# Volume form on Hamiltonian level surface

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

$$\iota(F(q,p))\Omega$$

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?