Periodic Oscillations in a 2N -Body Problem

Hip-hop solutions of the 2N-body problem are solutions that satisfy, at every instance of time, that the 2N bodies with the same mass m are at the vertices of two regular N-gons, and each one of these N-gons is at a plane that are equidistant from a fixed plane \Pi0 forming an antiprism. In this paper, we first prove that for every N and every m there exists a family of periodic hip-hop solutions. For every solution in these families the oriented distance to the plane \Pi0, which we call d(t), is an odd function that is also even with respect to t = T for some T > 0. For this reason we call solutions in these families double symmetric solutions. By exploring more carefully our initial set of periodic solutions, we numerically show that some of the branches established in our existence theorem have bifurcations that produce branches of solutions with the property that the oriented distance function d(t) is not even with respect to any T > 0; we call these solutions single symmetric solutions. We prove that no single symmetric solution is a choreography. We also display explicit double symmetric solutions that are choreographies.