Stability of symmetries for equilibrium configurations ofN particles in three dimensions

Abstract

We consider equilibrium configurations ofn identical particles in three dimensions interacting via two-body potentials depending only on the distance. The symmetry group of a given configuration is defined as the subgroup of isometries which leaves it invariant, up to permutations of the particles. We prove the stability of the symmetry in the following sense: the symmetry group of an equilibrium configuration is the same for the neighboring equilibria arising from any small enough perturbation of the initial potential. Furthermore, for a large class of realistic potentials, the existence of nontrivial symmetries is proved, thus giving a completely geometrical, although partial, approach to the classical crystal problem.