Sur les mouvements homographiques de N corps associés à des masses de signe quelconque, le cas particulier où la somme des masses est nulle, et une application à la recherche de chorégraphies perverses.

Abstract : The thesis was prepared at the Institut de Mécanique Céleste et de Calcul des Ephémérides of the Observatory of Paris, from September 2001 to April 2005, under the supervision of Prof. Alain Chenciner and Dr. Alain Albouy. It deals with the N-body problem, which is based on the study of solutions of Newton's equations. These equations describe the motion of N punctual particles interacting through gravitation. The thesis is more precisely linked to homographic solutions (the ratios between the mutual distances are constant) with positive and negative masses. Rigid motions (the mutual distances are constant) are studied. This problem is more difficult than the problem with positive masses because it is no more possible to define a scalar product thanks to the masses.

We are interested in the case where the total mass vanishes. The center of inertia becomes a vector, which is invariant under translation. This makes Newton's equations "more integrable". Thus, under an assumption on the initial velocities, the collinear three-body problem becomes integrable. This property enables to compute central configurations (configurations which generate a homothetic collapse onto a center) for masses x, -x, y, -y.

A property of absolute equilibria with vanishing total mass is applied to the problem of choreographies. A choreography is a solution such that the bodies chase each other on the same curve with the same phase shifts. It is shown that, for a logarithmic potential, the masses of a choreography are necessarily equal.
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Martin Celli. Sur les mouvements homographiques de N corps associés à des masses de signe quelconque, le cas particulier où la somme des masses est nulle, et une application à la recherche de chorégraphies perverses.. Mathématiques [math]. Université Paris-Diderot - Paris VII, 2005. Français. ⟨tel-00011790⟩

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