Stability of Poisson Equilibria and Hamiltonian Relative Equilibria by Energy Methods

TL;DR

This paper develops a comprehensive stability theory for Poisson and Hamiltonian systems with symmetries, extending classical methods to more general and complex cases including non-compact groups and non-Hausdorff orbit spaces.

Contribution

It generalizes the Energy-Casimir and Energy-Momentum methods, providing new stability criteria applicable to broader classes of Hamiltonian systems with symmetries.

Findings

Poisson equilibria are stable if isolated in a specific conserved set intersection.

Generalization of Energy-Momentum method to non-compact symmetry groups.

Relative equilibria stability characterized by stronger A-stability condition.

Abstract

We develop a general stability theory for equilibrium points of Poisson dynamical systems and relative equilibria of Hamiltonian systems with symmetries, including several generalisations of the Energy-Casimir and Energy-Momentum methods. Using a topological generalisation of Lyapunov's result that an extremal critical point of a conserved quantity is stable, we show that a Poisson equilibrium is stable if it is an isolated point in the intersection of a level set of a conserved function with a subset of the phase space that is related to the non-Hausdorff nature of the symplectic leaf space at that point. This criterion is applied to generalise the Energy-Momentum method to Hamiltonian systems which are invariant under non-compact symmetry groups for which the coadjoint orbit space is not Hausdorff. We also show that a $G$-stable relative equilibrium satisfies the stronger condition of being $A$-stable, where $A$ is a specific group-theoretically defined subset of $G$ which contains the momentum isotropy subgroup of the relative equilibrium.