Solutions quasi-périodiques et solutions de quasi-collision du problème spatial des trois corps

摘要

This thesis generalizes to the spatial three-body problem in the lunar case some studies about several families of quasiperiodic motions in the planar circular restricted three-body problem and in the planar three-body problem. As discovered by Harrington, if we develop the perturbing function of the system averaged over the fast angles in the powers of the ratio of the semi major axes, then the truncation at the first non-trivial order is integrable. This is the quadrupolar system. In a classical article, Lidov and Ziglin studied the dynamics of this system. We start by proving the existence of some quasi-periodic solutions of the spatial three-body problem by applying KAM theorems to this system. We then prove the existence of a family of quasi-periodic almost-collision solutions: These are solutions along which two bodies become arbitrarily close to one another but never collide: the lower limit of their distance is zero but the upper limit is strictly positive. After a change of time, these solutions are quasi-periodic in a regularized system. Such solutions were first discovered in the planar circular restricted three-body problem by Chenciner and Llibre, and afterwards, in the planar three-body problem by Féjoz. We show the existence of a positive measure of such solutions in the spatial three-body problem, which confirms rigorously a prediction of Marchal. The proof goes through the application of an equivariant KAM theorem to a regularization of the problem, here the Kustaanheimo-Stiefel regularization, and, as in Féjoz's work, it requires understanding the relation between the regularization and averaging.