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Ce travail s’inscrit dans l’étude des symétries d’espaces de dimension infinie. Il répond à des questions algébriques en suivant des méthodes analytiques. Plus précisément, nous étudions certaines représentations du groupe symplectique complexe dans des espaces fonctionnels. Elles sont caractérisées par leurs décompositions isotypiques relativement à un sous-groupe compact maximal. Ce travail décrit ces décompositions dans deux modèles : un modèle classique (dit compact) et un autre plus récent (dit non-standard). Nous montrons que cela établit un lien entre deux familles de fonctions spéciales (fonctions hypergéométriques et fonctions de Bessel) ; ces familles sont associées à des équations différentielles ordinaires d’ordre 2, fuchsiennes dans un cas et non fuchsiennes dans l’autre. Nous mettons aussi en évidence, dans le modèle non-standard, un lien avec certaines équations d'Emden-Fowler, ainsi qu’un opérateur différentiel simple qui agit sur les décompositions isotypiques. / The general setting of this work is the study of symmetry groups of infinite-dimensional spaces. We answer algebraic questions, using analytical methods. To be more specific, we study certain representations of the complex symplectic group in functional spaces. These representations are characterised by their isotypic decompositions with respect to a maximal compact subgroup. In this work, we describe these decompositions in two different models: a classical model (compact picture) and a more recent one (non-standard picture). We show that this establishes a connection between two families of special functions (hypergeometric functions and Bessel functions); these families correspond to second order differential equations, which are Fuchsian in one case and non-Fuchsian in the other. We also establish a link with certain Emden-Fowler equations and exhibit a simple differential operator that acts on the isotypic decompositions.

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.

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Cette thèse est divisée en trois parties. Dans la première partie, nous commençons par décrire des résultats très connus en théorie du contrôle géométrique tels que le théorème de Chow-Rashevsky, la condition de rang de Kalman, l'application Entrée-Sortie et le test linéaire. De plus, nous définissons et nous étudions brièvement la contrôlabilité locale au voisinage d'un contrôle de référence au premier et au second ordre. Dans la deuxième partie, nous donnons une preuve élémentaire du lemme de Franks linéaire pour les flots géodésiques qui utilise des techniques basiques de théorie du contrôle géométrique. Dans la dernière partie, étant donnée une variété Riemanienne compacte, nous prouvons un lemme de Franks uniforme au second ordre pour les flots géodésiques et on applique le résultat à la théorie de la persistance. Dans cette partie, nous introduisons avec plus de détails les notions de contrôlabilité locale au premier et au second ordre. En effet, nous donnons un résultat de contrôlabilité au second ordre dont la preuve est longue et technique. / This thesis is devided into three parts. In the first part we begin by describing some well known results in geometric control theory such as the Chow Rashevsky Theorem, the Kalman rank condition, the End-Point Mapping and the linear test. Moreover, we define and study briefly local controllability around a reference control at first and second order. In the second part we provide an elementary proof of the Franks lemma for geodesic flows using basic tools of geometric control theory. In the last part, given a compact Riemannian manifold, we prove a uniform Franks' lemma at second order for geodesic flows and apply the result in persistence theory. In this part we introduce with more details notions of local controllability at first and second order. In fact, we provide a second order controllability result whose proof is long and technical.