Choreographies on the restricted three-body problem

Abstract The three-body problem deals with point objects interacting mutually through Newton's gravitational force. For more than three centuries, the study of this type of system has led to the development and improvement of several mathematical techniques, both analytical and numerical, for the understanding of problems involving dynamical systems. This paper discusses some of these results applied to the restricted three-body problem formulated from Newton's second law and the law of universal gravitation. In particular, the behavior and stability of two important types of periodic solutions of this problem were studied: L. Euler's straight-line alignment and C. Moore's figure-eight choreography. Mathematica software was used to solve the dynamical system and to generate the images of movement of the bodies, as well as to calculate the set of Lyapunov exponents associated with each solution. Despite the inherently chaotic character of the three-body problem observed in the Lyapunov exponents, the figure-eight solution is linearly stable, as discussed in C. Simó's works.