 Marsden-Weinstein theorem

For a Hamiltonian group action, let $c$ be a regular value of the moment map fixed by the coadjoint action, and assume the action on ${\mu }^{-1}(c)$ is free and proper. The quotient is a symplectic manifold of dimension $\dim M-2\dim G$. Its unique reduced symplectic form is characterized by ${\pi }^{\ast }{\omega }_c={\iota }^{\ast }\omega$, where $\iota$ includes the level set and $\pi$ is the quotient projection. The kernel of the restricted two-form consists precisely of the orbit directions. More general noncentral levels use their stabilizer group; the stated central-level version suffices for circle group reduction.