Stability and bifurcations of symmetric tops

Eugene Lerman

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arXiv · 2111.04855

Stability and bifurcations of symmetric tops

Eugene Lerman

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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.

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Eugene Lerman. 2021-11-08. Stability and bifurcations of symmetric tops. https://arxiv.org/abs/2111.04855

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