Relative equilibria in the 3-dimensional curved n-body problem

TL;DR

This paper investigates the existence and properties of relative equilibria in the 3D curved n-body problem on spheres and hyperbolic spaces, classifying solutions and providing concrete examples including novel quasiperiodic orbits.

Contribution

It introduces a classification of relative equilibria in curved 3D spaces, derives existence criteria, and constructs explicit examples, including the first quasiperiodic solutions in this context.

Findings

Bodies move on circles or Clifford tori in spherical space.

Bodies move on circles or hyperbolic cylinders in hyperbolic space.

Existence of quasiperiodic relative equilibria in 3D curved n-body problem.

Abstract

We consider the 3-dimensional gravitational $n$-body problem, $n\ge 2$, in spaces of constant Gaussian curvature $\kappa\ne 0$, i.e.\ on spheres ${\mathbb S}_\kappa^3$, for $\kappa>0$, and on hyperbolic manifolds ${\mathbb H}_\kappa^3$, for $\kappa<0$. Our goal is to define and study relative equilibria, which are orbits whose mutual distances remain constant in time. We also briefly discuss the issue of singularities in order to avoid impossible configurations. We derive the equations of motion and define six classes of relative equilibria, which follow naturally from the geometric properties of ${\mathbb S}_\kappa^3$ and ${\mathbb H}_\kappa^3$. Then we prove several criteria, each expressing the conditions for the existence of a certain class of relative equilibria, some of which have a simple rotation, whereas others perform a double rotation, and we describe their qualitative behaviour. In particular, we show that in ${\mathbb S}_\kappa^3$ the bodies move either on circles or on Clifford tori, whereas in ${\mathbb H}_\kappa^3$ they move either on circles or on hyperbolic cylinders. Then we construct concrete examples for each class of relative equilibria previously described, thus proving that these classes are not empty. We put into the evidence some surprising orbits, such as those for which a group of bodies stays fixed on a great circle of a great sphere of ${\mathbb S}_\kappa^3$, while the other bodies rotate uniformly on a complementary great circle of another great sphere, as well as a large class of quasiperiodic relative equilibria, the first such non-periodic orbits ever found in a 3-dimensional $n$-body problem. Finally, we briefly discuss other research directions and the future perspectives in the light of the results we present here.