{U}(\theta )=(\gamma /4\pi )m_1{m}_2\ln (1-\cos \theta )$.

Remark 4.3 —

For a truly closed surface (i.e. compact and without boundaries or punctures), such as any surface conformal to the sphere [2,9,15,16], the source term γρ of the Poisson equation Δ_gΦ=γρ must satisfy an extra condition, called the Gauss condition,

$$\gamma {\iint }_M\rho \hspace{0.17em}\mathrm{d}\mathit{\text{r}}=0.$$ 4.6

This implies that the equation for the Green function must be generalized to

$${\mathrm{\Delta }}_{g}G(\mathit{\text{r}},{\mathit{\text{r}}}^{\prime })=\delta (\mathit{\text{r}}-{\mathit{\text{r}}}^{\prime })+C,$$ 4.7

where C is a compensating factor chosen so that the surface integral of δ(r−r′)+C is identically zero. C plays the role of a gauge.³ The simplest choice is to take C=−1/A, a constant [2,15,16]. Then, C represents a uniform background distribution of matter which links the local dynamics to the global geometry of the surface. The fact that this distribution is negative could be interpreted as an anti-matter distribution (see [2,5,16] for further remarks). Notably, with such a choice for C, the Green function of a sphere of radius R is

$$G(\mathit{\text{r}},{\mathit{\text{r}}}^{\prime })=\frac{1}{4\pi }\ln [2R^2(1-\cos \mathrm{\Theta })],$$

where Θ is the angular separation between r and r′. Note: $2R^2(1-\cos \mathrm{\Theta })$ is the chord distance between these two points. In G above, the radius of the sphere scales out—it contributes an unimportant constant. Dynamically, all spheres are equivalent to the unit sphere. The same is not true for the punctured sphere, as pointed out in remark 4.2.

Remark 4.4 —

It follows from remark 4.3 that, on a surface M conformal to the sphere, we cannot just consider a single point-like mass, because, in this case, the integral (4.6) would not be zero. There are many options for compensating the mass. Among the simplest is to add an equal but opposite (negative) mass, as suggested by Shchepetilov [17]. But perhaps the simplest choice of all is to add a uniform background mass, of negative sign, because this choice alone does not increase the dynamical degrees of freedom of the system [2]. Thus, if we consider a point-like distribution of matter, comprising N point-like masses, we must also consider a compensating term C_N,

$$\rho (\mathit{\text{r}},t)=\sum _{j=1}^N{m}_j\delta (\mathit{\text{r}}-{\mathit{\text{r}}}_j(t))+C_N.$$ 4.8

Here, we choose $C_N=-(\sum _{j=1}^N{m}_j)/A$. Then, inserting equation (4.8) into equation (4.4), we obtain the satellite potential Φ from

$$\mathrm{\Phi }(\mathit{\text{r}},t)=\gamma \sum _{j=1}^N{m}_{j}G(\mathit{\text{r}},{\mathit{\text{r}}}_j(t))+\text{const}.,$$

where G is the Green function found by solving equation (4.7) with C=−1/A.

5. The potential of N bodies on a surface (M,g)

The previous section explained how to deduce the satellite gravitational potential, assuming that the trajectories of the masses (the sources of the gravitational field) are known. Here we describe how to deduce the potential of the masses themselves. This relies on the following working hypothesis: each mass behaves as a satellite in the gravitational field generated by the other masses.

A similar hypothesis is necessary to formulate the dynamics of point vortices in an incompressible fluid. Following [2], we define the potential of the kth mass as

$$\mathrm{\Phi }({\mathit{\text{r}}}_k,t)=\underset{\mathit{\text{r}}\to {\mathit{\text{r}}}_k}{lim}\left(\mathrm{\Phi }(\mathit{\text{r}},t)-\gamma \frac{m_k}{2\pi }\ln d(\mathit{\text{r}},{\mathit{\text{r}}}_k)\right),$$

where d(r,r_k) is the geodesic distance between r and r_k. We thus obtain

$$\mathrm{\Phi }({\mathit{\text{r}}}_k,t)=\gamma \left(\sum _{j=1,\hspace{0.17em}j\ne k}^N{m}_{j}G({\mathit{\text{r}}}_k,{\mathit{\text{r}}}_j)+m_k\mathcal{R}({\mathit{\text{r}}}_k)\right),$$ 5.1

where $\mathcal{R}({\mathit{\text{r}}}_k)$ is the Robin function [2,10,15],

$$\mathcal{R}({\mathit{\text{r}}}_k)=\underset{\mathit{\text{r}}\to {\mathit{\text{r}}}_k}{lim}\left(G(\mathit{\text{r}},{\mathit{\text{r}}}_k)-\frac{1}{2\pi }\ln d(\mathit{\text{r}},{\mathit{\text{r}}}_k)\right).$$

It can then be inferred that the gravitational potential $\mathcal{U}$ takes the form

$$\mathcal{U}=\frac{1}{2}{\iint }_M\mathrm{\Phi }(\mathit{\text{r}},t)\rho (\mathit{\text{r}},t)\hspace{0.17em}\mathrm{d}{\mathrm{\Omega }}_{\mathit{\text{r}}}.$$

For a system of N point masses, m₁,…,m_N, located at positions r₁,…,r_N, this reduces to

$$\begin{array}{rl}\mathcal{U}& =\frac{1}{2}{\iint }_M\mathrm{\Phi }(\mathit{\text{r}},t)\sum _{j=1}^N{m}_j\left(\delta (\mathit{\text{r}}-{\mathit{\text{r}}}_j)-\frac{1}{A}\right)\mathrm{d}{\mathrm{\Omega }}_{\mathit{\text{r}}}\\ & =\frac{1}{2}\sum _{j=1}^N{m}_j\mathrm{\Phi }({\mathit{\text{r}}}_j,t)-\frac{m_{\mathrm{tot}}}{2A}{\iint }_M\mathrm{\Phi }(\mathit{\text{r}},t)\hspace{0.17em}\mathrm{d}{\mathrm{\Omega }}_{\mathit{\text{r}}}\end{array}$$ 5.2

where $m_{\mathrm{tot}}=\sum _j{m}_j$ is used henceforth to denote the sum of the masses. By direct analogy with the (excess) energy of a system of point vortices [2], we can now give the explicit form of the gravitational potential energy.

Proposition 5.1 —

The gravitational potential energy of a system of N point masses is

$$\mathcal{U}=\gamma \sum _{k=1}^N\sum _{j<k}^N{m}_j{m}_{k}G({\mathit{\text{r}}}_j,{\mathit{\text{r}}}_k)+\frac{\gamma }{2}\sum _{j=1}^N{m}_j^2\mathcal{R}({\mathit{\text{r}}}_j)-\frac{\gamma \hspace{0.17em}{m}_{\mathrm{tot}}}{2A}\sum _{j=1}^N{m}_j{\iint }_{M}G(\mathit{\text{r}},{\mathit{\text{r}}}_j)\hspace{0.17em}\mathrm{d}{\mathrm{\Omega }}_{\mathit{\text{r}}}.$$ 5.3

The proof follows immediately from equations (5.1) and (5.2).

