Let $M=\mathbb{R}^{2 n}=\left\{(\mathbf{q}, \mathbf{p}) \mid \mathbf{q}, \mathbf{p} \in \mathbb{R}^{n}\right\}$ be equipped with the standard symplectic form so that the Poisson bracket is given by:

$$\{f, g\}=\frac{\partial f}{\partial q_{j}} \frac{\partial g}{\partial p_{j}}-\frac{\partial f}{\partial p_{j}} \frac{\partial g}{\partial q_{j}}$$

for $f, g$ real-valued functions on $M$. Let $H=H(\mathbf{q}, \mathbf{p})$ be a Hamiltonian function.

(a) 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) State the Arnol'd-Liouville theorem.

(c) Define complex coordinates $z_{j}$ by $z_{j}=q_{j}+i p_{j}$, and show that if $f, g$ are realvalued functions on $M$ then:

$$\{f, g\}=-2 i \frac{\partial f}{\partial z_{j}} \frac{\partial g}{\partial \overline{z_{j}}}+2 i \frac{\partial g}{\partial z_{j}} \frac{\partial f}{\partial \bar{z}_{j}}$$

(d) For an $n \times n$ anti-Hermitian matrix $A$ with components $A_{j k}$, let $I_{A}:=\frac{1}{2 i} \overline{z_{j}} A_{j k} z_{k}$. Show that:

$$\left\{I_{A}, I_{B}\right\}=-I_{[A, B]},$$

where $[A, B]=A B-B A$ is the usual matrix commutator.

(e) Consider the Hamiltonian:

$$H=\frac{1}{2} \overline{z_{j}} z_{j}$$

Show that $(M, H)$ is integrable and describe the invariant tori.

[In this question $j, k=1, \ldots, n$, and the summation convention is understood for these indices.]