Let $(M,\omega)$ be a smooth 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. For $\xi\in\mathfrak g$, let $\xi_M\in\mathfrak X(M)$ denote the fundamental vector field
Assume the action is Hamiltonian with moment map $J:M\to\mathfrak g^*$, using the convention that, for every $\xi\in\mathfrak g$, the component function $J^\xi:M\to\mathbb R$ defined by $J^\xi(x)=J(x)(\xi)$ satisfies