Redução simplética de Hamiltonianos de Tonelli e aplicações ao problema de N corpos

Abstract

This work is devoted to study of Tonelli Hamiltonians with symmetries from the point of view of symplectic reduction. The goal is to apply techniques of this theory in the determination of minimizing invariant measures and weak KAM. At the same time we work with a generalization of the N body problem in the context of manifolds and obtained an existence theorem of weak KAM invariant by the diagonal action of the group of isometries of the manifold. As a consequence, we prove the validity of this theorem in the case of hyperbolic manifolds of constant sectional curvature with "Newtonian" potential usually found in the literature. Finally, we seek to provide examples to illustrate the results obtained. We devote the last chapter to the 2 body and restricted 3 body problems in the hyperbolic plane, which developed much of the symplectic reduction theory in the first case .