Symplectic Quotients
Suppose that the action of a compact Lie group G on the symplectic manifold (M, ω) is Hamiltonian, as defined above, with moment map . From the Hamiltonian condition it follows that is invariant under G.
Assume now that 0 is a regular value of μ and that G acts freely and properly on . Thus and its quotient are both manifolds. The quotient inherits a symplectic form from M; that is, there is a unique symplectic form on the quotient whose pullback to equals the pullback of ω to . Thus the quotient is a symplectic manifold, called the Marsden–Weinstein quotient, symplectic quotient or symplectic reduction of M by G and is denoted . Its dimension equals the dimension of M minus twice the dimension of G.
Read more about this topic: Moment Map