Remark 5.2 —

The Robin function represents the self-interaction of a single mass with the global geometry of the surface. In particular, with the gauge choice made for the compensating term—i.e. the uniform background mass distribution—the Robin function is directly related to variations of the Gaussian curvature K over the surface M [16,18],

$$\mathcal{R}(\mathit{\text{r}})=\frac{1}{2\pi }{\iint }_{M}G(\mathit{\text{r}},{\mathit{\text{r}}}^{\prime })K({\mathit{\text{r}}}^{\prime })\hspace{0.17em}\mathrm{d}{\mathit{\text{r}}}^{\prime }+c_1,$$ 5.4

where c₁ is a constant. For surfaces with constant Gaussian curvature such as the sphere, it follows that the Robin function is itself a constant [2], and therefore plays no role in the mass dynamics.

Remark 5.3 —

The expression (5.3)—as for the vortex Hamiltonian H (2.14) in [2]—holds for any closed, differentiable, genus zero surface (i.e. any surface topologically equivalent to a sphere).

Remark 5.4 —

The last term in equation (5.3) does not contribute to the dynamics, as the integral of Φ over the whole surface is a constant. We can then simplify the gravitational potential energy to

$$\mathcal{U}=\gamma \sum _{k=1}^N\sum _{j<k}^N{m}_j{m}_{k}G({\mathit{\text{r}}}_j,{\mathit{\text{r}}}_k)+\frac{\gamma }{2}\sum _{j=1}^N{m}_j^2\mathcal{R}({\mathit{\text{r}}}_j),$$ 5.5

where the Green function part describes the gravitational interaction between pairs of masses, while the Robin function part can be viewed as the gravitational potential describing the interaction of a single mass with its uniform compensating mass distribution over the surface. As shown in the example of an ellipsoid—see §8—it is through $\mathcal{R}$ that a single mass can still move on M. Explicit forms of $\mathcal{R}$ are given in [2].

6. The kinetic energy of a system of N point masses on (M,g)

It follows from the fact that we have a mechanical system whose phase space, mathematically speaking, is a cotangent bundle over the product manifold M^N, i.e. T*M^N [19], that we can deduce the mass dynamics as follows.

Proposition 6.1 —

Given a manifold M with metric g, the kinetic energy $\mathcal{K}$ of a mechanical system (i.e. a system which obeys Newton's Second Law) consisting of N point masses m₁,…,m_N is

$$\mathcal{K}=\sum _{j=1}^N\frac{1}{2m_j}\parallel {\mathit{\text{p}}}_j{\parallel }_{g^{-1}}^2,$$ 6.1

where $\parallel {\mathit{\text{p}}}_j{\parallel }_{g^{-1}}^2={\mathit{\text{p}}}_j^T{(g^{-1})}^T{\mathit{\text{p}}}_j,$ g⁻¹ is the inverse of the 2×2 metric tensor g, and p_j is the momentum of the jth mass.

Proof. —

There is a simple way to prove proposition 6.1 using the Lagrangian formulation and Legendre transformations [20]. The Lagrangian $\mathcal{L}$ of a system of particles of masses m₁,…,m_N at positions, respectively, r₁,…,r_N, moving at velocities v₁,…,v_N, is

$$\begin{array}{rl}\mathcal{L}=\mathcal{K}({\mathit{\text{v}}}_1,\dots ,{\mathit{\text{v}}}_N)-\mathcal{U}({\mathit{\text{r}}}_1,\dots ,{\mathit{\text{v}}}_N)& =\frac{1}{2}\sum _{j=1}^N{m}_j\parallel {\mathit{\text{v}}}_j{\parallel }_g^2-\mathcal{U}({\mathit{\text{r}}}_1,\dots ,{\mathit{\text{r}}}_N)\\ & =\frac{1}{2}\sum _{j=1}^N{m}_j{{\mathit{\text{v}}}_j}^{T}g{\mathit{\text{v}}}_j-\mathcal{U}({\mathit{\text{r}}}_1,\dots ,{\mathit{\text{r}}}_N).\end{array}$$

The momentum p_j is defined by

$${\mathit{\text{p}}}_j=\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }{\mathit{\text{v}}}_j}=m_j{{\mathit{\text{v}}}_j}^{T}g\hspace{0.17em}⟹\hspace{0.17em}{{\mathit{\text{v}}}_j}^T=\frac{1}{m_j}{\mathit{\text{p}}}_j{g}^{-1}\hspace{0.17em}⟹\hspace{0.17em}{\mathit{\text{v}}}_j={(g^{-1})}^T{\mathit{\text{p}}}_j^T.$$

It follows that the kinetic energy can be expressed as

$$\mathcal{K}=\frac{1}{2}\sum _{j=1}^N{m}_j{{\mathit{\text{v}}}_j}^{T}g{\mathit{\text{v}}}_j=\sum {j=1}^N\frac{1}{2m_j}{\mathit{\text{p}}}_j{(g^{-1})}^T{{\mathit{\text{p}}}_j}^T=\sum _{j=1}^N\frac{1}{2m_j}\mathit{\text{p}}{g}^{-1}{\mathit{\text{p}}}^T=\sum _{j=1}^N\frac{1}{2m_j}\parallel {\mathit{\text{p}}}_j{\parallel }_{g^{-1}}^2,$$

where g^T=g since g is a symmetric tensor. ▪

Example 6.2 —

For the sphere of radius R, where r_j=(φ_j,θ_j) (with φ_j the longitude and θ_j the co-latitude), the momentum is p_j=(p_jφ,p_jθ). The metric is $\mathrm{d}{s}^2=R^2[\mathrm{d}{\theta }^2+{\sin }^2\theta \hspace{0.17em}\mathrm{d}{\phi }^2]$, corresponding to which the metric tensor g and its inverse are given by

$$g=\left(\begin{array}{cc}{R}^2{\sin }^2\theta & 0\\ 0& R^2\end{array}\right)\text{and}{g}^{-1}=\left(\begin{array}{cc}\frac{1}{R^2{\sin }^2\theta }& 0\\ 0& \frac{1}{R^2}\end{array}\right).$$

