= Solution
For a one-dimensional closed energy orbit, define
$$
I(E)=\frac1{2\pi}\oint p(q,E)\,dq.
$$
After locally inverting this relation to write $E=H(I)$, use the Hamilton--Jacobi generating <function>
$$
W(q,I)=\int^q p(q',I)\,dq'
$$
and define the angle $\phi=\partial W/\partial I$ modulo $2\pi$. The transformation is canonical, the <Hamiltonian> depends only on $I$, and
$$
\dot I=0,
\qquad
\dot\phi=\frac{dH}{dI}=\omega(I).
$$
The adiabatic-invariance principle says that if an external parameter changes on a time scale much longer than the orbital period, and no separatrix is crossed, the action $I$ remains approximately constant. Its small within-cycle changes do not accumulate at leading adiabatic order.
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