Periodic Oscillations in the Restricted Hip-Hop 2N+1 - Body Problem

This manuscript investigates a dynamical system in which 2N primary particles of equal masses move in space under Newton’s law of gravitation forming the vertices of antiprisms while a particle of negligible mass moves along the common axis of symmetry of the antiprisms. This n-body problem that we call the restricted hip-hop (2N + 1)-body problem is an extension of the generalized Sitnikov problem studied in [17] for which the primaries remain in a plane. This work also relies on an early study [14] where certain families of periodic hip-hop solutions to a 2N–body problem were constructed. We prove the existence of a continuous symmetric family of solutions of the restricted hip-hop (2N + 1)-body problem for each family of symmetric and periodic hip-hop solutions of the primaries studied in [14]. The main tools for proving our results are the implicit function theorem and a compactness argument. In addition, we present some numerical periodic solutions to the restricted 7-body problem.