On the Liouville-Arnold theorem

Solution 1:

Let $M= \{ (p,q) \in \mathbb{R}^{n} \times \mathbb{R}^n \}$ ($p$ denotes the position variables and $q$ the corresponding momenta variables). Assume that $f_1, \cdots f_n$ are $n$ commuting first integrals then you get that $M_{z_1, \cdots, z_n} := \{ (p,q) \in M \; : \; f_1(p,q)=z_1, \cdots , f_n(p,q)=z_n \} $ with $z_i \in \mathbb{R}$ is a Lagrangian submanifold.

Observe that if the compactness and connectedness condition is satisfied then there exist action angle variables which means that the motion lies on an $n$-dimensional torus (which is a compact object).

The compactness condition is equivalent to that a position variable, $p_k$, or a momentum variable, $q_j$, cannot become unbounded for fixed $z_i$. Consequently, if the compactness condition is not satisfied there is no way you can expect to find action angle variables since action angle variable imply that the motion lies on a torus which is a compact object.