On the Existence of Collisionless Equivariant Minimizers for the Classical n-body Problem

We show that the minimization of the Lagrangian action functional on suitable classes of symmetric loops yields collisionless periodic orbits of the n-body problem, provided that some simple conditions on the symmetry group are satisfied. More precisely, we give a fairly general condition on symmetry groups G of the loop space Λ for the n-body problem (with potential of homogeneous degree −α, with α > 0) which ensures that the restriction of the Lagrangian action A to the space ΛG of G-equivariant loops is coercive and its minimizers are collisionless, without any strong force assumption. Many of the already known periodic orbits can be proved to exist by this result, and several new orbits are found with some appropriate choice of G. MSC Subj. Class: Primary 70F10 (Mechanics of particles and systems: n-body problems); Secondary 70F16 (Mechanics of particles and systems: Collisions in celestial mechanics, regularization), 37C80 (Dynamical systems and ergodic theory: Symmetries, equivariant dynamical systems), 70G75 (Mechanics of particles and systems: Variational methods).