On the stability in two degrees of freedom Hamiltonian systems under resonances.

摘要

Stability in Hamiltonian systems is an essential piece in the study of a number of problems in various scientific branches, such as Classic Mechanics, Celestial Mechanics, Atomic Physics, etc. Furthermore, it is a subject of high mathematical interest. Nevertheless, the problem is difficult to tackle even for systems with two degrees of freedom because, despite being the simplest case and the most studied one, there are still some special situations unsolved.;In spite of the existence of many application problems and particular results, no general theorem was enunciated until 1999 when Cabral and Meyer established a criterion to solve stability in Hamiltonian systems with two degrees of freedom in the presence of resonances that included most of the classical results.;The main contribution of this thesis is a theorem that considers softer hypotheses compared to previous one; therefore, this theorem generalizes it and allows to solve the stability issue under more general conditions. Moreover, we give a geometric interpretation of this result and establish a geometric criterion. From this one it is possible to obtain new stability results for some cases that Cabral and Meyer's theorem cannot solve, the so-called degenerate cases.;The process to draw these conclusions is complex and requires the use of the Hamiltonian normal form. This implies applying normalization techniques carried out in the so-called Lissajous variables. In this sense, another contribution is a compact characterization of the normal form in terms of the invariants related to the Lissajous variables.;Apart from the study of stability, the characterization of the phase flow is important given that the existence of relative equilibria is associated to the presence of families of periodic orbits. This subject is also studied in this thesis for a resonance of order 4 by characterizing the different types of phase flows. These types are determined by relative equilibria and their parametric bifurcations, whose calculation is related to the number of the real roots of a polynomial in a close interval.