A separating surface for Sitnikov-like \(n+1\)-body problems. (English) Zbl 1395.70018

Bakker, Lennard; Simmons, Skyler

J. Differ. Equations 258, No. 9, 3063-3087 (2015).

Summary: 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.

MSC:

70F15	Celestial mechanics
70H07	Nonintegrable systems for problems in Hamiltonian and Lagrangian mechanics

Keywords:

\(n\)-body problem; Sitnikov problem; hyperbolic escape orbits; parabolic escape orbits