Abstract:
General formulation of the one-time Lagrangian relativistic classical description of N directly interacting particles is developed. A representation of a continuous transformation group of the Minkowski space (in particular, Poincare group) by the Lie–Backlund tangent transformations of the configuration space of the system is constructed. By means of this representation the system of linear differential equations expressing the Poincare invariance of the Lagrangian formalism is obtained and corresponding restrictions on the form of the generalised Lagrangian are studied. The exact relativistic description of the interaction is shown to demand the dependence of the Lagrangian on the infinite order derivatives. The results will be used in the second part of the work for finding a general form of the approximate relativistic interaction Lagrangian and for working out the method of constructing the Poincare invariant Newton type equations of motion and their first integrals.
Citation:
R. P. Gaida, Yu. B. Klyuchkovskii, V. I. Tretyak, “Lagrangian classical relativistic mechanics of a system of directly interacting particles. I”, TMF, 44:2 (1980), 194–208; Theoret. and Math. Phys., 44:2 (1980), 687–697
Horace Crater, Luca Lusanna, “The Rest-Frame Darwin Potential from the Lienard–Wiechert Solution in the Radiation Gauge”, Annals of Physics, 289:2 (2001), 87
DAVID ALBA, LUCA LUSANNA, “THE LIENARD–WIECHERT POTENTIAL OF CHARGED SCALAR PARTICLES AND THEIR RELATION TO SCALAR ELECTRODYNAMICS IN THE REST-FRAME INSTANT FORM”, Int. J. Mod. Phys. A, 13:16 (1998), 2791
Roman Gaida, Volodymyr Tretyak, “Symmetries of the Fokker-Type Relativistic Mechanics in Various Forms of Dynamics”, JNMP, 3:3-4 (1996), 357
R. A. Moore, T. C. Scott, “Causality and quantization of time-delay systems: A two-body model problem”, Phys. Rev. A, 52:6 (1995), 4371
R. A. Moore, T. C. Scott, “Causality and quantization of time-delay systems: A model problem”, Phys. Rev. A, 52:3 (1995), 1831
R. P. Gaida, V. I. Tretyak, Yu. G. Yaremko, “Center-of-mass variables in the relativistic Lagrangian dynamics of a system of particles”, Theoret. and Math. Phys., 101:3 (1994), 1443–1453
R. A. Moore, T. C. Scott, “Quantization of second-order Lagrangians: The Fokker-Wheeler-Feynman model of electrodynamics”, Phys. Rev. A, 46:7 (1992), 3637
Thibault Damour, Gerhard Schäfer, “Redefinition of position variables and the reduction of higher-order Lagrangians”, Journal of Mathematical Physics, 32:1 (1991), 127
R. P. Gaida, Yu. B. Klyuchkovskii, V. I. Tretyak, “Group-theoretic approach to the construction of relativistic lagrangian mechanics of a system of particles”, Ukr Math J, 43:11 (1991), 1408
R. P. Gaida, Yu. B. Klyuchkovskii, V. I. Tretyak, “Relativistic theory of direct interactions and gravitational two-body problem in the second post-Newtonian approximation”, Soviet Physics Journal, 33:1 (1990), 40
R. P. Gaida, V. I. Tretyak, “Lagrangian of two charged gravitating bodies in the second post-Newton approximation”, Soviet Physics Journal, 33:7 (1990), 575
S. N. Sokolov, V. I. Tretyak, “Front form of relativistic Lagrangian dynamics in two-dimensional space-time and its connection with the Hamiltonian description”, Theoret. and Math. Phys., 67:1 (1986), 385–394
R. P. Gaida, Yu. B. Klyuchkovskii, V. I. Tretyak, “Forms of relativistic dynamics in a classical Lagrangian description of a system of particles”, Theoret. and Math. Phys., 55:1 (1983), 372–384
F. M. Lev, “On a Three-Body Problem in Relativistic Quantum Mechanics”, Fortschr. Phys., 31:2 (1983), 75
R. P. Gaida, Yu. B. Klyuchkovskii, V. I. Tretyak, “Lagrangian classical relativistic mechanics of a system of directly interacting particles. II”, Theoret. and Math. Phys., 45:2 (1980), 963–975