Regular Polygonal Equilibria on $\mathbb {S}^1$ and Stability of the Associated Relative Equilibria

Abstract

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