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The Sitnikov problem for several primary bodies configurations. (English) Zbl 1396.70014

Summary: In this paper we address an \(n+1\)-body gravitational problem governed by the Newton’s laws, where \(n\) primary bodies orbit on a plane \(\varPi \) and an additional massless particle moves on the perpendicular line to \(\varPi \) passing through the center of mass of the primary bodies. We find a condition for the described configuration to be possible. In the case when the primaries are in a rigid motion, we classify all the motions of the massless particle. We study the situation when the massless particle has a periodic motion with the same minimal period as the primary bodies. We show that this fact is related to the existence of a certain pyramidal central configuration.

MSC:

70F10 \(n\)-body problems
70G60 Dynamical systems methods for problems in mechanics
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