The First Law of Mechanics in General Relativity & Isochrone Orbits in Newtonian Gravity ; Première Loi de la Mécanique en Relativité Générale & Orbites Isochrones en Gravitation Newtonienne

The first part of the thesis focuses on the relativistic, two-body problem in the context of general relativity. More precisely, we derive a variational identity known as the ``First Law of Mechanics" that relates physical parameters of a binary system of compact objects, such as its total energy and angular momentum, to the characteristics of the objects themselves, such as their masses and spins. Our derivation is based on the gravitational skeleton formalism for compact objects at quadrupolar order, combined with an extended version of a general variational identity established for spacetimes endowed with a helical isometry describing circular orbits. We also propose a review of the various multipolar skeleton models and the different types of First Laws that exist in the literature, and discuss applications and physical implications of our results in the context of gravitational wave astronomy. The second part of the thesis deals with a classical problem of potential theory, in Newtonian gravity. In particular, we continue the exploration of isochrone potentials, introduced in the fifties by Michel Hénon. These potentials are defined by the property that any test particle orbits within it with a radial period that is independent of its angular momentum. After a complete classification of the isochrone potentials using nothing but euclidean geometry, we explore the dynamics in these potentials by classifying their orbits, providing analytical solutions to the equation of motion. Using a Hamiltonian treatment, we also derive action-angle coordinates for the isochrone problem, providing new insight of several well-known result of classical celestial mechanics to all isochrone orbits, such as the Kepler equation, Bertrand's theorem and Kepler's laws of motion, and a generalization of these to all isochrone orbits. Finally, we compute the Birkhoff normal form of the corresponding Hamiltonian for a generic potential, and derive the fundamental theorem of isochrony from the inspection of the Birkhoff invariants of ...