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