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Stability of Poisson Equilibria and Hamiltonian Relative Equilibria by Energy Methods

George W. Patrick·Mark Roberts·Claudia Wulff
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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.

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George W. Patrick, Mark Roberts, Claudia Wulff. 2002-01-24. Stability of Poisson Equilibria and Hamiltonian Relative Equilibria by Energy Methods. https://arxiv.org/abs/math/0201239

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