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 hiphop
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.