 Solution

A constant-energy phase-space trajectory of the harmonic oscillator is the ellipseIts area is $\pi \sqrt{2E/(m{\omega }^2)}\sqrt{2mE}=2\pi E/\omega$. Thus the action and slow energy change areThe action variable is an adiabatic invariant when $|\dot{\omega }|\ll {\omega }^2$, with $\omega$ staying positive. Averaging over many short oscillations therefore keeps $I$ approximately fixed; an increase in frequency increases the energy in the same proportion.