Stable motions of high energy particles interacting via a repelling potential (Q6564457)

The authors consider the motion of \(N\) particles interacting in a compact \(d\)-dimensional region under a smooth repelling potential. In the end the authors establish that such a system can be proved -- with some moderate assumptions -- to be non-ergodic for energies that are large. The authors also prove that a system like this, with a sufficiently large number \(N\) of repelling particles, can be bounded away from one another and avoid collisions with positive probability.\N\NSolutions are constructed that mirror the choreographic path of celestial mechanics; that is, particles move essentially synchronously along the same path or parallel paths with phase shifts between them that are nearly constant. It is clear then that these systems of repelling particles are non ergodic, and indeed have KAM-stable states. The authors also show that these stable sets persist at arbitrarily large energies for any finite number of particles.\N\NThe authors also prove that the motion of \(N\) repelling particles in a rectangular box at high energies is non-ergodic for a generic interacting potential. So there is a KAM-stable periodic motion where particles -- each on their own path -- move fast in just one direction but in synchrony with all the other moving particles.