I am interested in certain properties of measures evolving according to Hamiltonian mechanics. Say we have a point $z$ in phase space: $z = (p,q)$ where $p$ is a generalized momentum vector and $q$ is a generalized position vector. Given a Hamiltonian $H(p,q)$, $z$ evolves in time according to:
$\frac{\partial p_i}{\partial t} = -\frac{\partial H}{\partial q_i}$
$\frac{\partial q_i}{\partial t} = \frac{\partial H}{\partial p_i}$
Now if instead of a single point in phase space we have a probability measure $\mu$ over phase space, that measure evolves according to
$\frac{\partial \mu}{\partial t} = \sum_i \frac{\partial \mu}{\partial p_i}\frac{\partial H}{\partial q_i} - \frac{\partial \mu}{\partial q_i}\frac{\partial H}{\partial p_i}$
assuming $\mu$ has a differentiable pdf. This is the Liouville equation from classical mechanics. I am wondering if the functional
$\int dpdq \frac{\partial \mu}{\partial t}H(p,q) = \int dpdq \left(\sum_i \frac{\partial \mu}{\partial p_i}\frac{\partial H}{\partial q_i} - \frac{\partial \mu}{\partial q_i}\frac{\partial H}{\partial p_i}\right)H$
can ever be nonzero. My intuition is that it is always zero - since points move along curves of constant $H$ in phase space - however I can neither prove it nor find a counterexample.