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

A second Hamiltonian system has the Hamiltonian $G=G(\mathbf{p}, \mathbf{q})$. Neither $H$ nor $G$ contains the time explicitly. Show that the condition for $H(\mathbf{p}, \mathbf{q})$ to be invariant under the evolution of the coordinates and momenta generated by the Hamiltonian $G(\mathbf{p}, \mathbf{q})$ is that the Poisson bracket $[H, G]$ vanishes. Deduce that $G$ is a constant of the motion for evolution under $H$.

Show that, when $G=\alpha \sum_{k=1}^{n} p_{k}$, where $\alpha$ is constant, the motion it generates is a translation of each $q_{k}$ by an amount $\alpha t$, while the corresponding $p_{k}$ remains fixed. What do you infer is conserved when $H$ is invariant under this transformation?

(b) When $n=3$ and $H$ is a function of $p_{1}^{2}+p_{2}^{2}+p_{3}^{2}$ and $q_{1}^{2}+q_{2}^{2}+q_{3}^{2}$ only, find $\left[H, L_{i}\right]$ when

$$L_{i}=\epsilon_{i j k} q_{j} p_{k}$$