Non-planar minimizers and rotational symmetry in the [formula omitted]-body problem

Abstract

The Newtonian N -body problem admits uniformly rotating relative equilibrium solutions in the plane, but not in R 3 . When the bodies are allowed to interact in R 3 , 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 R 3 . 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.