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

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
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