(i) Consider a light rigid circular wire of radius $a$ and centre $O$. The wire lies in a vertical plane, which rotates about the vertical axis through $O$. At time $t$ the plane containing the wire makes an angle $\phi(t)$ with a fixed vertical plane. A bead of mass $m$ is threaded onto the wire. The bead slides without friction along the wire, and its location is denoted by $A$. The angle between the line $O A$ and the downward vertical is $\theta(t)$.

Show that the Lagrangian of the system is

$$\frac{m a^{2}}{2} \dot{\theta}^{2}+\frac{m a^{2}}{2} \dot{\phi}^{2} \sin ^{2} \theta+m g a \cos \theta .$$

Calculate two independent constants of the motion, and explain their physical significance.

(ii) A dynamical system has Hamiltonian $H(q, p, \lambda)$, where $\lambda$ is a parameter. Consider an ensemble of identical systems chosen so that the number density of systems, $f(q, p, t)$, in the phase space element $d q d p$ is either zero or one. Prove Liouville's Theorem, namely that the total area of phase space occupied by the ensemble is time-independent.

Now consider a single system undergoing periodic motion $q(t), p(t)$. Give a heuristic argument based on Liouville's Theorem to show that the area enclosed by the orbit,

$$I=\oint p d q$$

is approximately conserved as the parameter $\lambda$ is slowly varied (i.e. that $I$ is an adiabatic invariant).

Consider $H(q, p, \lambda)=\frac{1}{2} p^{2}+\lambda q^{2 n}$, with $n$ a positive integer. Show that as $\lambda$ is slowly varied the energy of the system, $E$, varies as

$$E \propto \lambda^{1 /(n+1)} .$$