Let $M=\mathbb{R}^{2 n}=\left\{(\mathbf{q}, \mathbf{p}) \mid \mathbf{q}, \mathbf{p} \in \mathbb{R}^{n}\right\}$ be equipped with its standard Poisson bracket.

(a) Given a Hamiltonian function $H=H(\mathbf{q}, \mathbf{p})$, write down Hamilton's equations for $(M, H)$. Define a first integral of the system and state what it means that the system is integrable.

(b) Show that if $n=1$ then every Hamiltonian system is integrable whenever

$$\left(\frac{\partial H}{\partial q}, \frac{\partial H}{\partial p}\right) \neq \mathbf{0}$$

Let $\tilde{M}=\mathbb{R}^{2 m}=\left\{(\tilde{\mathbf{q}}, \tilde{\mathbf{p}}) \mid \tilde{\mathbf{q}}, \tilde{\mathbf{p}} \in \mathbb{R}^{m}\right\}$ be another phase space, equipped with its standard Poisson bracket. Suppose that $\tilde{H}=\tilde{H}(\tilde{\mathbf{q}}, \tilde{\mathbf{p}})$ is a Hamiltonian function for $\tilde{M}$. Define $\mathbf{Q}=\left(q_{1}, \ldots, q_{n}, \tilde{q}_{1}, \ldots, \tilde{q}_{m}\right), \mathbf{P}=\left(p_{1}, \ldots, p_{n}, \tilde{p}_{1}, \ldots, \tilde{p}_{m}\right)$ and let the combined phase space $\mathcal{M}=\mathbb{R}^{2(n+m)}=\{(\mathbf{Q}, \mathbf{P})\}$ be equipped with the standard Poisson bracket.

(c) Show that if $(M, H)$ and $(\tilde{M}, \tilde{H})$ are both integrable, then so is $(\mathcal{M}, \mathcal{H})$, where the combined Hamiltonian is given by:

$$\mathcal{H}(\mathbf{Q}, \mathbf{P})=H(\mathbf{q}, \mathbf{p})+\tilde{H}(\tilde{\mathbf{q}}, \tilde{\mathbf{p}})$$

(d) Consider the $n$-dimensional simple harmonic oscillator with phase space $M$ and Hamiltonian $H$ given by:

$$H=\frac{1}{2} p_{1}^{2}+\ldots+\frac{1}{2} p_{n}^{2}+\frac{1}{2} \omega_{1}^{2} q_{1}^{2}+\ldots+\frac{1}{2} \omega_{n}^{2} q_{n}^{2}$$

where $\omega_{i}>0$. Using the results above, or otherwise, show that $(M, H)$ is integrable for $(\mathbf{q}, \mathbf{p}) \neq \mathbf{0}$.

(e) Is it true that every bounded orbit of an integrable system is necessarily periodic? You should justify your answer.