Central configurations of the N-body problem via equivariant Morse theory

Abstract

In this paper we use the equivariant Morse theory to give an estimate of the minimal number of central configurations in the N-body problem in ℝ3. In the case of equal masses we prove that the planar central configurations are saddle points for the potential energy. From this we deduce the presence of non-planar central configurations, for every N ≧ 4.The principal difficulty in applying Morse theory is that the potential function is defined on a manifold on which the group O(3) does not act freely. To avoid this problem the equivariant cohomology functor is applied in order to obtain the Morse inequalities.