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Gravitational Dynamical Systems

Abstract : Dynamical systems have a centuries-long history with roots going back to the mathematical development for astronomy. In the modern formalism, the present thesis investigates dynamical properties of gravitation at different astrophysical or cosmological scales.In potential theory, isochrony often refers to harmonic oscillations of pendulums. In 1959, the mathematician and astronomer Michel Hénon introduced an extended definition of isochrony to characterize orbital oscillations of stars around the center of the system to which they belong. In that case, the period of oscillations can depend on the energy of the star. Today, Michel Hénon’s isochrone potential is mainly used for its integrable property in numerical simulations, but is not widely known. In this thesis, we revisit his geometrical characterization of isochrony and complete the family of isochrone potentials in physics. The classification of this family under different mathematical group actions highlights a particular relation between the isochrones. The actual Keplerian nature of isochrones is pointed out and stands at the heart of the new isochronerelativity, which are presented together.The consequences of this relativity in celestial mechanics — a generalization of Kepler’sThird law, Bohlin or Levi-Civita transformation, Bertrand’s theorem — are applied to analyze the result of a gravitational collapse. By considering dynamical orbital properties, an isochrone analysis is developed to possibly characterize a quasi-stationary state of isolated self-gravitating systems, such as dynamically young stellar clusters or galaxies.At a cosmological scale, the dynamics of the universe depends on its energy content. Its evolution can be expressed as an ecological dynamical system, namely a conservative generalized Lotka-Volterra model. In this framework of a spatially homogeneous and isotropic spacetime, named Jungle Universe, the dynamical impact of a non-gravitational interaction between the energy components is analyzed. As a result, effective dynamical behaviors could account for an accelerated expansion of the universe without dark energy.
Keywords : Dynamics Cosmology
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Alicia Simon-Petit. Gravitational Dynamical Systems. Dynamical Systems [math.DS]. Université Paris-Saclay, 2018. English. ⟨NNT : 2018SACLY021⟩. ⟨tel-01997477⟩



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