O método da média e alguns resultados sobre órbitas periódicas

Abstract

In this dissertation, we study the averaging method, which stands out as one of the main tools in the theory of perturbations applied to ordinary differential equations, with special emphasis on Hamiltonian systems. The averaging method is based on the consideration of a non-autonomous first-order differential equation that is periodic in time, as wellas on the development of a change of coordinates that makes it possible to study the original equation by means of an autonomous differential equation, known as the averaging equation. This procedure provides a satisfactory approximation to the original equationover a limited time interval. In addition, the averaging method is used to identify periodic solutions in perturbed systems, which consist of an unperturbed part, which presents periodic solutions, and a perturbed part, which depends on a small parameter. We will show that the approach to finding periodic solutions in these systems refers to the search for non-degenerate equilibria of a given function. In particular, we present the results obtai ned with the averaging method regarding the existence of periodic solutions for perturbed Hamiltonian systems with two degrees of freedom. As an application of these results, we analyze the existence of periodic solutions in two problems that represent perturbations in the plane of the Kepler problem: the hydrogen atom problem under the influence of external forces and the anisotropic Kepler problem of the Manev type.