Applications and extensions of the Liouville theorem on constants of motion

Abstract

We give an elementary proof of the Liouville theorem, which allows us to obtain $n$ constants of motion in addition to $n$ given constants of motion in involution, for a mechanical system with $n$ degrees of freedom, and we give some examples of its application. For a given set of $n$ constants of motion that are not in involution with respect to the standard symplectic structure, there exist symplectic structures with respect to which these constants will be in involution and the Liouville theorem can then be applied. Using the fact that any second-order ordinary differential equation (not necessarily related to a mechanical problem) can be expressed in the form of the Hamilton equations, the knowledge of a first integral of the equation allows us to find its general solution.