Stability and bifurcations of symmetric tops

TL;DR

This paper analyzes the stability and bifurcations of symmetric tops, specifically Lagrange and Kirchhoff types, using reduction techniques and invariant potentials to understand their equilibrium behavior as a function of spin.

Contribution

It provides an elementary, systematic analysis of stability and bifurcations in symmetric tops through reduction and invariant potential methods, extending to general $SO(2)\times SO(2)$ invariant potentials.

Findings

Stability conditions depend on second and fourth derivatives of the potential.

Bifurcation points are characterized by changes in these derivatives.

The analysis applies to both classical and generalized symmetric tops.

Abstract

We study the stability and bifurcation of relative equilibria 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 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 ${\mathbb Z}_2$ invariant function on a circle, we analyze the stability and bifurcation of relative equilibria of the system in terms of the second and fourth derivative of the function.