Mapas simpléticos são bem conhecidos por preservarem o volume do espaço de fase em dinâmica Hamiltoniana e são particularmente apropriados para problemas que requerem longos tempos de integração. Nesta tese nós desenvolvemos abordagens baseadas em mapas simpléticos para o acoplamento de multi sub-sistemas/domínios astrofísicos/códigos de simulação, para integração eficiente de sistemas de N-corpos auto-gravitantes com grandes variações nas escalas de tempo características. Nós estabelecemos uma família de 48 novos mapas simpléticos baseados numa separação Hamiltoniana recursiva, que permite que o acoplamento ocorra de uma maneira hierárquica, contemplando assim todas as escalas de tempo das interações envolvidas. Nossa formulação é geral o suficiente para permitir que tal método seja utilizado como receita para combinar diferentes fenômenos físicos, que podem ser modelados independentemente por códigos especializados. Nós introduzimos também uma separação Hamiltoniana baseada em Hamiltonianos de Kepler, para resolver o problema gravitacional geral de N-corpos como uma composição de N2 problemas de 2-corpos. O método resultante é exato para cada problema de 2-corpos individual e produz resultados rápidos e precisos para sistemas de N-corpos quase- Keplerianos, como sistemas planetários ou um aglomerado de estrelas que orbita um buraco-negro supermassivo. O método é também apropriado para integração de sistemas de N-corpos com hierarquias intrínsecas, como um aglomerados de estrelas com binárias compactas. Nós apresentamos a implementação dos algoritmos mencionados e descrevemos o nosso código tupan, que está publicamente disponível na seguinte url: https://github.com/ggf84/tupan. / Symplectic maps are well know for preserving the phase space volume in Hamiltonian dynamics and are particularly suited for problems that require long integration times. In this thesis we develop approaches based on symplectic maps for the coupling of multi sub-systems/astrophysics domains/simulation codes for efficient integration of self-gravitating N-body systems with large variation in characteristic time-scales. We establish a family of 48 new symplectic maps based on a recursive Hamiltonian splitting, which allow the coupling to occur in a hierarchical manner, thus contemplating all time-scales of the involved interactions. Our formulation is general enough to allow that such method be used as a recipe to combine different physical phenomena which can be modeled independently by specialized simulation codes. We also introduce a Keplerian-based Hamiltonian splitting for solving the general gravitational Nbody problem as a composition of N2 2-body problems. The resulting method is precise for each individual 2-body solution and produces quick and accurate results for near-Keplerian N-body systems, like planetary systems or a cluster of stars that orbit a supermassive black-hole. The method is also suitable for integration of N-body systems with intrinsic hierarchies, like a star cluster with compact binaries. We present the implementation of the mentioned algorithms and describe our code tupan, which is publicly available on the following url: https://github.com/ggf84/tupan.

Il est bien établit théoriquement et observationnellement que les amas de galaxies ne sont pas des objets sphèriques, et qu'ils sont beaucoup mieux décrits par la géométrie triaxiale. Malgré cela, les travaux sur la forme tri-dimensionnelle des amas de galaxies sont encore trés rares. L'objet de cette thèse est de contribuer à cette problématique naissante. L'originalité de ce travail est d'aborder ce sujet théoriquement et observationnellement. J'ai mesuré la forme d'amas de galaxies simulés, proposant des prédictions sur la forme des haloes de matière noire. J'ai ensuite développé un algorithme qui se propose de combiner des données en lentilles gravitationnelles et en rayons X afin de contraindre un modèle de haloe triaxial. L'algorithme est testé sur des données simulées. Finalement, je présente l'analyse en rayons X de Abell 1703, qui, combinée avec l'analyse en lentilles gravitationnelles, permettra de déterminer la forme de Abell 1703. / It is well established both theoretically and observationally that galaxy clusters are not spherical objects and that they are much better approximated as triaxial objects. This thesis focusses on the three dimencional shape of galaxy clusters. The originality of my approach is to tackle the problem both theoretically and observationally. First, I have measured the shape of dark matter haloes in the Millenium XXL and Sbarbine simulations, providing predictions for dark matter halo shape over 5 order in magnitude in mass. Then, I have developed an algorithm aimed at fitting simultaneously lensing and X-ray data in order to constrain a triaxial mass distribution. The algorithm is tested and characterized on mock data sets. It is found to be able to recover the input parameters. Finally, I present the X-ray analysis of galaxy cluster Abell 1703, which will be combined with the existing lensing analysis in order to investigate its shape.

MOND is an observational rule for predicting the acceleration of stars and galaxies from the distribution of the visible matter. It possibly stems from a new law of physics. I list the theoretical aspects of MOND, its achievements and problems. MOND has been tested mainly in disc galaxies so far. Its tests in elliptical galaxies are rare because the MOND effects are small for them in the parts observable by the conventional methods. In the thesis, I explain the methods and ideas I developed for testing MOND in the ellipticals using stellar shells. Moreover, the shells enable us to test MOND for stars in radial orbits for the first time. The shells are results of galactic interactions. I discuss the shell formation mechanisms and summarize the findings from shell observations and simulations.