Stability of symmetric tops via one variable calculus

Abstract

We study the stability of symmetric trajectories of a particle on the Lie group $SO(3)$ whose motion is governed by an $SO(3)\times SO(2)$ invariant metric and an $SO(2)\times SO(2)$ invariant potential. Our method is to reduce the number of degrees of freedom at {\em singular} values of the $SO(2)\times SO(2)$ momentum map and study the stability of the equilibria of the reduced systems as a function of spin. The result is an elementary analysis of the fast/slow transition in the Lagrange and Kirchhoff tops. More generally, since an $SO(2)\times SO(2)$ invariant potential on $SO(3)$ can be thought of as ${\bf Z}_2$ invariant function on a circle, we get a condition on the second and fourth derivatives of the potential at the symmetric points that guarantees that the corresponding system gains stability as the spin increases.