Continuação de órbitas periódicas em perturbações simétricas do problema de Kepler no plano

Abstract

This dissertation aims to study symmetric periodic orbits in symmetric perturbations of the Kepler problem in the plane. The study begins with an introduction to Hamiltonian systems, symplectic coordinates, and the Kepler problem. It then explores the use of re ections with respect to coordinate axes (symmetries) to nd periodic solutions in perturbed planar Hamiltonian systems. The main focus is to establish su cient conditions for the existence of periodic solutions in symmetric perturbations of the Kepler problem, including the Kepler problem in rotating coordinates. Utilizing Poincaré's method of analytic continuation and suitable coordinates, the dissertation establishes su cient conditions for obtaining symmetric periodic solutions close to elliptical or circular solutions of the unperturbed Kepler problem. Furthermore, the dissertation presents two practical applications for the obtained results. The rst application involves a perturbed Kepler problem describing the interaction between atoms, considering a Coulomb potential with negative charge and a generalized anisotropic Buckingham potential, where anisotropy is treated as a small perturbing parameter. The second application addresses the problem of a particle attracted by the gravitational force of a non-homogeneous, uniformly rotating massive circular disk. This problem is described as a perturbation of the Kepler problem in rotating coordinates, where the radius is treated as a perturbing parameter. The dissertation contributes to the understanding of symmetric periodic orbits in perturbed gravitational systems and has signi cant applications in areas such as celestial dynamics.