On the stability of Hamiltonian relative equilibria with non-trivial isotropy

TL;DR

This paper develops a more general and easily computable criterion for the stability of Hamiltonian relative equilibria with non-trivial isotropy, extending previous work by removing the need for Lie algebra splitting.

Contribution

It introduces a new stability criterion that simplifies analysis and applies to systems with non-coadjoint equivariant momentum maps, broadening previous results.

Findings

Provides a unified stability criterion without Lie algebra splitting

Extends stability analysis to non-coadjoint equivariant systems

Simplifies stability computations for Hamiltonian systems with symmetry

Abstract

We consider Hamiltonian systems with symmetry, and relative equilibria with isotropy subgroup of positive dimension. The stability of such relative equilibria has been studied by Ortega and Ratiu and by Lerman and Singer. In both papers the authors give sufficient conditions for stability which require first determining a splitting of a subspace of the Lie algebra of the symmetry group, with different splittings giving different criteria. In this note we remove this splitting construction and so provide a more general and more easily computed criterion for stability. The result is also extended to apply to systems whose momentum map is not coadjoint equivariant.