REDUCED SPACES AND THEIR RELATION TO RELATIVE EQUILIBRIA

It is known that the Hamiltonian motion of a mechanical system with sy mmetry induces Hamiltonian flows on reduced phase spaces. In this pape r we apply Morse theory to study the relationship between the topology of the reduced space and the number of relative equilibria in the cor responding momentum level set. Our attention is restricted to simple m echanical systems with compact configuration space and compact symmetr y 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 fr om Morse theory to prove that the number of orbits of relative equilib ria with momentum in the coadjoint orbit of a given regular momentum v alue is bounded below by the the sum of Betti numbers of the correspon ding reduced space when the Hamiltonian is fibre quadratic and the red uced Hamiltonian is nondegenerate. In addition, for a certain class of group actions on the configuration manifold, it is shown that the abo ve result extends to Hamiltonians of the form potential plus kinetic.