Geometric characterization for the least Lagrangian action ofn-body problems

For n-body problems with quasihomogeneous potentials in ℝk (2[ n/2] ⩽ k) we prove that the minimum of the Lagrangian action integral defined on the zero mean loop space is exactly the circles with center at the origin and the configuration of the n-bodies is always a regular n - 1 simplex with fixed side length.