Periodic orbits near collision in a restricted four-body problem for the eight choreography

Abstract

We study orbits near collision in a non-autonomous restricted planar four-body problem. This restricted problem consists of a massless particle moving under the gravitational influence due to three bodies following the figure-eight choreography. We use regularized coordinates in order to deal numerically with motions near collision, and reversing symmetries to study both theoretically and numerically certain type of symmetric periodic orbits of the restricted system. The symmetric periodic orbits (initial conditions) were computed with the help of some boundary-value problems.