REGULAR CONTINUUM SYSTEMS OF POINT PARTICLES. I: SYSTEMS WITHOUT INTERACTION

Abstract

Normally in mathematics and physics only point particle systems, which are either finite or countable, are studied. We introduce new formal mathematical object called regular continuum system of point particles (with continuum number of particles). Initially each particle is characterized by the pair: (initial coordinate, initial velocity) in \(R^{2d}\). Moreover, all initial coordinates are different and fill up some domain in \(R^{d}\). Each particle moves via normal newtonian dynamics under influence of sone external force, but there is no interaction between particles. If the external force is bounded then trajectories of any two particles in the phase space do not intersect. More exactly, at any time moment any two particles have either different coordinates or different velocities. The system is called regular if there are no particle collisions in the coordinate space.

The regularity condition is necessary for the velocity of the particle, situated at a given time at a given space point, were uniquely defined. Then the classical Euler equation for the field of velocities has rigorous meaning. Though the continuum of particles is in fact a continuum medium, the crucial notion of regularity was not studied in mathematical literature.

It appeared that the seeming simplicity of the object (absence of interaction) is delusive. Even for simple external forces we could not find simple necessary and sufficient regularity conditions. However, we found a rich list of examples, one dimensional and many dimensional, where we get regularity conditions on different time intervals. In conclusion we formulate many perspective problems for regular systems with interaction.