Stability of symmetric tops via one variable calculus
Abstract
Description
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.
10 pages, LaTeX, uses epic and eepic style files
10 pages, LaTeX, uses epic and eepic style files