Reduced spaces and their relation to relative equilibria

Summary It is known that the Hamiltonian motion of a mechanical system with symmetry induces Hamiltonian flows on reduced phase spaces. In this paper we apply Morse theory to study the relationship between the topology of the reduced space and the number of relative equilibria in the corresponding momentum level set. Our attention is restricted to simple mechanical systems with compact configuration space and compact symmetry group. We begin by showing that the set of relative equilibria in a level set of the momentum map is compact. We then employ techniques from Morse theory to prove that the number of orbits of relative equilibria with momentum in the coadjoint orbit of a given regular momentum value is bounded below by the the sum of Betti numbers of the corresponding reduced space when the Hamiltonian is fibre quadratic and the reduced Hamiltonian is nondegenerate. In addition, for a certain class of group actions on the configuration manifold, it is shown that the above result extends to Hamiltonians of the form potential plus kinetic.