It follows from proposition 6.1 that the kinetic energy is

$$\mathcal{K}=\sum _{j=1}^N\frac{1}{2m_j{R}^2}(p_{j\phi },p_{j\theta })\left(\begin{array}{cc}\frac{1}{{\sin }^2{\theta }_j}& 0\\ 0& 1\end{array}\right)\left(\begin{array}{c}{p}_{j\phi }\\ p_{j\theta }\end{array}\right)=\sum _{j=1}^N\frac{1}{2m_j\hspace{0.17em}{R}^2}\left(\frac{p_{j\phi }^2}{{\sin }^2{\theta }_j}+p_{j\theta }^2\right).$$

It follows from proposition 6.1 that the configuration space of a system of N point masses is the product space M^N whose metric is $\mathrm{d}{s}_M^2=\sum _{j=1}^N{m}_{j}d{s}_j^2$. The corresponding metric tensor is given by

$$\mathcal{G}=\left(\begin{array}{ccc}{m}_{1}g& \dots & O\\ O& \ddots & O\\ O& \dots & m_{N}g\end{array}\right),$$

where O is a 2×2 matrix of zeros. Moreover, the motion of the point masses is described by the Hamiltonian system

$${\dot{\mathit{\text{r}}}}_j=-\frac{\mathrm{\partial }\mathcal{H}}{\mathrm{\partial }{\mathit{\text{p}}}_j},{\dot{\mathit{\text{p}}}}_j=\frac{\mathrm{\partial }\mathcal{H}}{\mathrm{\partial }{\mathit{\text{r}}}_j},\hspace{0.17em}j=1,\dots ,N,$$ 6.2

where

$$\mathcal{H}(\mathit{\text{Q}},\mathit{\text{P}})=\frac{\parallel \mathit{\text{P}}{\parallel }_{{\mathcal{G}}^{-1}}^2}{2}+\mathcal{U}(\mathit{\text{Q}}),$$ 6.3

with P=(p₁,…,p_N), Q=(r₁,…,r_N), $\parallel \mathit{\text{P}}{\parallel }_{{\mathcal{G}}^{-1}}^2={\mathit{\text{P}}}^T{\mathcal{G}}^{-1}\mathit{\text{P}}$ and $\mathcal{U}(\mathit{\text{Q}})=\mathcal{U}({\mathit{\text{r}}}_1,\dots ,{\mathit{\text{r}}}_n)$ is the potential in equation (5.5).

Remark 6.3 —

There is a difference between a satellite and a passive tracer advected by vortices in an incompressible fluid. The passive tracer genuinely has zero vorticity, whereas the satellite has a small, but non-zero, mass. So taking this into account, the satellite potential has a small (perturbative) self-interaction through the Robin function:

$${\mathrm{\Phi }}_0(\mathit{\text{r}},t)=\gamma \sum _{j=1}^N{m}_{j}G({\mathit{\text{r}}}_0,{\mathit{\text{r}}}_j(t))+\gamma m_0\mathcal{R}({\mathit{\text{r}}}_0),$$

with m₀≪m_j, j=1,…,N. The satellite kinetic energy is ${\mathcal{K}}_0=\parallel \mathit{\text{p}}{\parallel }_{g^{-1}}^2/(2m_0)$ (by proposition 6.1), and thus the satellite Hamiltonian is ${\mathcal{H}}_0(\mathit{\text{r}},\mathit{\text{p}},t)={\mathcal{K}}_0+m_0{\mathrm{\Phi }}_0(\mathit{\text{r}},t)$. Mathematically speaking, the configuration space is the surface M with metric g, and the phase space is the cotangent bundle T*M.

7. Symmetries and reduction

When studying the dynamics of N masses, the process of reduction makes use of the following fundamental symmetries:

• (a) The symmetries of the Hamiltonian. The groups of transformations that leave the Hamiltonian invariant. Such groups depend on the surface geometry. In the case of the plane, they are the group of translations and the group of rotations, while in the case of the sphere we only have the group of rotations.

• (b) The symmetries of the equations. The group of time transformations that leaves the equations of motion invariant. This group depends both on the degree of separability of the Hamiltonian and on the Robin function. We define a Hamiltonian as separable when it can be expressed as the sum of two distinct functions.

(a). Symmetries in common to all surfaces of revolution conformal to ${\mathbb{S}}^2$

Consider the N-body problem on a surface of revolution for which the masses are parametrized by r_j=(φ_j,θ_j), j=1,…,N. Let ds²=f(θ) dφ²+σ(θ) dθ² be the surface metric. Then it follows from proposition 6.1 that the corresponding kinetic energy is

$$\mathcal{K}=\sum _{j=1}^N\frac{1}{2m_j}\left(\frac{p_{j\phi }^2}{f({\theta }_j)}+\frac{p_{j\theta }^2}{\sigma ({\theta }_j)}\right).$$ 7.1

Thus, the kinetic energy does not depend upon the longitudes φ_j, j=1,…,N, as expected due to the axial symmetry of the surface geometry. Furthermore, the potential energy, see equation (5.5) and [2], can be decomposed as

$$\mathcal{U}=\gamma \sum _{j=1}^N\sum _{k>j}^N{m}_j{m}_{k}G(|{\phi }_j-{\phi }_k|,{\theta }_j,{\theta }_k)+\gamma \sum _{j=1}^N{m}_j^2\mathcal{R}({\theta }_j).$$ 7.2

Therefore, by Noether's theorem [19], we have conservation of angular momentum, $P_{\phi }=\sum _{j=1}^N{p}_{j\phi }$. In addition, for the pair of canonical variables (φ_j,p_jφ), j=1,…,N, the Hamiltonian equations of motion simplify to

$${\dot{\phi }}_j=\frac{\mathrm{\partial }\mathcal{K}}{\mathrm{\partial }{p}_{j\phi }},{\dot{p}}_{j\hspace{0.17em}\phi }=-\frac{\mathrm{\partial }\mathcal{U}}{\mathrm{\partial }{\phi }_j},\hspace{0.17em}j=1,\dots ,N,$$

which gives

$${\ddot{\phi }}_j=-\frac{\mathrm{\partial }\mathcal{U}}{\mathrm{\partial }{\phi }_j},j=1,\dots ,N.$$

Using equation (7.2), it can be verified that the equations above are invariant with respect to the time-varying coordinate transformation

$${\tilde{\phi }}_j={\phi }_j+\nu t.$$

All of the above can then be summarized in the following proposition.

Proposition 7.1 —

For a system of N-bodies on a surface of revolution, the total φ component of the momentum, $P_{\phi }=\sum _{j=1}^N{p}_{j\phi }$, is an integral of motion.

Furthermore, the equations of motion are invariant under the time-dependent transformation

$${\tilde{\phi }}_j={\phi }_j+\nu t\mathrm{\forall }\nu \in \mathbb{R}.$$ 7.3

Remark 7.2 —

Note that the time-varying symmetry group (7.3) for surfaces of revolution directly corresponds to the Galilean group for ${\mathbb{R}}^2$; see §9 and [19].

