Poincaré maps and near-collision dynamics for a restricted planar ([formula omitted])-body problem

Abstract

In this paper we perform the analysis of the following planar Newtonian restricted (n+1)-body problem: the first n particles have equal mass and are known as primaries. They are subject to their mutual gravitational attraction and are on the vertices of a regular polygon that rotates with an uniform angular velocity ω. The remaining particle has infinitesimal mass and moves between the primaries. We limit our inquiry to energy levels that allow us to break down the motion of the infinitesimal in an appropriate family of Poincaré maps. Due to the special features of the problem we can deal with the possible collisions and their numerical complexities working in Hill regions, doing several symplectic changes of coordinates and using regularization techniques similar to Birkhoff’s. The method involves several symplectic changes of coordinates to get a workable form. We show several numerical simulations of the Poincaré maps to display the dynamics of the problem.