(i) (a) Write down Hamilton's equations for a dynamical system. Under what condition is the Hamiltonian a constant of the motion? What is the condition for one of the momenta to be a constant of the motion?

(b) Explain the notion of an adiabatic invariant. Give an expression, in terms of Hamiltonian variables, for one such invariant.

(ii) A mass $m$ is attached to one end of a straight spring with potential energy $\frac{1}{2} k r^{2}$, where $k$ is a constant and $r$ is the length. The other end is fixed at a point $O$. Neglecting gravity, consider a general motion of the mass in a plane containing $O$. Show that the Hamiltonian is given by

$$H=\frac{1}{2} \frac{p_{\theta}^{2}}{m r^{2}}+\frac{1}{2} \frac{p_{r}^{2}}{m}+\frac{1}{2} k r^{2},$$

where $\theta$ is the angle made by the spring relative to a fixed direction, and $p_{\theta}, p_{r}$ are the generalised momenta. Show that $p_{\theta}$ and the energy $E$ are constants of the motion, using Hamilton's equations.

If the mass moves in a non-circular orbit, and the spring constant $k$ is slowly varied, the orbit gradually changes. Write down the appropriate adiabatic invariant $J\left(E, p_{\theta}, k, m\right)$. Show that $J$ is proportional to

$$\sqrt{m k}\left(r_{+}-r_{-}\right)^{2},$$

where

$$r_{\pm}^{2}=\frac{E}{k} \pm \sqrt{\left(\frac{E}{k}\right)^{2}-\frac{p_{\theta}^{2}}{m k}}$$

Consider an orbit for which $p_{\theta}$ is zero. Show that, as $k$ is slowly varied, the energy $E \propto k^{\alpha}$, for a constant $\alpha$ which should be found.

[You may assume without proof that

$$\left.\int_{r_{-}}^{r_{+}} d r \sqrt{\left(1-\frac{r^{2}}{r_{+}^{2}}\right)\left(1-\frac{r_{-}^{2}}{r^{2}}\right)}=\frac{\pi}{4 r_{+}}\left(r_{+}-r_{-}\right)^{2} \cdot\right]$$