Almost Circular Orbits in Classical Action-at-a-Distance Electrodynamics

C. M. Andersen; Hans C. von Baeyer

doi:10.1103/PhysRevD.5.802

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Almost Circular Orbits in Classical Action-at-a-Distance Electrodynamics

C. M. Andersen and Hans C. von Baeyer

Department of Physics, College of William and Mary, Williamsburg, Virginia 23185

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Phys. Rev. D 5, 802 – Published 15 February, 1972

DOI: https://doi.org/10.1103/PhysRevD.5.802

Abstract

The motion of two fully relativistic classical spinless point particles interacting electromagnetically is studied in the special case of almost circular orbits. The equations of motion are difference-differential equations with half-retarded plus half-advanced Liénard-Wiechert potentials. As expected on both physical and mathematical grounds we find multiple stable solutions and conclude that ordinary (Newtonian) initial conditions are not sufficient to determine the trajectories. In addition to the stable solutions we find an infinity of divergent solutions. Alternatives for dealing with the extraneous solutions are discussed. The exact equations of motion can in certain limits be approximated by differential equations. Our solutions serve to delimit the range of applicability of the approximations.

References (22)

J. A. Wheeler and R. P. Feynman, Rev. Mod. Phys. 17, 157 (1945) ibid.21, 425 (1949)
A. Schild, Phys. Rev. 131, 2762 (1963)
C. M. Andersen and H. C. von Baeyer, Ann. Phys. (N.Y.) 60, 67 (1970)
C. G. Darwin, Phil. Mag. 39, 537 (1920)
J. L. Anderson, Principles of Relativity Physics (Academic, New York, 1967), p. 225
E. Breitenberger, Am. J. Phys. 36, 505 (1968)
J. L. Synge, Proc. Roy. Soc. (London) A177, 118 (1940)
A. Sommerfeld, Ann. Physik 51, 1 (1915) J. L. Synge, Relativity: The Special Theory (Interscience, New York, 1956), p. 396
E. H. Kerner, J. Math. Phys. 3, 35 (1962) ibid.6, 1218 (1965)
F. J. Kennedy, J. Math. Phys. 10, 1349 (1969)
R. D. Driver, Phys. Rev. 178, 2051 (1969)
G. N. Plass, Ph.D. thesis, Princeton, 1947 (unpublished)
D. Leiter, Am. J. Phys. 38, 207 (1970)
R. E. Bellman and K. L. Cooke, Differential-Difference Equations (Academic, New York, 1963)
A. Staruszkiewicz, Acta Phys. Polon. 33, 1007 (1968)
A. Staruszkiewicz, Ann. Physik 23, 66 (1969)
F. Rohrlich, Classical Charged Particles (Addison-Wesley, Reading, Mass., 1965), p. 134
H. Van Dam and E. P. Wigner, Phys. Rev. 138, B1576 (1965) ibid.142, 838 (1966)
[1]
Omitted endnote
Chap. 12 of [14]
[15]

[c1] J. A. Wheeler and R. P. Feynman, Rev. Mod. Phys. 17, 157 (1945) ibid.21, 425 (1949)

[c2] A. Schild, Phys. Rev. 131, 2762 (1963)

[c3] C. M. Andersen and H. C. von Baeyer, Ann. Phys. (N.Y.) 60, 67 (1970)

[c4] C. G. Darwin, Phil. Mag. 39, 537 (1920)

[c5] J. L. Anderson, Principles of Relativity Physics (Academic, New York, 1967), p. 225

[c6] E. Breitenberger, Am. J. Phys. 36, 505 (1968)

[c7] J. L. Synge, Proc. Roy. Soc. (London) A177, 118 (1940)

[c8] A. Sommerfeld, Ann. Physik 51, 1 (1915) J. L. Synge, Relativity: The Special Theory (Interscience, New York, 1956), p. 396

[c9] E. H. Kerner, J. Math. Phys. 3, 35 (1962) ibid.6, 1218 (1965)

[c10] F. J. Kennedy, J. Math. Phys. 10, 1349 (1969)

[c11] R. D. Driver, Phys. Rev. 178, 2051 (1969)

[c12] G. N. Plass, Ph.D. thesis, Princeton, 1947 (unpublished)

[c13] D. Leiter, Am. J. Phys. 38, 207 (1970)

[c14] R. E. Bellman and K. L. Cooke, Differential-Difference Equations (Academic, New York, 1963)

[c15] A. Staruszkiewicz, Acta Phys. Polon. 33, 1007 (1968)

[c16] A. Staruszkiewicz, Ann. Physik 23, 66 (1969)

[c17] F. Rohrlich, Classical Charged Particles (Addison-Wesley, Reading, Mass., 1965), p. 134

[c18] H. Van Dam and E. P. Wigner, Phys. Rev. 138, B1576 (1965) ibid.142, 838 (1966)

[c19] [1]

[c20] Omitted endnote

[c21] Chap. 12 of [14]

[c22] [15]

Physical Review D

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Almost Circular Orbits in Classical Action-at-a-Distance Electrodynamics

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