= Solution
A constant-energy <phase-space trajectory> of the <harmonic oscillator> is the ellipse
$$
\frac{q^2}{2E/(m\omega^2)}+\frac{p^2}{2mE}=1.
$$
Its area is $\pi\sqrt{2E/(m\omega^2)}\sqrt{2mE}=2\pi E/\omega$. Thus \b[the action and slow energy change are]
$$
\boxed{I=E/\omega,\qquad E(t)\simeq I\omega(t).}
$$
The <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.
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