I am interested in properties of the action-angle variables obtained from a Hamiltonian system.
Specifically, given Hamiltonian $H(x,y)$ in a region without critical points, we can switch from Cartesian coordinates $(x,y)$ to "action/angle" coordinates $(I,\phi)$ (using Arnold's notation), where $I$ is the "action" variable and $\phi$ as the "angle" variable.
Now, I would like to prove (if it is in fact true) the following property of the $\phi$, "angle," variable.
$$\oint_{\{(x,y)|H(x,y)=h\}} \Delta_{x,y}\phi \,d\phi=0$$
That is, that the Cartesian Laplacian of $\phi$ integrates to zero around a level set of the Hamiltonian.
I am able to confirm this for polar coordinates by direct computation (in which case $I=r^2$ and $\phi=\theta$), but I am having trouble proving this for a general 2D Hamiltonian system.
Any suggestions or leads are greatly appreciated!