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 → R be a G-invariant Hamiltonian function. A relative equilibrium is a point of phase space with Hamiltonian evolution coincident with a one parameter orbit of the symmetry group G. Relative equilibria correspond to fixed points of the flow on the Poisson or symplectic reduced phase spaces [1][3]. Verifying the nonlinear stability of relative equilibria is generally equated with establishing the Liapunov stability of the corresponding fixed point of the flow on the reduced phase space [3][6][9]. A defect of this approach is the absence of a fundamental interpretation of nonlinear stability in terms of the dynamics on the unreduced phase spaces. The most obvious candidate for a definition of stability in this context is orbital stability: the evolution obtained from an initial condition near enough to a given relative equilibrium remains in any specified open neighborhood of the orbit of that relative equilibrium. In general, however, orbital stability of relative equilibria in Hamiltonian systems with symmetry cannot be expected. For example, thinking about the motion of a single rigid body rotating on its longest or shortest principle axis of inertia, then perturbing this motion in such a way that the body only rotates more quickly, you can see there results two MSC: 58F10, 58F05.