A Solvable Many-Body Problem in the Plane

Abstract

A solvable many-body problem in the plane is exhibited. It is characterized by rotation-invariant Newtonian (“acceleration equal force”) equations of motion, featuring one-body (“external”) and pair (“interparticle”) forces. The former depend quadratically on the velocity, and nonlinearly on the coordinate, of the moving particle. The latter depend linearly on the coordinate of the moving particle, and linearly respectively nonlinearly on the velocity respectively the coordinate of the other particle. The model contains 2n ² arbitrary coupling constants, n being the number of particles. The behaviour of the solutions is outlined; special cases in which the motion is confined (multiply periodic), or even completely periodic, are identified.