El problema de 2 cossos a l’esfera

Abstract

[en] This manuscript presents a study of the two-body problem in different contexts. First, the problem is considered in the euclidean space and its reduction to the the Kepler’s problem is analysed. Next, we study how the gravitational potential is obtained as a solution of the Poisson equation, by deriving the expression of the potencial for the two-body problem in $\mathbb{R}^{2}$ and $\mathbb{R}^{3}$ . The relevance of this approach is that the corresponding Laplace’s operator reflects the intrinsic geometry of the space under consideration, which allows it to systematically generalize gravitational potencial to other geometries. In this work, we focus on performing the same study on a generic space of positive and constant curvature, the unit sphere $\mathbb{S}^{2}$ , also giving the result of this problem to the unit hypersphere $\mathbb{S}^{3}$ . Finally, some simulations of the motion of two bodies on $\mathbb{S}^{2}$ are included.