 Solution

The action variable for a regular periodic one-degree-of-freedom Hamiltonian orbit is $I(E)=(2\pi {)}^{-1}\oint pdq$, with orientation along the motion. Along its energy curve, implicit differentiation gives $\partial p/\partial E=1/H_p=1/\dot{q}$. Differentiating the loop integral branchwise givesAt simple turning points the two momentum branches join and their endpoint contributions cancel, so the integrable turning-point singularities do not spoil this differentiation. The formula concerns regular periodic levels, not a separatrix of infinite period.