Proof of Arnold-Liouville's Thm: movement in angular coordinates conditionally periodic 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.
 A: By flowing along the Hamiltonian vector fields corresponding to the Poisson commuting functions $f:= (f_1 = H, f_2,\ldots, f_n)$ and restricting attention to a level set $M_f$, we obtain an action of $\Phi:\mathbb{R}^n\times M_f \to M_f$. Let's fix some point $x_0 \in M_f$ and consider the orbit map $\Phi_{x_0}:\mathbb{R}^n \to M_f$ defined by $\Phi_{x_0}(v) = \Phi(v,x_0)$. The stabilizer of this map is a rank $n$ subgroup $\Gamma$ of $\mathbb{R}^n$. 
Let $e_1,\ldots, e_n$ be the standard basis of $\mathbb{R}^n$. Note that the flow $\psi^H_t$ corresponding to the Hamiltonian $H$ is given by $\psi^H_t(x)\mapsto \Phi(te_1,x)$. 
Now let $v_1,\ldots, v_n$ be a basis for the free abelian group $\Gamma$, and let $A:\mathbb{R}^n\to \mathbb{R}^n$ be the invertible matrix defined by $A e_i = v_i$. The map $\Phi_{x_0}\circ A$ has stabilizer $\mathbb{Z}^n\leq \mathbb{R}^n$ and descends through the quotient to a diffeomorphism $\mathbb{T}^n = \mathbb{R}^n/\mathbb{Z}^n \to M_f$ which defines the circle-valued coordinates $\varphi:=(\varphi^1,\ldots,\varphi^n)$. Since the $\Phi_{x_0}$ is surjective, for any $x \in M_f$ there exists $w \in \mathbb{R}^n$ such that $\Phi_{x_0}(w) =x$. Using this fact and the previous paragraph, we obtain
$$\psi_t^H(x) = \Phi(te_1,x) = \Phi(te_1,\Phi(w,x_0)) = \Phi_{x_0}(te_1 + w) =\Phi_{x}(AA^{-1}(te_1+w)) = [\Phi_{x}\circ A]( tA^{-1}e_1 + A^{-1}w) = [\Phi_{x}\circ A]( \omega t + A^{-1}w),$$
where the vector $\omega:=A^{-1}e_1$. Therefore, $$\dot{\varphi} = \omega$$
as desired. 
