Gravity in one dimension: Stability of a three particle system

The autonomous dynamical system consisting of parallel planes of constant mass density moving under their mutual gravitational attraction is of interest for testing diverse astrophysical models of gravitational relaxation. The simplest non-integrable system consists of three sheets. In this paper the dynamics and stability of this one-dimensional solar system is systematically investigated. A linear transformation of the coordinates reduces the problem to that of a falling body constrained by oblique boundaries. By constructing a Poincare surface of section, it is found that regions of stability and chaos coexist. This behavior is predicted by the KAM theorem for systems having smoother trajectories in phase space, but does not apply here because of the discontinuity in acceleration experienced by each sheet during an encounter. The results of numerical experiments indicate that chaotic regions may be associated with trajectories which contain nearly triple collisions of the three particles.