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.     

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.     

Bogoliubov group variables in relativistic field theory

We define the Bogoliubov variables for strongly coupled systems that are invariant under the Poincaré group in (1+1)-dimensional space-time. This allows us to achieve a compatibility between taking the conservation laws into account exactly and developing a regular perturbation theory. We perform the secondary quantization in terms of the Bogoliubov variables and discuss the problem of reducing the number of states of the field. We also discuss the conditions for validity of the perturbation theory. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 111, No. 2, pp. 242–251, May, 1997.     

Analytic dynamics of a one-dimensional system of particles with strong interaction

We consider the dynamics of a system of N particles on the circle with interaction of nearest neighbors, a Coulomb potential, and an analytic external force. The trajectories are real analytic functions of time. However, the series for them converge only for sufficiently small times. For zero initial velocities and a uniform initial location of particles, we prove N-dependent estimates on the coefficients of this series.     

Motion of particles in a centrally symmetric field in the relativistic theory of gravitation

Theoretical and Mathematical Physics -     

A model of relativistic dynamics

A model of relativistic dynamics is proposed for classical (nonquantum) multiparticle systems within the Lagrangian formalism on the space of world lines.     

Lagrangian stability of a nonlinear quasi-periodic system

In this paper, we prove the Lagrangian stability of the quasi-periodic system d²x/dt²+G_x(x,t)=0, where G is quasi-periodic in both x and t, respectively.     

Kinetic schemes for the relativistic gas dynamics

Summary. A kinetic solution for the relativistic Euler equations is presented. This solution describes the flow of a perfect gas in terms of the particle density n, the spatial part of the four-velocity u and the inverse temperature . In this paper we present a general framework for the kinetic scheme of relativistic Euler equations which covers the whole range from the non-relativistic limit to the ultra-relativistic limit. The main components of the kinetic scheme are described now. (i) There are periods of free flight of duration _M, where the gas particles move according to the free kinetic transport equation. (ii) At the maximization times t_n=n_M, the beginning of each of these free-flight periods, the gas particles are in local equilibrium, which is described by Jüttners relativistic generalization of the classical Maxwellian phase density. (iii) At each new maximization time t_n>0 we evaluate the so called continuity conditions, which guarantee that the kinetic scheme satisfies the conservation laws and the entropy inequality. These continuity conditions determine the new initial data at t_n. iv If in addition adiabatic boundary conditions are prescribed, we can incorporate a natural reflection method into the kinetic scheme in order to solve the initial and boundary value problem. In the limit _M0 we obtain the weak solutions of Eulers equations including arbitrary shock interactions. We also present a numerical shock reflection test which confirms the validity of our kinetic approach. Mathematics Subject Classification (1991):65M99, 76Y05This work is supported by the project Long-time behaviour of nonlinear hyperbolic systems of conservation laws and their numerical approximation, contract # DFG WA 633/7-2.     

Two-point boundary value problems in relativistic dynamics

For gyro systems of relativistic type, we obtain solvability conditions for the two-point boundary value problem. We use the geodesic modeling method, in which the original problem is reduced to studying the existence of isotropic geodesic curves of the Kaluts-O. Klein Lorentz metric joining two fibers of a bundle over the configuration manifold of the system. As an example, we consider problems on the motion of charged test particles in an arbitrary electromagnetic field and in the outer Reissner-Nordstrem space-time in the field of a charged black hole and some external electromagnetic field.Translated fromMatematicheskie Zametki, Vol. 59, No. 3, pp. 437–449, March, 1996.This research was partially supported by the Russian Foundation for Basic Research under grant No. 94-01-00492a and by the International Science Foundation under grant No. NP4000.