Remark 7.3 —

The group of transformations (7.3) leaves the equations of motion invariant but not the Hamiltonian. From equation (7.1), we have

$${\dot{\phi }}_j=\frac{\mathrm{\partial }\mathcal{K}}{\mathrm{\partial }{p}_{j\phi }}=\frac{p_{j\phi }}{m_{j}f({\theta }_j)}.$$

It follows that, under the transformation (7.3), the φ component of the momentum changes to

$${\tilde{p}}_{j\tilde{\phi }}=m_{j}f({\theta }_j)({\dot{\tilde{\phi }}}_j+\nu ),$$

which in turn changes the kinetic energy (7.1).

Remark 7.4 —

Observe that, in Euclidean spaces such as ${\mathbb{R}}^2$ and ${\mathbb{R}}^3$, the reduction procedure above can be viewed as introducing a fictitious centre of mass on the surface M, though the real configuration space is M^N. On a general surface, we can no longer use the notion of centre of mass, at least not as in Euclidean spaces (since it would not generally lie on M). Nevertheless, the above coordinate transformations are legitimate on the configuration space M^N.

(b). The unit sphere ${\mathbb{S}}^2$

Consider a system of N masses, m₁, m₂,…,m_N, at the positions r₁,r₂,…,r_N on the unit sphere ${\mathbb{S}}^2$. In spherical coordinates r_j=(φ_j,θ_j), j=1,…,N, the metric of the configuration space is

$$\mathrm{d}{s}^2=\sum _{j=1}^N{m}_j({\sin }^2{\theta }_j\hspace{0.17em}\mathrm{d}{\phi }_j^2+\mathrm{d}{\theta }_j^2).$$

The Green function for the unit sphere problem [2,9,12,21] is

$$G({\mathit{\text{r}}}_j,{\mathit{\text{r}}}_k)=G(|{\phi }_j-{\phi }_k|,{\theta }_j,{\theta }_k)=\frac{1}{4\pi }\log (1-d_{jk}),$$

where

$$d_{jk}=\cos {\theta }_j\cos {\theta }_k+\sin {\theta }_j\sin {\theta }_k\cos ({\phi }_j-{\phi }_k).$$

It follows from equations (6.1) and (5.5) that the Hamiltonian is only partially separable and

$$H=\sum _{j=1}^N\frac{1}{m_j}\left(\frac{p_{j\phi }^2}{{\sin }^2{\theta }_j}+p_{j\theta }^2\right)+\gamma \sum _{j=1}^N\sum _{k>j}^N{m}_j{m}_{k}G(|{\phi }_j-{\phi }_k|,{\theta }_j,{\theta }_k),$$ 7.4

where γ is the gravitational constant of the unit sphere and, as for the plane, the Robin function is a constant [2]. The corresponding equations of motion are

$${\dot{\phi }}_j=\frac{\mathrm{\partial }H}{\mathrm{\partial }{p}_{j\phi }},{\dot{\theta }}_j=\frac{\mathrm{\partial }H}{\mathrm{\partial }{p}_{j\theta }},{\dot{p}}_{j\phi }=-\frac{\mathrm{\partial }H}{\mathrm{\partial }{\phi }_j}\mathrm{and}{\dot{p}}_{j\theta }=-\frac{\mathrm{\partial }H}{\mathrm{\partial }{\theta }_j}.$$ 7.5

Note, the Hamiltonian is invariant under

• — rotations with respect to the three axes, which gives the conservation of the total angular momentum L=(L_x,L_y,L_z); these three integrals do not all commute;

• — time translations, which is related to the conservation of the Hamiltonian H (or excess energy [2]).

In addition, owing to the partial separability of the Hamiltonian, as stated in proposition 7.1, we have the group of symmetries (7.3)

$${\tilde{\phi }}_j={\phi }_j+\nu t,j=1,\dots ,N,$$

i.e. time-dependent transformations which leave equations (7.5) invariant.

The symmetry of the sphere also leads to a simple Cartesian-coordinate formulation of the equations of motion. In Cartesian coordinates, centred at the origin of the sphere, each position vector is a unit vector: |r_j|=1, j=1,2,…,N. Using ${\mathit{\text{r}}}_j=(\sin {\theta }_j\cos {\phi }_j,\sin {\theta }_j\sin {\phi }_j,\cos {\theta }_j)$ in (7.5), we find, after some manipulation,

$${\dot{\mathit{\text{r}}}}_k={\mathit{\text{u}}}_k$$

and

$${\dot{\mathit{\text{u}}}}_k=[\gamma (\tilde{M}-{\tilde{m}}_k)-|{\mathit{\text{u}}}_k{|}^2]{\mathit{\text{r}}}_k+\gamma \sum _{j\ne k}{\tilde{m}}_j\frac{{\mathit{\text{r}}}_j-{\mathit{\text{r}}}_k}{1-{\mathit{\text{r}}}_j\mathbf{\cdot }{\mathit{\text{r}}}_k},$$

where ${\tilde{m}}_k=m_k/4\pi$ and $\tilde{M}=\sum _{k=1}^n{\tilde{m}}_k$. One can verify that the identity $\mathrm{d}({\mathit{\text{r}}}_k\mathbf{\cdot }{\mathit{\text{u}}}_k)/\mathrm{d}t={\dot{\mathit{\text{r}}}}_k\mathbf{\cdot }{\mathit{\text{u}}}_k+{\mathit{\text{r}}}_k\mathbf{\cdot }{\dot{\mathit{\text{u}}}}_k=0$ is satisfied, as required.

The conserved total (kinetic plus potential) energy E is obtained from

$$\frac{E}{4\pi }=\frac{1}{2}\sum _{k=1}^n{\tilde{m}}_k|{\mathit{\text{u}}}_k{|}^2+\gamma \sum _{k=2}^n\sum _{j=1}^{k-1}{\tilde{m}}_j{\tilde{m}}_k\ln (1-{\mathit{\text{r}}}_j\mathbf{\cdot }{\mathit{\text{r}}}_k).$$

(i). The two-body problem

In the case of two masses, the corresponding system of equation (7.5) has eight degrees of freedom (four Hamiltonian degrees of freedom). Using conservation of L_x and L_y, together with the freedom in choosing the orientation of the z-axis (which amounts to choosing L_x=L_y=0), the further conservation of L_z and the time transformation (7.3) reduce the original system to a system having four degrees of freedom. Furthermore, using the conservation of Hamiltonian H itself, we can further reduce this to three. To ensure integrability [19], we need one additional integral of motion.

Following the analysis of Santoprete [22], Schaefer [6] proved the following theorem.

