In this thesis, central configurations, regularization of Simultaneous binary collision, linear stability of Kepler orbits, and index theory for symplectic path are studied. The history of their study is summarized in section 1. Section 2 deals with the following problem: given a collinear configuration of 4 bodies, under what conditions is it possible to choose positive masses which make it central. It is always possible to choose three positive masses such that the given three positions with the masses form a central configuration. However, for an arbitrary configuration of 4 bodies, it is not always possible to find positive masses forming a central configuration. An expression of four masses is established depending on the position x and the center of mass u, which gives a central configuration in the collinear four body problem. Specifically it is proved that there is a compact region in which no central configuration is possible for positive masses. Conversely, for any configuration in the complement of the compact region, it is always possible to choose positive masses to make the configuration central. The singularities of simultaneous binary collisions in collinear four-body problem is regularized by explicitly constructing new coordinates and time transformation in section 3. The motion in the new coordinates and time scale across simultaneous binary collision is at least C^2. Furthermore, the behavior of the motion closing, across and after the simultaneous binary collision, is also studied. Many different types of periodic solutions involving single binary collisions and simultaneous binary collisions are constructed. In section 4, the linear stability is studied for the Kepler orbits of the rhombus four-body problem. We show that, for given four proper masses, there exists a family of periodic solutions for which each body with the proper mass is at the vertex of a rhombus and travels along an elliptic Kepler orbit. Instead of studying the 8 degrees of freedom Hamilton system for planar four-body problem, we reduce this number by means of some symmetry to derive a two degrees of freedom system which then can be used to determine the linear instability of the periodic solutions. After making a clever change of coordinates, a two dimensional ordinary differential equation system is obtained, which governs the linear instability of the periodic solutions. The system is surprisingly simple and depends only on the length of the sides of the rhombus and the eccentricity e of the Kepler orbit. In section 5, index theory for symplectic paths introduced by Y.Long is applied to study the stability of a periodic solution x for a Hamiltonian system. We establish a necessary and sufficient condition for stability of the periodic solution x in two and four dimension.

Le théorème de l'indice d'Atiyah et Singer, démontré en 1963, est un résultat qui a permis de relier des thématiques mathématiques variées, allant des équations aux dérivées partielles a la topologie et la géométrie différentielle. Plus précisément, il fait le lien entre la dimension de l'espace des solutions d'une équation aux dérivées partielles elliptique et des invariants topologiques du type (co)homologie, et a des applications importantes, regroupant plusieurs théorèmes majeurs venant de divers domaines (géométrie algébrique, topologie différentielle, analyse fonctionnelle). D'un autre cote, les fonctions zêta associées à des opérateurs pseudo différentiels sur une variété riemannienne close contiennent dans leurs propriétés analytiques des informations intéressantes. On peut par exemple retrouver dans les résidus le théorème de Weyl sur l asymptotique du nombre de valeurs propres d'un laplacien, et en particulier le volume de la variété. En se plaçant dans le cadre de la géométrie différentielle non commutative développée par Connes, on peut pousser cette idée plus loin. Plus précisément, on peut obtenir, en combinant des techniques de renormalisation zêta avec la propriété d'excision en cohomologie cyclique, des théorèmes d'indice dans l'esprit de celui d'Atiyah-Singer. L'intérêt de ce point de vue réside dans sa généralisation possible à des situations géométriques plus délicates. La présente thèse établit des résultats dans cette direction / The index theorem of Atiyah and Singer, discovered in 1963, is a striking result which relates many different fields in mathematics going from the analysis of partial differential equations to differential topology and geometry. To be more precise, this theorem relates the dimension of the space of some elliptic partial differential equations and topological invariants coming from (co)homology theories, and has important applications. Many major results from different fields (algebraic topology, differential topology, functional analysis) may be seen as corollaries of this result, or obtained from techniques developed in the framework of index theory. On another side, zeta functions associated to pseudodifferential operators on a closed Riemannian manifold contain in their analytic properties many interesting informations. For instance, the Weyl theorem on the asymptotic number of eigenvalues of a Laplacian may be recovered within the residues of the zeta function. This gives in particular the volume of the manifold, which is a geometric data. Using the framework of noncommutative geometry developed by Connes, this idea may be pushed further, yielding index theorems in the spirit of the one of Atiyah Singer. The interest in this viewpoint is to be suitable for more delicate geometrical situations. The present thesis establishes results in this direction

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