(a) A Hamiltonian system with $n$ degrees of freedom has Hamiltonian $H=H(\mathbf{p}, \mathbf{q})$, where the coordinates $\mathbf{q}=\left(q_{1}, q_{2}, q_{3}, \ldots, q_{n}\right)$ and the momenta $\mathbf{p}=\left(p_{1}, p_{2}, p_{3}, \ldots, p_{n}\right)$ respectively.

Show from Hamilton's equations that when $H$ does not depend on time explicitly, for any function $F=F(\mathbf{p}, \mathbf{q})$,

$$\frac{d F}{d t}=[F, H],$$

where $[F, H]$ denotes the Poisson bracket.

For a system of $N$ interacting vortices

$$H(\mathbf{p}, \mathbf{q})=-\frac{\kappa}{4} \sum_{\substack{i=1 \\ N}}^{N} \sum_{\substack{j=1 \\ j \neq i}}^{N} \ln \left[\left(p_{i}-p_{j}\right)^{2}+\left(q_{i}-q_{j}\right)^{2}\right]$$

where $\kappa$ is a constant. Show that the quantity defined by

$$F=\sum_{j=1}^{N}\left(q_{j}^{2}+p_{j}^{2}\right)$$

is a constant of the motion.

(b) The action for a Hamiltonian system with one degree of freedom with $H=H(p, q)$ for which the motion is periodic is

$$I=\frac{1}{2 \pi} \oint p(H, q) d q .$$

Show without assuming any specific form for $H$ that the period of the motion $T$ is given by

$$\frac{2 \pi}{T}=\frac{d H}{d I}$$

Suppose now that the system has a parameter that is allowed to vary slowly with time. Explain briefly what is meant by the statement that the action is an adiabatic invariant. Suppose that when this parameter is fixed, $H=0$ when $I=0$. Deduce that, if $T$ decreases on an orbit with any $I$ when the parameter is slowly varied, then $H$ increases.