New Periodic Orbits for the $n$-Body Problem

Since the discovery of the figure-eight orbit for the three-body problem [Moore, C., 1993, Phys. Rev. Lett., 70, pp. 3675–3679] a large number of periodic orbits of the $n$-body problem with equal masses and beautiful symmetries have been discovered. However, most of those that have appeared in the literature are either planar or are obtained from perturbations of planar orbits. Here we exhibit a number of new three-dimensional periodic $n$-body orbits with equal masses and cubic symmetry, including some whose moment of inertia tensor is a scalar. We found these orbits numerically, by minimizing the action as a function of the trajectories’ Fourier coefficients. We also give numerical evidence that a planar three-body orbit first found in [Hénon, M., 1976, Celest. Mech., 13, pp. 267–285], rediscovered by [Moore, 1993], and found to exist for different masses by [Nauenberg, M., 2001, Phys. Lett., 292, pp. 93–99], is dynamically stable.