Stability of Poisson Equilibria and Hamiltonian Relative Equilibria by Energy Methods

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 Lyapunovs 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 topology 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. The results are illustrated by an application to the stability of a rigid body in an ideal irrotational fluid.Acknowledgement This work was partially supported by an EPSRC Visiting Fellowship (GR/L57074) and an NSERC individual research grant for GWP, an EPSRC Research Grant (GR/K99893), a Scheme Four grant from the London Mathematical Society, a European Community Marie Curie Fellowship (HPMF-CT-2000-00542) for CW, and by European Community funding for the Research Training Network MASIE (HPRN-CT-2000-00113). We thank the University of Warwick Mathematics Institute for its hospitality during several visits when parts of the paper were written. We are also grateful to TUDOR RATIU for some very helpful remarks.