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Assume we have a Hamiltonian system on $(\mathbb{R}^{2n},\omega)$ with Hamiltonian $H = H(q,p)$. In a paper I read, it says, without clarification, that the natural Liouville measure $\mu$ obtained by the volume form $\Omega = \omega \wedge \cdots \wedge \omega $ restricts on the energy level surfaces $N = \left\{(q,p) \in \mathbb{R}^{2n}: H(q,p) = c\right\}$ to


where F(q,p) is a function that satisfies $\omega(X_{H},F(q,p)) = dH(F(q,p)) = 1$ and $\iota(\cdot)$ is the interior product. A possible choice would be $F(q,p) = \frac{grad H}{\|grad H\|}$.

Why is this so?

Thanks in advance!

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It's often helpful to look at the original source; could you give a reference to the paper? –  user53153 Feb 11 '13 at 1:21

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