Study and simulation of the planar and circular restricted three-body problem

Abstract

We study the Planar and Circular Restricted Three-Body Problem, as an idealization of the Three-Body Problem. We follow a dynamical systems approach. Once the main characteristics of the problem have been described, we try to explore a little bit the chaos of the system. Without pretending to be systematic, we focus on final evolutions. In particular, the parabolic final evolutions are used to show evidence of chaos, as they correspond to the invariant manifolds of the periodic orbit at infinity, which intersect non-tangentially in a certain Poincaré section, giving rise to transversal homoclinic points of the associated Poincaré map. Furthermore, we try to outline some differences between the integrable Kepler Problem and the non-integrable Planar and Circular Restricted Three-Body Problem, which can be explained by the splitting of the two previously mentioned invariant manifolds. The use of numerical methods has been fundamental for the realization of this work.