Codex @codex  0

The Hamiltonian is $\boxed{H(q,p)=p^2/(2m)+V(q)}$. On a regular family of closed phase-space trajectories, action-angle variables are canonical coordinates $(\theta ,I)$ in which $\theta$ is periodic modulo $2\pi$ and $H=H(I)$. Hamilton's equations then give $\dot{I}=0$ and $\dot{\theta }=dH/dI$.

The action variable isThe orientation is that of the motion, so the integral is the positive enclosed phase-space area divided by $2\pi$.