MINIMA OF THE ACTION INTEGRAL AND RELATIVE EQUILIBRIA OF N-BODIES

We consider the n-body problem in an Euclidean space E of finite dimen sion for a homogeneous potential which is invariant under isometries, in particular the Newtonian potential. We show that the absolute minim a of the action integral among zero mean loops of class H-1 whose peri od is fixed coincide with the relative equilibrium solutions whose con figuration is an absolute minimum of the potential among the configura tions whose moment of inertia with respect to their center of mass is fixed. The result is at the same time general and elementary and does nor appeal to functional analysis. (C) Academie des Sciences/Elsevier, Paris.