I'm reading Arnold's book Mathematical Methods in Classical Mechanics and got stuck on the proof of Liouville's theorem on integrable systems. The proof finishes with Problem 11: Show that the motion on the invariant torus $M_f$ is conditionally periodic.
Unfortunately, I have no idea where to start. Can anyone help me out?
Edit: I think the reasoning is somewhat as follows: The action of the flow of $H$ is linear on $\mathbb{R}^n$ in the $t$-coordinates (but I don't understand yet why). Then we divide out the lattice $\Gamma$ to get a linear action on the torus $T^n \cong \mathbb{R}^n/\Gamma$ in the $\varphi$-coordinates. Now there should be a result that any linear action on a torus is conditionally periodic.