Let $(M^{2n},\omega)$ be a symplectic manifold, let $H:M\to\mathbb R$ be a smooth function, and let $c\in\mathbb R$ be a regular value of $H$. Set
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\begin{align*}
\Sigma:=H^{-1}(c).
\end{align*}
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Let $X_H\in\mathfrak X(M)$ be the Hamiltonian vector field defined by
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\begin{align*}
\omega(X_H,\cdot)=dH.
\end{align*}
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For each $x\in\Sigma$, let
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\begin{align*}
\mathcal L_{\Sigma,x}:=\ker\left(\omega_x|_{T_x\Sigma}\right)=\{v\in T_x\Sigma:\omega_x(v,w)=0\text{ for every }w\in T_x\Sigma\}
\end{align*}
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denote the characteristic line of $\Sigma$ at $x$. Then $X_H(x)\in T_x\Sigma$ and
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\begin{align*}
\mathbb R X_H(x)=\mathcal L_{\Sigma,x}
\end{align*}