Clasificación topológica del flujo hamiltoniano de algunos problemas roto-traslatorios

Abstract

[SPA] Esta tesis está compuesta de seis capítulos principales y algunos apéndices diferentes. Los principales resultados obtenidos y expuestos en (Balsas, Jiménez y Vera (2007) [7], (2008) [8]; Balsas, Jiménez, Vera y Vigueras (2009) [9], (2016) [10]; Balsas, Jiménez, Guirao y Vera (2009) [11]) así como en esta memoria, versan sobre el estudio topológico de la dinámica de varios sistemas hamiltonianos derivados de problemas de carácter rototraslatorio. Consideraremos el problema general del movimiento de n cuerpos, en particular, el caso del satélite girostático en interacción newtoniana con un sólido rígido esférico, poniendo así de manifiesto la influencia de los movimientos internos que no modifican la distribución de masas sobre el movimiento de rotación de la parte rígida del satélite. En cuanto al estudio cualitativo, utilizaremos la formulación hamiltoniana, las variedades invariantes del movimiento, los teoremas de Liouville-Arnold y otras técnicas específicas.

[ENG] This thesis is made up of six main chapters and some different appendices. Let us see briefly their most important results: In the chapter called introduction, the methodology which is going to be used is described, as well as the principal objectives we want to achieve in this thesis. In chapter 2 we introduce some important concepts which are needed to understand this thesis. Thus, the Classical Mechanics is described; besides, it comes from the restricted Relativity Theory. First of all, the Lagrangian and Hamiltonian notation is introduced and so the Dynamical systems. Secondly, the Langrange and Hamiltonian equations are given. Thirdly, the canonical transformations and their characteristical functions are studied too. Later, the Hamilton-Jacobi equations are given and also the Liouville and Liouville-Arnold theorems. Then the Lie groups, the Lie algebras and the Lie actions are presented. As they are quite important for us, the simplectic and Poisson manifolds are studied too, as well as the momentum maps and some reduction theorems, which will allow us to reduce significantly the problems we will study during this work. Through chapter 3, the non-canonical Hamiltonian dynamics of a gyrostat in Newtonian interaction with n spherical rigid bodies is considered. By using the symmetries of the system we obtain two reductions. Then, working in the reduced problem, we calculate the expression for the potential and the equations of motion, a Casimir function of the system and the equations that determine the relative equilibria. Some global conditions for the existence of relative equilibria are given. Furthermore, we give the variational characterization of these equilibria and three invariant manifolds of the problem; being calculated the equations of motion in these manifolds, which are described by means of a canonical Hamiltonian system. Lastly, the equations of motion for a planar motion are described in one of these invariant manifolds. In chapter 4 we describe the Hamiltonian dynamics, in some invariant manifolds of the motion of a gyrostat in Newtonian interaction with a spherical rigid body. Considering a first integrable approximation of this roto-translatory problem, by means of Liouville-Arnold theorem and some specifics techniques, we obtained a complete topological classification of the phase ow associated to this system. The action-angle variables regions are obtained and the Delauny variables too. Thanks to the action-angle variables, we are able to calculate the modified Keplerian elements of this problem, useful to elaborate a perturbation theory. And last but not least, for one specific case the orbits are classified. We must take into account that the results of this work have a direct application to the study of two body roto-translatory problems, where the rotation of one of them influences strongly in the orbital motion of the system. In particular, we can apply these results to binary asteroids. In chapter 5 we have considered the Manev systems in a rotating reference frame. We describe the Hamiltonian dynamics, in the invariant manifolds Eh, Jk and Ihk by means of Liouville-Arnold theorem and some specific techniques. We also obtain a complete topological classification of the phase ow associated to this system. Finally, the actionangle variables are obtained. These variables allow us to calculate the modified Keplerian elements of this problem useful to elaborate a perturbation theory. In chapter 6 we consider the Kepler problem with a perturbation; this is an approximation to the Main Problem of the artificial satellite. We make an analytical, numerical and topological study of Hamiltonian dynamics for a simplified case where we only considered the first one and second dominant term of the gravitational potential. Using the Liouville-Arnold theorem and a particular analysis of the momentum map in its critical points we obtain a complete topological classification of the different invariant sets of the phase ow associated to this problem. Finally, during the appendices we can see the algorithms and calculations which have been done to obtain the critical points and the relative equilibria, to study the topology of Ihk and to classify the orbits of one of the main cases.