Let $H$ be a smooth function on a $2 n$-dimensional phase space with local coordinates $\left(p_{j}, q_{j}\right)$. Write down the Hamilton equations with the Hamiltonian given by $H$ and state the Arnold-Liouville theorem.

By establishing the existence of sufficiently many first integrals demonstrate that the system of $n$ coupled harmonic oscillators with the Hamiltonian

$$H=\frac{1}{2} \sum_{k=1}^{n}\left(p_{k}^{2}+\omega_{k}^{2} q_{k}^{2}\right)$$

where $\omega_{1}, \ldots, \omega_{n}$ are constants, is completely integrable. Find the action variables for this system.