On the many-body problem with short-range interaction

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Abstract

The classical problem of the interaction of charged particles is considered in the framework of the concept of short-range interaction. Difficulties in the mathematical description of short-range interaction are discussed, for which it is necessary to combine two models, a nonlinear dynamic system describing the motion of particles in a field, and a boundary value problem for a hyperbolic equation or Maxwell’s equations describing the field. Attention is paid to the averaging procedure, that is, the transition from the positions of particles and their velocities to the charge and current densities. The problem is shown to contain several parameters; when they tend to zero in a strictly defined order, the model turns into the classical many-body problem. According to the Galerkin method, the problem is reduced to a dynamic system in which the equations describing the dynamics of particles, are added to the equations describing the oscillations of a field in a box. This problem is a simplification, different from that leading to classical mechanics. It is proposed to be considered as the simplest mathematical model describing the many-body problem with short-range interaction. This model consists of the equations of motion for particles, supplemented with equations that describe the natural oscillations of the field in the box. The results of the first computer experiments with this short-range interaction model are presented. It is shown that this model is rich in conservation laws.

About the authors

Mark M. Gambaryan

Peoples’ Friendship University of Russia (RUDN University)

Author for correspondence.
Email: gamb.mg@gmail.com
ORCID iD: 0000-0002-4650-4648

PhD student of Department of Applied Probability and Informatics

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation

Mikhail D. Malykh

Peoples’ Friendship University of Russia (RUDN University); Meshcheryakov Laboratory of Information Technologies Joint Institute for Nuclear Research

Email: malykh-md@rudn.ru
ORCID iD: 0000-0001-6541-6603

Doctor of Physical and Mathematical Sciences, Assistant Professor of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia (RUDN University); Researcher in Meshcheryakov Laboratory of Information Technologies, Joint Institute for Nuclear Research

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation; 6, Joliot-Curie St., Dubna, Moscow Region, 141980, Russian Federation

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