GEOMETRIC AND ALGEBRAIC REDUCTION FOR SINGULAR MOMENTUM MAPS

This paper concerns the reduction of singular constraint sets of symplectic manifolds. It develops a "geometric" reduction procedure, as well as continues the work of Sniatycki and Patrick on reduction a la Dirac. The relationships among the Dirac, geometric, and Sniatycki-Weinstein algebraic reduction procedures are studied. Primary emphasis is placed on the case where the constraints are given by the vanishing of a (singular) momentum map associated to the Hamiltonian action of a compact Lie group. Specifically, an explicit local normal form is given for the momentum map which is used to show that these various reductions are all well defined, and a necessary and sufficient condition for them to agree is derived. Related conditions are investigated and shown to be sufficient in the case of torus actions. Numerous examples are computed, illustrating the results. Some discussion and examples are given for the noncompact case. © 1990.