Dynamics near Relative Equilibria: Nongeneric Momenta at a 1:1 Group-Reduced Resonance

Abstract

An interesting situation occurs when the linearized dynamics of the shape of a formally stable Hamiltonian relative equilibrium at nongeneric momentum 1:1 resonates with a frequency of the relative equilibrium's generator. In this case some of the shape variables couple to the group variables to first order in the momentum perturbation, and the first order perturbation theory implies that the relative equilibrium slowly changes orientation in the same way that a charged particle with magnetic moment moves on a sphere under the influence of a radial magnetic monopole. In the course of showing this a normal form is constructed for linearizations of relative equilibria and for Hamiltonians near group orbits of relative equilibria.