Moon tides: generalizing classical gravity to an oscillating sphere. A hodge decomposition point of view

Abstract

In Part I of this thesis, we focus on how to generalize the notion of a gravitational field not only for a given fixed metric but also in the case of an oscillating one. The problem was raised as a toy model for Moon tides. We follow a generalization of Hodge decomposition in a relativistic background which enables a deduction of the dynamics of point masses, using in part recently derived results for point vortices on closed differentiable surfaces M endowed with a metric g. In Part II of the thesis, the connection between braids and Hamiltonian systems is explored based on the works of Boyland, Aref and Stremler. A connection between the braids formed by a point particle system and its Liouville integrability is found by the construction of an integrability notion coming from Braid Theory. A first theorem relating this braid integrability to the already know Liouville one is given.