Define an integrable system with $2 n$-dimensional phase space. Define angle-action variables.

Consider a two-dimensional phase space with the Hamiltonian

$$H=\frac{p^{2}}{2 m}+\frac{1}{2} q^{2 k}$$

where $k$ is a positive integer and the mass $m=m(t)$ changes slowly in time. Use the fact that the action is an adiabatic invariant to show that the energy varies in time as $m^{c}$, where $c$ is a constant which should be found.