Abstract

The interest of the present project is the study of Hamiltonian dynamical systems combining different techniques, with particular emphasis on design and implementation of appropriate normal forms to each problem, especially in $N$-body problems. We aim to develop theoretical and applied aspects related to the normal forms of Hamiltonian functions in order to understand important dynamics aspects, since they are essential tools to address different problems in a diversity of fields. For example, in aerospace engineering, to study the attitude dynamics and control of space vehicles, in Celestial Mechanics, to analyze the capture of asteroids or the existence of invariants tori and periodic solutions and almost periodic solutions of the $N$-body problems, or in Chemistry and Physics, to understand the reaction dynamics molecular ionization processes or the control in the orientation of cold molecules by laser pulses.\\ A common aspect to all these problems is to rebuild the dynamics from the original system extracted from the problem in its normal form. We will focus mainly in qualitative aspects: stability of solutions, analysis of parametric bifurcations and approximations of special invariant structures by means of the Implicit Function theorems and KAM's theorems. We are also interested in the quantitative part: validity of the approaches within the phase space and time or the globalization of the approximated local invariants.\\ The purpose of the project is the study of certain Hamiltonian dynamical systems by means of qualitative techniques, in general those related to the construction of normal forms appropriated to different systems in analysis. On one side, we will try to find general properties, valid for a special class of Hamiltonian systems and, secondly, we will try to find concrete properties of a specific Hamiltonian system corresponding to a problem of interest in Celestial Mechanics, or, in the not too distant future, the implementation in some chemistry and physics problems related to reaction dynamics in certain molecules.\\ Among the issues to be discussed we have the construction of generalized normal forms, time dependent or not, and their application to obtain invariant manifolds. The stability of these manifolds and the possible bifurcations of them will be analyzed by various techniques such as invariant theory.\\ In order to achieve the objectives of this project both a theoretical study as an important work of symbolic programming are needed. By means of the theoretical study we will analyze the qualitative aspects of the problems that we want to study and we will propose algorithms and strategies to address them. The symbolic procedures are essential to carry out the qualitative aspects of analysis because the normal form usually contains many terms.\\ Finally, it is important to note that to achieve the objectives of the present project, the responsible researcher will have the collaboration of the Dr. Jesus Palacián of Universidad Pública de Navarra, España and his research group. I will also have the collaboration of Dr. Fabio dos Santos (my PhD student) from Universidade Federal de Sergipe, Brazil and José Edmundo Mansilla from Universidad de los Lagos, Chile. Moreover, together Dr. Jean Pierre Marco from Universit'e Paris 6, France and his research group, recently we have started a collaboration as consequence of the project Math-Amsud 2010-2011 between Chile-Brazil-France.