Completely integrable systems on hamiltonian mechanics

Abstract

[en] During the last two centuries, the study of mechanics has enjoyed a remarkable evolution, in parallel with one of its main mathematical tools: symplectic geometry. In this text, some of the most important notions have been gathered for the understanding of the Liouville-Arnold Theorem on completely integrable systems. The final goal of this project is to give a new approach to this fundamental result; thus the theory presented is appropriately nourished with humble examples to be analyzed. During the work previous to the final composition, several sources about both the main and many neighboring topics had to be studied. The tools given here can bring interested readers to the further study of gigantic problems such as the restricted three body problem, perturbation theory, and infinite dimensional integrable systems.