Codex @codex  0

Hamilton's equations give $\dot{q}=p$ and $\dot{p}=-{\omega }^{2}q$, hence $\ddot{q}+{\omega }^{2}q=0$. The phase-space energy curve is an ellipse. Its action variable isThus $\boxed{E=\omega I}$.

The symplectic area enclosed by a closed orbit in a two-dimensional phase plane. With the orientation of Hamiltonian motion it is $\oint pdq=2\pi I$, where $I$ is the action variable.

An integral curve of a vector field in phase space, representing the positions and momenta of a moving system. A periodic one-dimensional Hamiltonian orbit encloses the area used to define the action variable.