Regular polygonal equilibrium configurations on S^1 and stability of the associated relative equilibria

TL;DR

This paper investigates regular polygonal equilibrium configurations in the curved n-body problem on S^3, establishing conditions for equilibrium and analyzing the stability of associated relative equilibria on S^1 and S^2.

Contribution

It characterizes when regular polygonal configurations are equilibria in S^3 and analyzes their stability depending on angular velocity.

Findings

Equilibrium configurations occur only for odd n with equal masses.

Relative equilibria are Lyapunov stable on S^1.

Stability on S^2 depends on the angular velocity, being stable for larger values and unstable for smaller ones.

Abstract

For the curved n-body problem in S^3, we show that a regular polygonal configuration for n masses on a geodesic is an equilibrium configuration if and only if n is odd and the masses are equal. The equilibrium configuration is associated with a one-parameter family (depending on the angular velocity) of relative equilibria, which take place on S^1 embedded in S^2. We then study the stability of the associated relative equilibria on two invariant manifolds, T^*((\S^1)^n\D) and T^*((\S^2)^n\D). We show that they are Lyapunov stable on S^1, they are Lyapunov stable on S^2 if the absolute value of angular velocity is larger than a certain value, and that they are linearly unstable on S^2 if the absolute value of angular velocity is smaller than that certain value.