A Separating Surface for Sitnikov-like n+1-body Problems

TL;DR

This paper constructs a geometric surface in phase space that separates escaping and non-escaping orbits of a massless particle in symmetric n+1-body Newtonian systems, demonstrated through numerical examples.

Contribution

It introduces a novel geometric construction of a separating surface in phase space for Sitnikov-like problems, enhancing understanding of particle escape dynamics.

Findings

The surface effectively distinguishes between escaping and bounded orbits.

Numerical examples validate the geometric construction.

The method applies to symmetric periodic configurations in Newtonian mechanics.

Abstract

We consider the restricted n + 1-body problem of Newtonian mechanics. For periodic, planar configurations of n bodies which is symmetric under rotation by a fixed angle, the z-axis is invariant. We consider the effect of placing a massless particle on the z-axis. The study of the motion of this particle can then be modelled as a time-dependent Hamiltonian System. We give a geometric construction of a surface in the three-dimensional phase space separating orbits for which the massless particle escapes to infinity from those for which it does not. The construction is demonstrated numerically in a few examples