The classical analog of intertwining operators in the relativistic Hamiltonian mechanics of a system of particles

In the context of nonquantum Hamiltonian formalism of the relativistic theory of direct interaction we construct a canonical transformation of the collective variables of center of mass type which transforms the canonical generators of the Poincaré algebra in one form of dynamics into the corresponding generators in another form of dynamics.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 62–65.     

Spatially distributed classical Lagrangian mechanics

It is well known that the existence of two nontrivial integrals of the motion makes it possible to parametrize the motion of a Lagrangian rigid body by two variables. On the basis of this fact it is shown that certain combinations of the quantities that characterize the trajectory of such a body satisfy well-known nonlinear equations: sine—Gordon, Korteweg—de Vries, Klein-Gordon, and nonlinear Schrödinger equation.Elabuga State Pedagogical Institute. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 101, No. 3, pp. 369–373, December, 1994.     

Group-theoretic approach to the construction of relativistic lagrangian mechanics of a system of particles

Mathematical aspects of the Lagrangian formalism of relativistic mechanics of a system of interacting particles are considered. A geometric definition of a form of relativistic dynamics possessing the spacelike or isotropic foliation of Minkowski space is introduced. A realization of the Lie algebra of the Poincaré group by means of Lie-Bäcklund vector fields on a general jet continuation of the configuration space is constructed. Invariance conditions of Lagrangian relativistic mechanics are formulated and investigated; the characteristic features of this formalism, which arises as a consequence of the demands of Poincaré invariance, are described.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 11, pp. 1516–1521, November, 1991.     

Statistical mechanics of classical particles with logarithmic interactions

The inhomogeneous mean-field thermodynamic limit is constructed and evaluated for both the canonical thermodynamic functions and the states of systems of classical point particles with logarithmic interactions in two space dimensions. The results apply to various physical models of translation invariant plasmas, gravitating systems, as well as to planar fluid vortex motion. For attractive interactions a critical behavior occurs which can be classified as an extreme case of a second-order phase transition. To include in particular attractive interactions a new inequality for configurational integrals is derived from the arithmetic-geometric mean inequality. The method developed in this paper is easily seen to apply as well to systems with fairly general interactions in all space dimensions. In addition, it also provides us with a new proof of the Trudinger-Moser inequality known from differential geometry – in its sharp form.     

Integrable system of generalized relativistic interacting tops

Theoretical and Mathematical Physics - We describe a family of integrable $$GL(NM)$$ models generalizing classical spin Ruijsenaars–Schneider systems (the case $$N=1$$ ) on one hand and...     

Thermodynamic pressure and its fluctuations in a classical ideal gas of relativistic particles

A full and consecutive analysis of the dynamic and thermodynamic properties of an ideal gas of relativistic particles with Lorentz?CEinstein dispersion law and arbitrary number of translational degrees of freedom is carried out. Gibbs statistical mechanics is used along with Bogolyubov??s concept of quasiaverages and the generalized version of the Bogolyubov?CZubarev theorem in the classical regime well beyond the temperature of the quantum degeneracy. General expressions for a pair of equations of state, namely thermic (for the pressure) and caloric (for the inner energy) are found; the fluctuations of these quantities are also found: the compressibility and heat capacity, respectively. All expressions are found in closed form and studied in low- and high-temperature limits.     

The macroscopic system of Einstein-Maxwell equations for a system of interacting particles

We derive the macroscopic Einstein—Maxwell equations up to the second-order terms, in the interaction for systems with dominating electromagnetic interactions between particles (e.g., radiation-dominated cosmological plasma in the expanding Universe before the recombination moment). The ensemble averaging of the microscopic Einstein and Maxwell equations and of the Liouville equations for the random functions of each type of particle leads to a closed system of equations consisting of the macroscopic Einstein and Maxwell equations and the kinetic equations for one-particle distribution functions for each type of particle. The macroscopic Einstein equations for a system of electromagnetically and gravitationally interacting particles differ from the classical Einstein equations in having additional terms in the lefthand side due to the interaction. These terms are given by a symmetric rank-two traceless tensor with zero divergence. Explicitly, these terms are represented as momentum-space integrals of the expressions containing one-particle distribution functions for each type of particle and have much in common with similar terms in the left-hand side of the macroscopic Einstein equations previously obtained for a system of self-gravitating particles. The macroscopic Maxwell equations for a system of electromagnetically and gravitationally interacting particles also differ from the classical Maxwell equations in having additional terms in the left-hand side due to simultaneous effects described by general relativity and the interaction effects. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 125, No. 1, pp. 107–131, October, 2000.     

Smoothing effects for classical solutions of the relativistic Landau-Maxwell system

The relativistic Landau-Maxwell system is one of the most fundamental and complete models for describing the dynamics of a dilute hot plasma in which particles interact through Coulomb collisions and their self-consistent electromagnetic field. In this work, we prove that the classical solutions obtained by Strain and Guo become immediately smooth with respect to all variable under the extra assumption of the electromagnetic field. As a by-product, we also prove that the classical solutions to the relativistic Landau-Poisson system and the relativistic Landau equation have the same property without any extra assumption.     

Thermodynamic limit for a classical system of particles with hard cores

Theoretical and Mathematical Physics -