A separating surface for Sitnikov-like n+1n+1-body problems

We consider the generalized Sitnikov problem of Newtonian mechanics. For periodic, planar configurations of n bodies which are 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 modeled 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. A procedure for removing the periodicity condition of the planar configuration is outlined. The construction of the surface is demonstrated numerically in a few examples.