Regular continuum systems of point particles. I: systems without interaction

TL;DR

This paper introduces the concept of regular continuum systems of point particles without interaction, establishing conditions for their regularity and exploring their properties in various dimensions, with implications for the Euler equations.

Contribution

It formalizes the notion of regular continuum particle systems without interaction and investigates the conditions under which their trajectories remain collision-free.

Findings

Trajectories of particles do not intersect if external forces are bounded.

Regularity conditions are complex and not fully characterized even in simple cases.

Numerous examples of regular systems are provided across different dimensions.

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 some 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 mufti-dimensional, where we could get regularity conditions on different time intervals. In conclusion we formulate many unsolved problems for regular systems with interaction.