**Title: **A Separating Surface for Sitnikov-like Problems

**Abstract:** The Sitnikov Problem is a famous example of a restricted 3-body system in celestial mechanics. It involves two equal masses whose orbits trace out ellipses in the xy plane, and a third zero-mass particle that moves along the z-axis. The particle remains on the z-axis due to the symmetry of the system. More generally, planar n-body configurations which are periodic and symmetric under rotation by a fixed angle maintain the invariance of the z-axis, providing a natural generalization of the Sitnikov Problem. The study of the motion of the massless particle can then be modelled as a time-dependent system of differential equations. I will give a geometric construction of a surface in phase space separating orbits for which the massless particle escapes to infinity from those for which it does not, and discuss how this surface varies depending upon the planar configuration. The construction is demonstrated numerically in a few examples. This is joint work with Lennard Bakker.