Compactness and index of relative equilibria for the curved n-body problem

Shuqiang Zhu

Published 2020-03-15Version 1

For the curved n-body problem, we show that the set of relative equilibrium is away from most singular configurations in H^3, and away from a subset of singular configurations in S^3. We also show that each of the n!/2 geodesic relative equilibria for n masses has Morse index n-2. Then we get a direct corollary that there are at least (3n-4)(n-1)!/2 relative equilibria for given n masses if all relative equilibria of these masses are non-degenerate.