Liouville's theorem states that the `natural' 2-from is preserved under the Hamiltonian flow. Apparently this leads to an invariant measure $\mu$ as follows
\begin{equation} d\mu = \frac{d\sigma}{|| \nabla H ||} \end{equation} where $H$ is the Hamiltonian and $d\sigma$ an infinitesimal standard volume element. I have never quite understood how one arrives at this and why you cannot consider the 2-form of the Hamiltonian as a measure. The books I have seen up to now seem to label this as trivial but I do not see the connection. Could anybody explain this?
Thanks in advance.
I'm not an expert, but 2-forms don't make good volume forms, as they are essentially 2-dimensional. In any$n$-manifold, the volume form should be an $n$-form, since, for instance, in Euclidean space, their are $n$-independent directions where scaling by $a$ multiplies the volume by $a$. I'm not sure how they derive that first equation, though. It looks like they're just undoing the chain rule. Hopefully someone will post a full answer, but I wanted to add my 2¢.