The motion of N particles interacting by a smooth repelling potential and
confined to a compact d-dimensional region is proved to be, under mild
conditions, non-ergodic for all sufficiently large energies. Specifically,
choreographic solutions, for which all particles follow approximately the same
path close to an elliptic periodic orbit of the single-particle system, are
proved to be KAM stable in the high energy limit. Finally, it is proved that
the motion of N repelling particles in a rectangular box is non-ergodic at high
energies for a generic choice of interacting potential: there exists a
KAM-stable periodic motion by which the particles move fast only in one
direction, each on its own path, yet in synchrony with all the other parallel
moving particles. Thus, we prove that for smooth interaction potentials the
Boltzmann ergodic hypothesis fails for a finite number of particles even in the
high energy limit at which the smooth system appears to be very close to the
Boltzmann hard-sphere gas.