Talk:Liouville–Arnold theorem

"and the energy level set is compact" in the introduction

That assumption is written in a confusing way: Arnol'd-Liouville theorem assumes that the level set of the whole momentum map is compact. If only the energy level set is compact, it can still be true that the Hamiltonian trajectory evolves linearly with time, although there are no invariant tori anymore. We therefore do not lie inside the classical framework of Arnol'd-Liouville theorem anymore.

As a conclusion, instead of:

"In dynamical systems theory, the Liouville–Arnold theorem states that if, in a Hamiltonian dynamical system with n degrees of freedom, there are also n independent, Poisson commuting first integrals of motion, and the energy level set is compact, then there exists a canonical transformation to action-angle coordinates in which the transformed Hamiltonian is dependent only upon the action coordinates and the angle coordinates evolve linearly in time."

I suggest:

"In dynamical systems theory, the Liouville–Arnold theorem states that if, in a Hamiltonian dynamical system with n degrees of freedom, there are also n independent, Poisson commuting first integrals of motion, and the level set of all first integrals are compact, then there exists a canonical transformation to action-angle coordinates in which the transformed Hamiltonian is dependent only upon the action coordinates and the angle coordinates evolve linearly in time." 134.76.82.102 (talk) 08:58, 3 September 2024 (UTC)