(i) A point mass $m$ with position $q$ and momentum $p$ undergoes one-dimensional periodic motion. Define the action variable $I$ in terms of $q$ and $p$. Prove that an orbit of energy $E$ has period

$$T=2 \pi \frac{d I}{d E} .$$

(ii) Such a system has Hamiltonian

$$H(q, p)=\frac{p^{2}+q^{2}}{\mu^{2}-q^{2}}$$

where $\mu$ is a positive constant and $|q|<\mu$ during the motion. Sketch the orbits in phase space both for energies $E \gg 1$ and $E \ll 1$. Show that the action variable $I$ is given in terms of the energy $E$ by

$$I=\frac{\mu^{2}}{2} \frac{E}{\sqrt{E+1}} .$$

Hence show that for $E \gg 1$ the period of the orbit is $T \approx \frac{1}{2} \pi \mu^{3} / p_{0}$, where $p_{0}$ is the greatest value of the momentum during the orbit.