Let $(M,\theta)$ 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 $M$ by symplectomorphisms. For each $\xi\in\mathfrak g$, let $\xi_M\in\mathfrak X(M)$ denote the fundamental vector field of the action, defined by
paragraph
admin
\begin{align*}
(\xi_M)_p=\left.\frac{d}{dt}\right|_{t=0}\exp(t\xi)\cdot p
\end{align*}
latex_env
admin
for every $p\in M$. Then there exists a smooth map
paragraph
admin
\begin{align*}
\mu:M\to\mathfrak g^*
\end{align*}
latex_env
admin
such that, for every $\xi\in\mathfrak g$, the smooth function
paragraph
admin
\begin{align*}
\mu^\xi:M&\to\mathbb R
\end{align*}