Action-Angle Variables

For a bound one-degree-of-freedom system the natural coordinates are not position and momentum but the action $J=\tfrac1{2\pi}\oint p\,dq$ (the phase-space area the orbit encloses, divided by $2\pi$) and an angle $\theta$ that simply winds at the constant rate $\dot\theta=\omega(J)=\partial H/\partial J$. The left panel is the phase orbit with that enclosed area shaded; the right is the action-angle picture, where the harmonic orbit becomes a circle of radius $\sqrt{2J}$ swept uniformly. The harmonic oscillator is isochronous ($\omega=\omega_0$ for every amplitude); the pendulum is anharmonic ($\omega$ falls as the swing grows). Ramp $\omega_0$ slowly and $J$ barely moves while the energy tracks $\omega$: adiabatic invariance.