The Restricted 3-Body Problem on S1 : regularization and a particular solution

Abstract

We study a special case of the two body problem when the particles are restricted to move in the space S1. We regularize all collisions using a symplectic transformation and classify the trajectories in four families. After that, we add a third infinitesimal body, getting four restricted three body problems on S1 corresponding to each one of the previous families. Then, we apply only one symplectic transformation that regularizes all the singularities due to binary collisions between the infinitesimal body with each primary. We show the global dynamics in one of the restricted problems, when the primaries are fixed at the poles of S1. We exhibit a particular set of solutions which takes place when the primaries perform hyperbolic motions only.