 Solution

Liouville integrability on a $2n$-dimensional symplectic manifold means that there are $n$ functionally independent first integrals, including $H$, whose pairwise Poisson brackets vanish. Near a compact connected regular common level set, the Arnold-Liouville theorem gives action-angle variables $(J_i,{\theta }_i)$ with $J_i=(2\pi {)}^{-1}\oint pdq$, ${\theta }_i$ periodic modulo $2\pi$, $H=H(J)$, ${\dot{J}}_i=0$ and ${\dot{\theta }}_i=\partial H/\partial J_i$.

For a frozen mass, the turning point is $q_{\ast }=(2E{)}^{1/(2k)}$. The action variable isAdiabatic invariance of the action holds to leading order when the mass varies slowly compared with an orbital period, away from degenerating orbits. Keeping $J$ fixed givesFor $k=1$ this agrees with $E=\omega J$ and $\omega \propto m^{-1/2}$.