Theorem 7.5 —

Consider the Kepler problem on the unit sphere ${\mathbb{S}}^2$. One body of mass m₂ is held fixed at the north pole and the dynamics of the second body, of mass m₁ and position r=(φ,θ), is described by the Hamiltonian equations

$$\dot{\phi }=\frac{\mathrm{\partial }H}{\mathrm{\partial }{p}_{\phi }},\dot{\theta }=\frac{\mathrm{\partial }H}{\mathrm{\partial }{p}_{\theta }},\dot{p_{\phi }}=-\frac{\mathrm{\partial }H}{\mathrm{\partial }\phi }=0\mathrm{and}\dot{p_{\theta }}=-\frac{\mathrm{\partial }H}{\mathrm{\partial }\theta },$$

where θ represents also the geodesic distance between the two bodies and

$$H(\theta ,p_{\phi },p_{\theta })=\frac{p_{\phi }^2}{2m_1{\sin }^2\theta }+\frac{p_{\theta }^2}{2m_1}+\gamma m_1{m}_2\log [1-\cos (\theta )].$$

The only integrals which are quadratic in the momentum components, i.e. integrals of the type

$$I=a(\phi ,\theta )p_{\theta }^2+2b(\phi ,\theta )p_{\phi }{p}_{\theta }+c(\phi ,\theta )p_{\phi }^2+g(\phi ,\theta ),$$

are

$$I=2m_1{C}_{1}H+C_2{p}_{\phi }^2+C_2,\mathrm{\forall }{C}_1,\hspace{0.17em}{C}_2\in \mathbb{R}.$$

Remark 7.6 —

The extra integrals encountered in the theorem above are linear combinations of known integrals (plus a constant), and, therefore, they do not provide us with a truly new extra integral, as in the corresponding planar problem (see §9), where the corresponding theorem (theorem 9.2) provides us with the components of the Laplace–Runge–Lenz vector.

Remark 7.7 —

In the theorem above, the dynamics is viewed from within the sphere's intrinsic geometry, while in Santoprete [22] the dynamics is viewed as a sphere embedded in ${\mathbb{R}}^3$. Consequently, the potential Φ is different from the intrinsic one, as discussed in §4.

8. Dynamics of one mass

For illustration, let us consider a single mass on a ellipsoid of revolution M, with R=1. As discussed in §3, the metric of M is $\mathrm{d}{s}^2=({\cos }^2\theta +b^2{\sin }^2\theta )\hspace{0.17em}\mathrm{d}{\theta }^2+{\sin }^2\theta \hspace{0.17em}\mathrm{d}{\phi }^2,$ corresponding to which the metric tensor g and its inverse are given by

$$g=\left(\begin{array}{cc}{\sin }^2\theta & 0\\ 0& {\cos }^2\theta +b^2{\sin }^2\theta \end{array}\right)\text{and}{g}^{-1}=\left(\begin{array}{cc}\frac{1}{{\sin }^2\theta }& 0\\ 0& \frac{1}{{\cos }^2\theta +b^2{\sin }^2\theta }\end{array}\right).$$

It follows from proposition 6.1 that the configuration space of a system of one point mass M has the metric $\mathrm{d}{s}_M^2=m\hspace{0.17em}\mathrm{d}{s}^2$. The corresponding metric tensor is given by $\mathcal{G}=mg$. As in equation (6.2), the motion of one point mass is described by the Hamiltonian system

$$\dot{\mathit{\text{r}}}=-\frac{\mathrm{\partial }\mathcal{H}}{\mathrm{\partial }\mathit{\text{p}}}\mathrm{and}\dot{\mathit{\text{p}}}=\frac{\mathrm{\partial }\mathcal{H}}{\mathrm{\partial }\mathit{\text{q}}},$$ 8.1

where r=(φ,θ) and p=(p_φ,p_θ), together with

$$\mathcal{H}(\mathit{\text{q}},\mathit{\text{p}})=\frac{\parallel \mathit{\text{p}}{\parallel }_{{\mathcal{G}}^{-1}}^2}{2}+\mathcal{U}(\mathit{\text{r}}),$$

where

$$\parallel \mathit{\text{p}}{\parallel }_{{\mathcal{G}}^{-1}}^2={\mathit{\text{p}}}^T{\mathcal{G}}^{-1}\mathit{\text{p}}=\frac{p_{\phi }^2}{2m{\sin }^2\theta }+\frac{p_{\theta }^2}{2m({\cos }^2\theta +b^2{\sin }^2\theta )}$$

and

$$\mathcal{U}(\mathit{\text{r}})=\frac{\gamma m^2}{2}\mathcal{R}(\mathit{\text{r}})$$

from equation (5.5). Here, γ is the gravitational constant for the ellipsoid and $\mathcal{R}(\mathit{\text{r}})$ is the Robin function [10], a pure function of θ for a surface of revolution [2].

Then from equation (8.1), a single mass on an ellipsoid evolves according to

$$\begin{array}{rl}\dot{\phi }& =\frac{\mathrm{\partial }\mathcal{H}}{\mathrm{\partial }{p}_{\phi }}=\frac{p_{\phi }}{m{\sin }^2\theta },\dot{\theta }=\frac{\mathrm{\partial }\mathcal{H}}{\mathrm{\partial }{p}_{\theta }}=\frac{p_{\theta }}{m({\cos }^2\theta +b^2{\sin }^2\theta )},\\ {\dot{p}}_{\phi }& =-\frac{\mathrm{\partial }\mathcal{H}}{\mathrm{\partial }\phi }=0,{\dot{p}}_{\theta }=-\frac{\mathrm{\partial }\mathcal{H}}{\mathrm{\partial }\theta }=\frac{p_{\theta }^2(b^2-1)\sin 2\theta }{2m{({\cos }^2\theta +b^2{\sin }^2\theta )}^2}+\frac{p_{\phi }^2\cos \theta }{m{\sin }^3\theta }-\frac{\gamma m^2}{2}\frac{\mathrm{d}\mathcal{R}}{\mathrm{d}\theta }.\end{array}$$

An explicit form for $\mathrm{d}\mathcal{R}/\mathrm{d}\theta$ may be found by combining eqns (4.18) and (5.10) in [2], giving

$$\frac{\mathrm{d}\mathcal{R}}{\mathrm{d}\theta }=-\frac{1}{2\pi \sin \theta }\left(\cos \theta -q(\theta )\frac{\mu (\theta )}{\mu (0)}\right),$$

where $q(\theta )=\sqrt{{\cos }^2\theta +b^2{\sin }^2\theta }$ and

$$\mu (\theta )=\frac{q\cos \theta }{2}+\left\{\begin{array}{cc}\frac{b^2}{2\sqrt{1-b^2}}\ln \left(\frac{q+\sqrt{1-b^2}\cos \theta }{b}\right)& :b<1\\ \frac{b^2}{2\sqrt{b^2-1}}{\sin }^{-1}\left(\frac{\sqrt{b^2-1}\cos \theta }{b}\right)& :b>1.\end{array}\right.$$ 8.2

Note that 4πμ(0) gives the total surface area A of the ellipsoid.

