Hamiltonian Relative Equilibria with Continuous Isotropy

TL;DR

This paper investigates the structure, stability, and bifurcations of Hamiltonian relative equilibria with continuous isotropy in systems where the symmetry group action is non-free, extending existing theories to broader contexts.

Contribution

It provides a systematic framework using bundle equations to analyze non-free symmetry actions, extending known results and introducing new insights into stability and bifurcation phenomena.

Findings

Extended the geometric understanding of relative equilibria with continuous isotropy.

Developed a systematic approach using bundle equations for non-free actions.

Identified conditions for stability, persistence, and bifurcations in these systems.

Abstract

In symmetric Hamiltonian systems, relative equilibria usually arise in continuous families. The geometry of these families in the setting of free actions of the symmetry group is well-understood. Here we consider the question for non-free actions. Some results are already known in this direction, and we use the so called bundle equations to provide a systematic treatment of this question which both consolidates the known results, extending the scope of the results to deal with non-compact symmetry groups, as well as producing new results. Specifically we address questions about the stability, persistence and bifurcations of these relative equilibria.