A Universal Reduction Procedure for Hamiltonian Group Actions

We give a universal method of inducing a Poisson structure on a singular reduced space from the Poisson structure on the orbit space for the group action. For proper actions we show that this reduced Poisson structure is nondegenerate. Furthermore, in cases where the Marsden-Weinstein reduction is well-defined, the action is proper, and the preimage of a coadjoint orbit under the momentum mapping is closed, we show that universal reduction and Marsden-Weinstein reduction coincide. As an èxample, we explicitly construct the reduced spaces and their Poisson algebras for the spherical pendulum.