As shown in figure 2 (for b=0.5), a mass initially at rest starts moving along a meridian (φ= const.). This is caused by the interaction with the uniform negative mass spread over its surface, as expressed through the Robin function.

In particular, if the mass is initially at rest p_φ(0)=p_θ(0)=0, and located at φ(0)=φ₀, θ(0)≠{0,π/2,π}, then the motion is purely along a meridian. This, in fact, occurs on any surface of revolution—see [5,2] for more details.

9. The planar two-body problem revisited

Consider a system of N masses m₁,m₂,…,m_N with corresponding positions r₁,r₂,…,r_N on the plane ${\mathbb{R}}^2$. We consider Cartesian coordinates, r_j=(x_j,y_j), j=1,…,N. The metric of the configuration space is then simply

$$\mathrm{d}{s}^2=\sum _{j=1}^N{m}_j(\mathrm{d}{x}_j^2+\mathrm{d}{y}_j^2).$$

The Green function for the planar problem [12,2] is

$$G({\mathit{\text{r}}}_j,{\mathit{\text{r}}}_k)=G(\parallel {\mathit{\text{r}}}_j-{\mathit{\text{r}}}_k\parallel )=\frac{1}{4\pi }\log \parallel {\mathit{\text{r}}}_j-{\mathit{\text{r}}}_k{\parallel }^2.$$

It follows from equations (6.1) and (5.5) that the Hamiltonian is separable and

$$H=\sum _{j=1}^N\frac{1}{m_j}(p_{jx}^2+p_{jy}^2)+\gamma \sum _{j=1}^N\sum _{k>j}^N{m}_j{m}_{k}G(\parallel {\mathit{\text{r}}}_j-{\mathit{\text{r}}}_k\parallel ),$$

since in this case the Robin function is a constant [10,2].

Note, the Hamiltonian is invariant under:

• — rotations with respect to the axis perpendicular to the plane, which implies conservation of total angular momentum L;

• — translations with respect to the x- and y-axes, which implies conservation of the total linear momentum P=(P_x,P_y), where $P_x=\sum _{j=1}^N{p}_{j,x},\hspace{0.17em}{P}_y=\sum _{j=1}^N{p}_{j,y}$; and

• — time translations, which implies conservation of the Hamiltonian H (excess energy).

Owing to the separability of the Hamiltonian—i.e. $H=\mathcal{K}(\mathit{\text{P}})+\mathcal{U}(\mathit{\text{Q}})$ with Q=(r₁,…,r_N) and P=(p₁,…,p_N)—the Hamiltonian equations (6.2) simplify to

$${\dot{\mathit{\text{r}}}}_j=\frac{\mathrm{\partial }\mathcal{K}}{\mathrm{\partial }{\mathit{\text{p}}}_j}\mathrm{and}{\dot{\mathit{\text{p}}}}_j=-\frac{\mathrm{\partial }\mathcal{U}}{\mathrm{\partial }{\mathit{\text{r}}}_j},$$ 9.1

and the first-order system of equations above is equivalent to the second-order system

$${\ddot{\mathit{\text{r}}}}_j=-\frac{\mathrm{\partial }\mathcal{U}}{\mathrm{\partial }{\mathit{\text{r}}}_j},j=1,\dots ,N.$$ 9.2

It follows that the equations of motion have an additional symmetry property: they are invariant with respect to the time-varying coordinate transformations — the Galilean group [19]—

$${\tilde{x}}_j=x_j+v_{x}t,{\tilde{y}}_j=y_j+v_{y}t,\hspace{0.17em}j=1,\dots ,N.$$

(a). The two-body problem and the Laplace–Runge–Lenz vector

In the case of two masses, the corresponding system of equations (9.1) has eight degrees of freedom (four Hamiltonian degrees of freedom). Using conservation of linear momentum P=(P_x,P_y) and the Galilean group above, the original system reduces to a system of four degrees of freedom. Furthermore, using the invariance under rotations and fixing the angular momentum L to a specific value, we further reduce this to two degrees of freedom. Finally, using the conservation of the Hamiltonian H, the system is reduced to a single degree of freedom. The two-body problem in the plane is therefore integrable.

Remark 9.1 —

Following the analysis of Santoprete [22], Schaefer [6,23] proved the following theorem.

Theorem 9.2 —

Consider the Kepler problem on the plane ${\mathbb{R}}^2$. One body of mass m₂ is held fixed at the origin of a chosen reference system and the dynamics of the second body, has, respectively, mass m₁ and position r=(r,φ), where r and φ are the usual polar coordinates. The Kepler problem is described by the Hamiltonian equations

$$\dot{\phi }=\frac{\mathrm{\partial }H}{\mathrm{\partial }{p}_{\phi }},\dot{r}=\frac{\mathrm{\partial }H}{\mathrm{\partial }{p}_r},{\dot{p}}_{\phi }=-\frac{\mathrm{\partial }H}{\mathrm{\partial }\phi }=0\mathrm{and}{\dot{p}}_r=-\frac{\mathrm{\partial }H}{\mathrm{\partial }r},$$ 9.3

where r represents also the geodesic distance between the two bodies and

$$H(r,p_{\phi },p_r)=\frac{p_{\phi }^2}{2m_1{r}^2}+\frac{p_r^2}{2m_1}+\gamma m_1{m}_2\log (r).$$

The only integrals which are quadratic in the momentum variables, i.e. integrals of the type

$$I=a(r,\varphi )p_r^2+2b(r,\phi )p_{\phi }{p}_r+c(r,\phi )p_{\phi }^2+\chi (r,\phi ),$$

are

$$I_1=\sin \phi p_r{p}_{\phi }+\frac{\cos \phi }{r}{p}_{\phi }^2-\gamma m_1{m}_2\cos \phi$$ 9.4

and

$$I_2=-\cos \phi p_r{p}_{\phi }+\frac{\sin \phi }{r}{p}_{\phi }^2-\gamma m_1{m}_2\sin \phi .$$ 9.5

Remark 9.3 —

In the literature, the vector I=(I₁,I₂) is also called the Laplace–Runge–Lenz vector [20].

Remark 9.4 —

Schaefer and co-workers [6,23] proved that three of the four integrals H, p_φ, I₁ and I₂ are independent integrals and in involution. Therefore, the components of the Laplace–Runge–Lenz vector provide us with an extra integral in the case of planar dynamics.

