Florin Diacu Relative Equilibria in the 3-Dimensional Curved $n$-Body Problem

Annotation

The author considers 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 $\kappa<0$. His goal is to define and study relative equilibria, which are orbits whose mutual distances remain constant in time. He also briefly discusses the issue of singularities in order to avoid impossible configurations. He derives the equations of motion and defines six classes of relative equilibria, which follow naturally from the geometric properties of ${\mathbb S}_\kappa^3$ and ${\mathbb H}_\kappa^3$. Then he proves 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 he describes their qualitative behaviour.