Abstracts Category : Other

Stability in the plane planetary three-body problem : Stabilit dans le problme trois corps plantaire plan

by Thibaut Castan

Arnold a dmontr l'existence de solutions quasipriodiques dans le problme plantaire trois corps plan, sous rserve que la masse de deux des corps, les plantes, soit petite par rapport celle du troisime, le Soleil. Cette condition de petitesse dpend de faon cache de la largeur d'analyticit de l'hamiltonien du problme, dans des coordonnes transcendantes. Hnon ex- plicita un rapport de masses minimal ncessaire l'application du thorme de Arnold. L'objectif de cette thse sera de donner une condition suffisante sur les rapports de masses. Une premire partie de mon travail consiste estimer cette largeur d'analyticit, ce qui passe par l'tude prcise de l'quation de Kepler dans le complexe, ainsi que celle des singularits complexes de la fonction perturbatrice. Une deuxime partie consiste mettre l'hamiltonien sous forme normale, dans l'optique d'une application du thorme KAM (du nom de Kolmogorov-Arnold-Moser). Il est ncessaire d'tudier le hamiltonien sculaire pour le mettre sous une forme normale adquate. On peut alors quantifier la non-dgnrescence de l'hamiltonien sculaire, ainsi qu'estimer la perturbation. Enfin, il faut dmontrer une version quantitative fine du thorme KAM, inspire de Pschel, avec des constantes explicites. A l'issue de ce travail, il est montr que le thorme KAM peut tre appliqu pour des rapports de masses entre plantes et toile de l'ordre de 10^(-85). Arnold showed the existence of quasi-periodic solutions in the plane planetary three-body prob- lem, provided that the mass of two of the bodies, the planets, is small compared to the mass of the third one, the Sun. This smallness condition depends in a sensitive way on the analyticity widths of the Hamiltonian of the three-body problem, expressed with the help of some tran- scendental coordinates. Hnon gave a minimal ratio of masses necessary to the application of Arnolds theorem. The main objective of this thesis is to determine a sufficient condition on this ratio. A first part of this work consists in estimating these analyticity widths, which requires a precise study of the complex Kepler equation, as well as the complex singularities of the disturb- ing function. A second part consists in reworking the Hamiltonian to put it under normal form, in order to apply the KAM theorem (KAM standing for Kolmogorov-Arnold-Moser). In this aim, it is essential to work with the secular Hamiltonian to put it under a suitable normal form. We can then quantify the non-degeneracy of the secular Hamiltonian, as well as estimate the perturbation. Finally, it is necessary to derive a quantitative version of the KAM theorem, in order to identify the hypotheses necessary for its application to the plane three-body problem. After this work, it is shown that the KAM theorem can be applied for a ratio of masses that is close to 10^(85) between the planets and the star.Advisors/Committee Members: Fjoz, Jacques (thesis director).