(a) A Hamiltonian system with $n$ degrees of freedom is described by the phase space coordinates $\left(q_{1}, q_{2}, \ldots, q_{n}\right)$ and momenta $\left(p_{1}, p_{2}, \ldots, p_{n}\right)$. Show that the phase-space volume element

$$d \tau=d q_{1} d q_{2} \ldots . d q_{n} d p_{1} d p_{2} \ldots . d p_{n}$$

is conserved under time evolution.

(b) The Hamiltonian, $H$, for the system in part (a) is independent of time. Show that if $F\left(q_{1}, \ldots, q_{n}, p_{1}, \ldots, p_{n}\right)$ is a constant of the motion, then the Poisson bracket $[F, H]$ vanishes. Evaluate $[F, H]$ when

$$F=\sum_{k=1}^{n} p_{k}$$

and

$$H=\sum_{k=1}^{n} p_{k}^{2}+V\left(q_{1}, q_{2}, \ldots, q_{n}\right)$$

where the potential $V$ depends on the $q_{k}(k=1,2, \ldots, n)$ only through quantities of the form $q_{i}-q_{j}$ for $i \neq j$.

(c) For a system with one degree of freedom, state what is meant by the transformation

$$(q, p) \rightarrow(Q(q, p), P(q, p))$$

being canonical. Show that the transformation is canonical if and only if the Poisson bracket $[Q, P]=1$.