Let $(M,\theta)$ be a symplectic manifold, and let $G$ be a Lie group acting smoothly on $M$ on the left by symplectomorphisms. For $\xi\in\mathfrak g$, let $\xi_M\in\mathfrak X(M)$ be the fundamental vector field defined by
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\begin{align*}
\xi_M(x)=\frac{d}{dt}\bigg|_{t=0}\exp(t\xi)\cdot x
\end{align*}