Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)3032610.22363/2658-4670-2022-30-1-52-61Research ArticleOn the many-body problem with short-range interactionGambaryanMark M.<p>PhD student of Department of Applied Probability and Informatics</p>gamb.mg@gmail.comhttps://orcid.org/0000-0002-4650-4648MalykhMikhail D.<p>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</p>malykh-md@rudn.ruhttps://orcid.org/0000-0001-6541-6603Peoples’ Friendship University of Russia (RUDN University)Meshcheryakov Laboratory of Information Technologies Joint Institute for Nuclear Research01042022301526125022022Copyright © 2022, Gambaryan M.M., Malykh M.D.2022<p style="text-align: justify;">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.</p>many-body problemGalerkin methodshort-range interactionзадача многих телметод Галёркинаблизкодействие[C. K. Birdsall and L. A. Bruce, Plasma physics via computer simulation. Bristol, Philadelphia and New York: Adam Hilger, 1991.][R. W. Hockney and J. W. Eastwood, Computer simulation using particles. Bristol, Philadelphia and New York: Adam Hilger, 1988.][V. P. Tarakanov, User’s manual for code KARAT. Springfield, VA: Berkley Research, 1999.][E. Hairer, G. Wanner, and S. P. Nørsett, Solving Ordinary Differential Equations, 3rd ed. New York: Springer, 2008, vol. 1.][G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics. Berlin: Springer-Verlag, 1976.][H. Qin, S. Zhang, J. Xiao, J. Liu, and Y. Sun, “Why is Boris algorithm so good?” Physics of Plasmas, vol. 20, p. 084503, 2013. DOI: 10.1063/1.4818428.][A. G. Sveshnikov, A. N. Bogolyubov, and K. V. V., Lectures on Mathematical Physics [Lektsii po matematicheskoy fizike]. Moscow: MGU, 1993, in Russian.][I. I. Vorovich, “On some direct methods in the nonlinear theory of oscillations of shallow shells [O nekotorykh pryamykh metodakh v nelineynoy teorii kolebaniy pologikh obolochek],” Izvestiya Akademii Nauk USSR, Seriya Matematicheskaya, vol. 21, no. 6, pp. 747-784, 1957, in Russian.][P. G. Ciarlet, The finite element method for elliptic problems. NorthHolland, 1978.][N. G. Afendikova, “The history of Galerkin’s method and its role in M.V. Keldysh’s work [Istoriya metoda Galerkina i yego rol’ v tvorchestve M.V.Keldysha],” Keldysh Institute preprints, no. 77, 2014, in Russian.][G. Hellwig, Partial differential equations. An introduction. Leipzig: Teubner, 1960.][G. Hellwig, Differential operators of Mathematical Physics. Reading, MA: Addison-Wesley, 1967.][A. N. Tikhonov, “Systems of differential equations containing small parameters at derivatives [Sistemy differentsial’nykh uravneniy, soderzhashchiye malyye parametry pri proizvodnykh],” Mat. Sb., vol. 31, no. 3, pp. 575-586, 1952, in Russian.][A. B. Vassilieva and V. F. Butuzov, Asymptotic methods in singular perturbation theory [Asimptoticheskiye metody v teorii singulyarnykh vozmushcheniy]. Moscow: Vysshaya shkola, 1990, in Russian.]