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.