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 let $G$ act smoothly on $M$ on the left by symplectomorphisms. Let
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\begin{align*}
\exp:\mathfrak g\to G
\end{align*}
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denote the Lie group exponential map. For each $g\in G$, let
denote the coadjoint action, defined by $\operatorname{Ad}^*_g(\alpha)=\alpha\circ\operatorname{Ad}_{g^{-1}}$ for every $\alpha\in\mathfrak g^*$. Let
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\begin{align*}
J:M\to\mathfrak g^*
\end{align*}
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be an equivariant moment map for the action, meaning $J(g\cdot p)=\operatorname{Ad}^*_g(J(p))$ for every $g\in G$ and $p\in M$, with the convention that for every $\xi\in\mathfrak g$,
where $J^\xi:M\to\mathbb R$ is defined by $J^\xi(p)=J(p)(\xi)$ and $\xi_M\in\mathfrak X(M)$ is the fundamental vector field
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\begin{align*}
(\xi_M)_p=\left.\frac{d}{dt}\right|_{t=0}\exp(t\xi)\cdot p.
\end{align*}
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Let $\mu\in\mathfrak g^*$, set
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\begin{align*}
Z=J^{-1}(\mu),
\end{align*}
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and suppose the Marsden-Weinstein-Meyer reduction hypotheses hold at $\mu$: $\mu$ is a regular value of $J$, the stabilizer $G_\mu=\{g\in G:\operatorname{Ad}^*_g\mu=\mu\}$ acts freely and properly on $Z$, and the quotient
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\begin{align*}
M_\mu=Z/G_\mu
\end{align*}
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is a [smooth manifold](/page/Smooth%20Manifold) with quotient projection $\pi:Z\to M_\mu$ and reduced symplectic form $\omega_\mu\in\Omega^2(M_\mu)$ satisfying
Let $H\in C^\infty(M;\mathbb R)$ be $G$-invariant, and let $X_H\in\mathfrak X(M)$ be its Hamiltonian vector field, defined by
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\begin{align*}
\iota_{X_H}\omega=dH.
\end{align*}
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Let $I\subset\mathbb R$ be an interval, and assume that each integral curve of $X_H$ with initial point in $Z$ considered on $I$ exists on all of $I$. Then $X_H$ is tangent to $Z$. Moreover, there exists a unique smooth function $H_\mu\in C^\infty(M_\mu;\mathbb R)$ satisfying
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\begin{align*}
H|_Z=\pi^*H_\mu,
\end{align*}
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and for every integral curve $\gamma:I\to Z$ of $X_H$, the projected curve $\gamma_\mu:I\to M_\mu$ defined by $\gamma_\mu(t)=\pi(\gamma(t))$ is an integral curve of the Hamiltonian vector field $X_{H_\mu}\in\mathfrak X(M_\mu)$ determined by