Stability and bifurcations of symmetric tops

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) × SO(2) invariant metric and an SO(2) × SO(2) invariant potential. Our method is to reduce the number of degrees of freedom at singular values of the SO(2) × 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) × SO(2) invariant potential on SO(3) can be thought of as Z₂ 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.