(b). The two-body problem and Kepler's Laws

In ${\mathbb{R}}^2$, as discussed in the previous section, the two-body problem is integrable. Furthermore, it is easy to prove that all orbits are limited (figure 4). Nevertheless, Kepler's Laws are no longer valid in such a geometry [19,6], as the two-body problem admits a unique elliptic orbit, the circular one. To see this, consider the equations of motion of the corresponding Kepler problem (equation (9.3)),

$$\dot{r}=\frac{p_r}{m_1},{\dot{p}}_r=\frac{p_{\phi }^2}{m_1{r}^3}-\frac{\gamma m_1{m}_2}{r}$$ 9.6

and

$$\dot{\phi }=\frac{p_{\phi }}{m_1{r}^2},{\dot{p}}_{\phi }=0.$$ 9.7

We show next that no elliptic orbit exists apart from the circular one. First of all, we have p_φ(t)=p_φ(0) so that at least a circular orbit exists. Now suppose, by contradiction, that a non-circular elliptic orbit exists, i.e.

$$r(\phi )=\frac{a(1-e^2)}{1+e\cos \phi },$$ 9.8

where a is the major semi-axis of the ellipse and e is its eccentricity (0<e<1). Then, from this and equations (9.7) (in particular dφ/dt=p_φm₁r²), we have that

$$\frac{\mathrm{d}r}{\mathrm{d}t}=\frac{\mathrm{d}r}{\mathrm{d}\phi }\frac{\mathrm{d}\phi }{\mathrm{d}t}=\frac{e\sin \phi p_{\phi }}{m_{1}a(1-e^2)}.$$

On the other hand, from equations (9.6), we have dr/dt=p_r/m₁ so that

$$p_r(\phi )=\frac{e\sin \phi p_{\phi }}{a(1-e^2)}.$$

Differentiating this with respect to time, we obtain

$${\dot{p}}_r=\frac{\mathrm{d}p}{\mathrm{d}\phi }\frac{\mathrm{d}\phi }{\mathrm{d}t}=\frac{e\cos \phi p_{\phi }^2}{ma(1-e^2)r^2}.$$

After some algebra, the equation above, together with the equation of motion ${\dot{p}}_r=p_{\phi }^2/m_1{r}^3-\gamma m_1{m}_2/r$, gives

$$\cos \phi =\frac{m{a}^2{(1-e^2)}^2\gamma m_1{m}_2-p_{\phi }^2}{e{p}_{\phi }^2},$$

which leads to a contradiction: the right-hand side is a constant, while the left-hand side is not—φ varies between 0 and 2π. It follows that Kepler's Laws are no longer valid on the plane when viewed from the intrinsic geometry point of view—as opposed to the motion of two bodies in ${\mathbb{R}}^3$—which, by angular momentum conservation, is restricted to a plane.

Figure 4.

(a) Effective potential for the planar case in its intrinsic geometry, $U_{\mathrm{ef}}(r)=p_{\phi }^2/(2m_1{r}^2)+\gamma m_1{m}_2\log (r)$; (b) effective potential for the planar case as embedded in ${\mathbb{R}}^3$, $U_{\mathrm{ef}}(r)=p_{\phi }^2/(2m_1{r}^2)-\gamma m_1{m}_2/r$. (Online version in colour.)

As a final remark, the following theorem (whose proof can be found in [6,19,24]) may be used to conclude that, for the gravitational potential U(r)∝log(r), in a neighbourhood of the circular orbit, there are no periodic orbits of any kind apart from the circular one.

Theorem 9.5 Bertrand's theoremBertrand's theorem —

The only central forces for which all limited orbits are also closed orbits are the forces generated by the potentials U(r)∝1/r and U(r)∝r².

10. Conclusion

We have shown how to formulate the dynamics of point masses on closed surfaces. A key aspect of the analysis is to properly account for the mathematical requirement that the mass integrated over the surface must vanish. This leads to significant differences from previous formulations, developed for punctured surfaces (not truly closed). For example, on a sphere, the radius of the sphere scales out of the gravitational potential, but on a punctured sphere it does not. This has a profound influence on the resulting equations of motion.

An interesting feature of our formulation of point mass dynamics on closed surfaces is that Newton's famous law of inertia does not hold generally (on any surface with variable Gaussian curvature). This means that a particle at rest can begin to move. This is caused by the interaction with the geometry, specifically with the uniform (negative) mass spread uniformly over the surface. Moreover, Kepler's Laws, originally formulated for three-dimensional Euclidean space, generally do not hold. They do not hold even for two-dimensional motion on the plane in its intrinsic geometry. Moreover, on the sphere the Kepler problem does not admit an extra independent integral, quadratic in the momentum components, as occurs for the Kepler problem on the plane.

Finally, and perhaps unexpectedly, variations of Gaussian curvature on closed surfaces generate dynamics. This may be viewed as the classical analogue of the equivalence principle of general relativity, where the curvature of space–time is equivalent to a force field.

Appendix A. Hodge decomposition

We begin by discussing the Hodge decomposition theorem for vector fields in ${\mathbb{R}}^2$ and ${\mathbb{R}}^3$. We closely follow Baird [25] and Chorin & Marsden [26]. Any well-behaved (at least twice differentiable) vector field v on ${\mathbb{R}}^2$ or ${\mathbb{R}}^3$ can be decomposed in three components: transverse v₁ (rotational and non-divergent), radial v₂ (irrotational and divergent) and Laplacian v₃ (irrotational and non-divergent), i.e.

$$\mathit{\text{v}}={\mathit{\text{v}}}_1+{\mathit{\text{v}}}_2+{\mathit{\text{v}}}_3,$$

with

$$\mathrm{div}({\mathit{\text{v}}}_1)=0,\mathrm{div}({\mathit{\text{v}}}_2)=\mathrm{\Lambda },\mathrm{div}({\mathit{\text{v}}}_3)=0$$ A 1

and

$$\mathrm{curl}({\mathit{\text{v}}}_1)=\mathrm{\Sigma },\mathrm{curl}({\mathit{\text{v}}}_2)=0,\mathrm{curl}({\mathit{\text{v}}}_3)=0.$$ A 2

In ${\mathbb{R}}^2$, using Cartesian coordinates, equations div(v₁)=0 and curl(v₂)=0 permit one to introduce, respectively, the functions Ψ and Φ such that the components v₁ and v₂ of the vector field can be written as

$${\mathit{\text{v}}}_1=J\mathrm{\nabla }\mathrm{\Psi }\mathrm{and}{\mathit{\text{v}}}_2=-\mathrm{\nabla }\mathrm{\Phi },$$ A 3

where J is the usual co-symplectic matrix $J=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$ and ∇=(∂/∂x,∂/∂y)^T is the usual gradient.

