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

Research paper by **V. N. Chubarikov, A. A. Lykov, V. A. Malyshev**

Indexed on: **08 Nov '16**Published on: **08 Nov '16**Published in: **arXiv - Mathematical Physics**

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#### 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.