Non-planar minimizers and rotational symmetry in the N-body problem

The Newtonian N-body problem admits uniformly rotating relative equilibrium solutions in the plane, but not in R3. When the bodies are allowed to interact in R3, is it preferable, in terms of action, to leave the plane and follow a non-planar trajectory? We use the variational techniques of Chenciner and Venturelli (Celestial Mech. Dyn. Astro. 77 (2000) 139) to show that for an open set of masses, there is a class of collision-free, action-minimizing orbits of certain rotational symmetry in the four-body problem which are non-coplanar, i.e. the planar relative equilibrium is not the least-action solution among orbits in R3. Both periodic and quasi-periodic solutions are constructed in this way. We also discuss constructing collision-free action-minimizing solutions possessing d-rotational symmetry along with various other symmetry constraints.