Poisson Structure and Reduction by Stages of the Full Gravitational N-Body Problem

This paper deals with the full gravitational N-body problem. We express the Newton-Euler equations of motion into the Hamiltonian formalism employing a noncanonical Poisson structure. For this system, we identify the full-symmetry group given by the Galilean group. We carry out a partial reduction in two stages: translational and rotational symmetries. Moreover, we identify the Poisson structure at each stage of the reduction process. A characterization of relative equilibria is provided. It allows for classification and shows that the centers of mass of all bodies move in parallel planes. The case of N > 2 reveals a rather different scenario from N = 2. Precisely, the relative equilibria are classified as follows: Lagrangian equilibria, in which all the bodies are moving in the same plane; nonLagrangian equilibria, in which each body is in a different plane; and semi-Lagrangian equilibria, in which some of the bodies share a common plane but not all of them are in the same one. The main novelty in the equilibria classification is that the plane of motion of the center of mass does not need to be parallel to the plane determined by the total angular momentum. In our analysis, we specify sufficient conditions ensuring the planes of motion are parallel to the total angular momentum plane.