Persistence of relative equilibria in Hamiltonian systems with non-compact symmetry

Abstract

We prove new results on the persistence of Hamiltonian relative equilibria with generic velocity–momentum pairs in the case of non-compact non-free group actions and taking into account time reversibility. Our starting point is a relative equilibrium which is non-degenerate modulo isotropy which, in the case of a generic momentum implies persistence of the given relative equilibrium to all nearby momentum values with the same isotropy. We show that the analysis of the persistence problem involves the study of a singular algebraic variety which is determined solely by the symmetry group of the problem. We present persistence results for relative equilibria with velocity–momentum pairs which are regular points of this variety and give sufficient conditions for a velocity–momentum pair to be regular. We apply our results to relative equilibria of Euclidean equivariant systems, including models of rigid bodies in fluids.