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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
Author
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.
Keywords
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