Celestial mechanics and the theory of dynamic systems: a revision of the circular-restricted-three-body problem

In this work we present the approximation of the circular restricted three-body problem that interact by gravitation force, the so-called circular restricted three-body problem (CR3BP). Such model is reviewed starting from the formalism and concepts from the theory of dynamic systems, focusing on resonance phenomena, state-space representation, Poincare's maps and the KAM theorem. We have discussed mathematic stability aspects of the problem, its equilibria points – Lagrangean points – and how the integrability condition is applied to do so. Results from computational simulations are analyzed and some are presented, e.g., simulations that explore regions of chaotic behavior on the Poincare's section, directly correlated to the vacancies on asteroids belts and planet's rings from our solar system, as some possible orbit maneuvers that expends less energy.

Keywords: three-body problem; dynamic systems theory; simulations; stability