Reduction of Hamiltonian systems with symmetry

Abstract

This paper presents an alternative phase space reduction process for Hamiltonian systems with symmetry using a Lie algebra of infinitesimal symmetries instead of a Lie group of symmetries. This approach avoids the use of the various conditions on the action of the group or of the isotropy subgroup necessary for the usual construction of a reduced phase space (quotient) with a globally defined symplectic manifold structure. The existence of such a reduced phase space is proved without recourse to the existence of an equivariant moment map. Instead it is assumed that the orbits of the action associated to the Lie algebra can be described as the level sets of some analytic map, an assumption that holds in many of the known examples. The relation between the two reduction processes is discussed; in particular, some cases where the Marsden-Weinstein process leads to a quotient space which is not a smooth manifold are shown to be amenable to the method now proposed. The natural control theoretic interpretation of the techniques used is also discussed.