Pure and Applied Mathematics Quarterly

Volume 19 (2023)

Number 4

Special Issue in honor of Victor Guillemin

Guest Editors: Yael Karshon, Richard Melrose, Gunther Uhlmann, and Alejandro Uribe

Stability and bifurcations of symmetric tops

Pages: 2037 – 2065

DOI: https://dx.doi.org/10.4310/PAMQ.2023.v19.n4.a12


Eugene Lerman (Department of Mathematics , University of Illinois at Urbana-Champaign, Urbana, Il., U.S.A.)


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.


bifurcation, stability, finite-dimensional Hamiltonian systems

2010 Mathematics Subject Classification

Primary 37J20, 37J25. Secondary 58A40.

Received 9 November 2021

Received revised 6 March 2022

Accepted 5 April 2022

Published 20 November 2023