Let $(M,\omega)$ be a finite-dimensional symplectic manifold, let $G$ be a finite-dimensional Lie group with [Lie algebra](/page/Lie%20Algebra) $\mathfrak g$, and suppose that $G$ acts smoothly on $M$ on the left by Hamiltonian symplectomorphisms. For $\xi\in\mathfrak g$, let $\xi_M\in\mathfrak X(M)$ denote the fundamental vector field
for every $\xi\in\mathfrak g$. For $g\in G$, let $\operatorname{Ad}_g^*:\mathfrak g^*\to\mathfrak g^*$ denote the coadjoint action, defined by $\operatorname{Ad}_g^*\alpha=\alpha\circ\operatorname{Ad}_{g^{-1}}$ for $\alpha\in\mathfrak g^*$. For $\xi\in\mathfrak g$, let $\operatorname{ad}_\xi^*:\mathfrak g^*\to\mathfrak g^*$ denote its infinitesimal coadjoint action, defined by
denote the Lie algebra of $G_\mu$, and let $G_\mu\cdot x=\{g\cdot x:g\in G_\mu\}$ be the orbit of $x$ under the restricted action of $G_\mu$ on $M$. Then, for every $x\in J^{-1}(\mu)$,