To determine Ψ and Φ, we then substitute equations (A.3) into the equations curl(v₁)=Σ and div(v₂)=Λ to obtain

$$\mathrm{\Delta }\mathrm{\Psi }=\mathrm{\Sigma }\mathrm{and}\mathrm{\Delta }\mathrm{\Phi }=-\mathrm{\Lambda }.$$

Remark A.1 —

Observe that vector fields belonging to the third class, i.e. verifying

$$\mathrm{div}({\mathit{\text{v}}}_3)=0\text{and}\mathrm{curl}({\mathit{\text{v}}}_3)=0,$$

can be expressed as v₃=−∇Φ_h, where Φ_h is a harmonic function, i.e. satisfying ΔΦ_h=0.

Remark A.2 —

The solution of the Poisson equation ΔΦ=−Λ is not unique because we can always add a solution of the corresponding harmonic equation ΔΦ_h=0. To restrict the class of harmonic functions to constant functions, extra conditions are necessary. For vector fields in ${\mathbb{R}}^2$, the extra condition is to require that, far from any sources, the velocity field tends to zero (v(r)→0 as $|\mathit{\text{r}}|\to \mathrm{\infty })$. In general, each surface has its own set of extra conditions but these are not known in general.

Remark A.3 —

There is a more general version of the Hodge decomposition theorem that can be used for any simply connected domain D of ${\mathbb{R}}^2$, as well as for simply connected surfaces such as the sphere, surfaces conformal to the sphere, and the hyperbolic plane. This generalized theorem is called the ‘one-form decomposition theorem’ [6,23,27,28]. Given a surface M with metric g, we can always associate a one-form field with a given vector field by using the metric tensor. In fact, if a is a vector field on M the corresponding one-form field is σ=g(a,⋅)—or equivalently σ=a^Tg in local coordinates.

Remark A.4 —

Let v=u be the velocity field of a fluid particle—also called a passive tracer. A fluid is said to be incompressible if

$$\mathrm{div}(\mathit{\text{u}})=0.$$ A 4

Given the vorticity field defined as ω=curl(u), incompressible fluids belong to the first class of vector fields above. For the class of surfaces we are considering, using the coordinates of the area form [12,2], equation (A.4) allows us to re-write the velocity field as u=J∇Ψ, where Ψ is a suitably regular function, called the streamfunction, to be determined. Substituting the equation above into curl(u)=ω, we obtain Δ_gΨ=ω, which is the equation that defines Ψ for a given metric g and a given vorticity field ω. For more details, see [12,2].

Remark A.5 —

Let v=a be the acceleration field of a test particle of mass m_o. The corresponding force field F=m_oa is a central force field if

$$\mathrm{curl}(\mathit{\text{a}})=0$$

and div(a)=±γρ, where γ is a constant and ρ is the density of the sources generating the central force. The plus or minus sign in div(a)=±γρ refers to the fact that we can have either a repulsive or an attractive force. It follows that we can re-write a=−∇Ψ, where Ψ is a solution of Δ_gΨ=∓γρ.

Appendix B. The mathematical structure of the N-vortex and N-body phase spaces

From a differential geometry point of view, the phase spaces of the N-vortex and N-body problems are symplectic manifolds. In fact:

• — Consider a system of N vortices of positions r₁,…,r_N and circulations Γ₁,…,Γ_N. The vortex dynamics is governed by a Hamiltonian system of equations and the corresponding phase space is a symplectic manifold.

  More precisely, if M is the surface with metric g on which the vortices are restricted to move, take Ω=dq∧dp to be the area form associated with g. Then the configuration space (M^N) can be viewed as a symplectic manifold endowed with the symplectic form

  $${\mathrm{\Omega }}_v=\sum _{j=1}^N{\mathrm{\Gamma }}_j{\mathrm{\Omega }}_j=\sum _{j=1}^N{\mathrm{\Gamma }}_j\hspace{0.17em}\mathrm{d}{q}_j\wedge \mathrm{d}{p}_j.$$

  The equations describing the dynamics are a system of the first-order equations

  $$\frac{\mathrm{d}{q}_j}{\mathrm{d}t}=\frac{\mathrm{\partial }{H}_v}{\mathrm{\partial }{p}_j}\mathrm{and}\frac{\mathrm{d}{p}_j}{\mathrm{d}t}=-\frac{\mathrm{\partial }{H}_v}{\mathrm{\partial }{q}_j},$$

  where

  $$H_v=\sum _{j=1}^N\sum _{k>j}^N{\mathrm{\Gamma }}_j{\mathrm{\Gamma }}_{k}G({\mathit{\text{r}}}_j,{\mathit{\text{r}}}_k),$$

  r_j=(q_j,p_j) and G(r_j,r_k) is the corresponding Green function. Several papers have already been published on this topic; for further details, the reader may refer to [2,9,10].
• — Consider a system of N point-like bodies of positions r₁,…,r_N and masses m₁,…,m_N. The phase space is the cotangent bundle of the product manifold M^N, specifically T*M^N, which once more is, by its very nature, a symplectic manifold endowed with the symplectic form

  $${\mathrm{\Omega }}_b=\sum _{j=1}^N\mathrm{d}{q}_j\wedge \mathrm{d}{p}_j,$$

  where (q_j,p_j), j=1,…,N are pairs of Darboux coordinates (i.e. canonical variables). As for the vortex case, the equations describing the dynamics are a system of the first-order equations

  $$\frac{\mathrm{d}{q}_j}{\mathrm{d}t}=\frac{\mathrm{\partial }{H}_b}{\mathrm{\partial }{p}_j}\mathrm{and}\frac{\mathrm{d}{p}_j}{\mathrm{d}t}=-\frac{\mathrm{\partial }{H}_b}{\mathrm{\partial }{q}_j},$$

  where, in this case, the Hamiltonian is

  $$H_b=\sum _{j=1}^N\frac{\parallel \mathbf{\text{p}}{\parallel }_{g^{-1}}^2}{2m_j}+\gamma \sum _{j=1}^N\sum _{k>j}^N{m}_j{m}_{k}G({\mathit{\text{r}}}_j,{\mathit{\text{r}}}_k).$$

  Observe that, for the case of N masses in ${\mathbb{R}}^3$, γ is the usual gravitational constant and G(r_k,r_j)=1/∥r_k−r_j∥². For further references on the Euclidean case, see [19].

Authors' contributions

Overall, all authors contributed equally to all aspects of the work. S.B. developed most of the geometrical formulation. D.G.D. developed the numerical code and prepared most of the figures. R.G.S. carried out the study of the existence of extra integrals for the problem of two bodies on the sphere and on the plane.

Competing interests

We declare we have no competing interests.

Funding

R.G.S. was supported by FAPERJ (Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro) through the fellowship FAPERJ Nota 10. S.B. acknowledges funding from CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior) and FAPERJ. D.G.D. gratefully acknowledges support for this research from CNPq (Conselho Nacional de Desenvolvimento Cientifico e Tecnologico) and FINEP (Inovação e Pesquisa) in Brazil, and from the UK Engineering and Physical Sciences Research Council (grant no. EP/H001794/1).