Let $(M,\theta)$ be a symplectic manifold, let $G$ be a Lie group with [Lie algebra](/page/Lie%20Algebra) $\mathfrak g$, and suppose that $G$ acts smoothly on $M$ on the left by symplectomorphisms. For $\xi\in\mathfrak g$, let $\xi_M\in\mathfrak X(M)$ denote the fundamental vector field
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\begin{align*}
(\xi_M)_x=\left.\frac{d}{ds}\right|_{s=0}\exp(s\xi)\cdot x
\end{align*}
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for $x\in M$. Suppose the action is Hamiltonian with moment map $\mu:M\to\mathfrak g^*$, using the convention that, for every $\xi\in\mathfrak g$, the smooth function $\mu^\xi:M\to\mathbb R$ defined by
Let $H\in C^\infty(M;\mathbb R)$ be $G$-invariant, meaning
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\begin{align*}
H(g\cdot x)=H(x)
\end{align*}
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for every $g\in G$ and every $x\in M$. Let $X_H\in\mathfrak X(M)$ be the Hamiltonian vector field determined by the sign convention
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\begin{align*}
dH=\iota_{X_H}\theta.
\end{align*}
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If $\varphi:D\to M$ is the maximal flow of $X_H$, with $D\subset\mathbb R\times M$ its open flow domain and $\varphi_t(p)=\varphi(t,p)$, then for every $\xi\in\mathfrak g$, every $p\in M$, and every $t\in\mathbb R$ such that $(t,p)\in D$,