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

On the Stability of Hamiltonian Relative Equilibria with Non-trivial Isotropy

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 [7] and by Lerman and Singer [3]. In both papers the authors give sufficient conditions for stability which require first determining a splitting of a subalgebra of g, with different splittings giving ...

Stability of Hamiltonian Relative Equilibria by Energy Methods

Relative equilibria in Hamiltonian systems: The dynamic interpretation of nonlinear stability on a reduced phase space

Stability of relative equilibria for Hamiltonian systems is generally equated with Liapunov stability of the corresponding fixed point of the flow on the reduced phase space. Under mild assumptions, a sharp interpretation of this stability is given in terms of concepts on the unreduced space. Suppose that (P, ω) is a symplectic manifold on which a Lie group G acts symplectically, and let H : P ...

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 Lyapunov’s result that an extremal critical point of a conserved quantity is stable, we show that a Poisson equilibrium ...

The transversal relative equilibria of a Hamiltonian system with symmetry

Let P be a symplectic manifold with a free symplectic action of a connected compact Lie group G. We show that, given a certain transversality condition, the set of relative equilibria E nearpe ∈ E of aG-invariant Hamiltonian system onP is locally Whitney-stratified by the conjugacy classes of the isotropy subgroups (under the product of the coadjoint and adjoint actions) of the momentum-generat...