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On the formulation of initial-value problems for systems consisting of relativistic particles


We discuss questions related to the well-posedness of problems on the motion of relativistic many-body systems. For one-dimensional relativistic motion of N similar charges, we prove that an ordinary Cauchy problem usual in Newton mechanics can be stated; this is done in the framework of microscopic Maxwell-Lorentz electrodynamics (including a model with self-action) or Wheeler-Feynman theory.

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  1. 1.

    C. M. Andersen and H. C. von Baeyer, “Almost circular orbits in classical action-at-a-distance electrodynamics,” Phys. Rev. D, 5, No. 4, 802–813 (1972).

    Article  Google Scholar 

  2. 2.

    W. E. Baylis and J. Huschilt, “Numerical solutions to two-body problems in classical electrodynamics: straight-line motion with retarded fields and no radiation reaction,” Phys. Rev. D, 7, No. 10, 2844–2850 (1973).

    Article  Google Scholar 

  3. 3.

    C. G. Darvin, Philos. Mag., 39, 537 (1920).

    Google Scholar 

  4. 4.

    P. A. M. Dirac, “Classical theory of radiating electrons,” Proc. Roy. Soc. London, Ser. A, 167, 148–168, (1938).

    Article  Google Scholar 

  5. 5.

    R. D. Driver, “A ‘backwards’ two-body problem of classical relativistic electrodynamics,” Phys. Rev., 178, No. 5, 2051–2057 (1969).

    Article  MathSciNet  Google Scholar 

  6. 6.

    R. D. Driver, “Can the future influence the present?” Phys. Rev. D, 19, No. 4, 1098–1107 (1979).

    Article  MathSciNet  Google Scholar 

  7. 7.

    R. D. Driver and D. K. Hsing, in: Dynamical Systems, Proc. Int. Symp.Univ. Florida, Academic Press, New York (1977), pp. 427–430.

    Google Scholar 

  8. 8.

    J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Appl. Math. Sci., 99, Springer-Verlag, New York (1993).

    MATH  Google Scholar 

  9. 9.

    D. K. Hsing, “Existence and uniquence theorem for the one-dimentional backwards two-bodies problem of electrodynamics,” Phys. Rev. D, 16, 974–982 (1977).

    Article  MathSciNet  Google Scholar 

  10. 10.

    A. D. Myshkis, Linear Differential Equations with Retarded Arguments [in Russian], Nauka, Moscow (1972).

    Google Scholar 

  11. 11.

    G. N. Plass, “Classical electrodynamic equations of motion with radiative reaction,” Rev. Mod. Phys., 33, No. 1, 37–62 (1961).

    Article  MathSciNet  Google Scholar 

  12. 12.

    A. Schild, “Electromagnetic two-body problem,” Phys. Rev., 131, No. 6, 2762–2766 (1963).

    Article  MathSciNet  MATH  Google Scholar 

  13. 13.

    J. L. Synge, “On the electromagnetic two-bodies problem,” Proc. Roy. Soc. London, Ser. A, 177, 118–199, (1941).

    MathSciNet  Google Scholar 

  14. 14.

    H. Van Dam and E. P. Wigner, “Classical relativistic mechanics of interacting point particles,” Phys. Rev., 138, No. 6b, 1576–1582 (1965).

    Article  MathSciNet  Google Scholar 

  15. 15.

    J. A. Wheeler and R. P. Feynman, “Interaction with the absorber as the mechanism of radiation,” Rev. Mod. Phys., 17, 157–181 (1945).

    Article  Google Scholar 

  16. 16.

    J. A. Wheeler and R. P. Feynman, “Classical electrodynamics in terms of direct interparticle action,” Rev. Mod. Phys., 21, 425–433 (1949).

    Article  MathSciNet  MATH  Google Scholar 

  17. 17.

    V. I. Zhdanov, “On the one-dimentional symmetric two-body problem of classical electrodynamics,” Int. J. Theor. Phys., 15, No. 2, 157–167 (1976).

    Article  MathSciNet  Google Scholar 

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Translated from Fundamental’naya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 11, No. 1, Geometry, 2005.

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Kirpichev, S.B., Polyakov, P.A. On the formulation of initial-value problems for systems consisting of relativistic particles. J Math Sci 141, 1051–1061 (2007).

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  • Cauchy Problem
  • Circular Orbit
  • Central Collision
  • Classical Electrodynamic
  • Advanced Potential