State the Arnold-Liouville theorem.

Consider an integrable system with six-dimensional phase space, and assume that $\nabla \wedge \mathbf{p}=0$ on any Liouville tori $p_{i}=p_{i}\left(q_{j}, c_{j}\right)$, where $\nabla=\left(\partial / \partial q_{1}, \partial / \partial q_{2}, \partial / \partial q_{3}\right)$.

(a) Define the action variables and use Stokes' theorem to show that the actions are independent of the choice of the cycles.

(b) Define the generating function, and show that the angle coordinates are periodic with period $2 \pi$.