Formulation of the Post-Newtonian Equations of Motion of the Restricted Three Body Problem

ABSTRACT

In the present work the geodesic equation represents the equations of motion of the particles along the geodesics is derived. The deviation of the curved space-time metric tensor from that of the Minkowski tensor is considered as a perturbation. The quantities is expanded in powers of*c*^{-2}. The equations of motion of the relativistic three body problem in the PN formalism are obtained.

In the present work the geodesic equation represents the equations of motion of the particles along the geodesics is derived. The deviation of the curved space-time metric tensor from that of the Minkowski tensor is considered as a perturbation. The quantities is expanded in powers of

Cite this paper

nullF. Abd El-Salam and S. El-Bar, "Formulation of the Post-Newtonian Equations of Motion of the Restricted Three Body Problem,"*Applied Mathematics*, Vol. 2 No. 2, 2011, pp. 155-164. doi: 10.4236/am.2011.22018.

nullF. Abd El-Salam and S. El-Bar, "Formulation of the Post-Newtonian Equations of Motion of the Restricted Three Body Problem,"

References

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[13] F. W. Lucas, “Chaotic Amplification in the Relativistic Restricted Three-Body Problem,” Verlag der Zeitschrift für Naturforschung, Tübingen, Vol. 58a, No. 1, 2003, pp. 13-22.

[14] C. W. Misner, K. S. Thorne and J. A. Wheeler, “Gravitation,” Freeman and Company, San Francisco, 1973, p. 840.

[15] J. Foster and J. D. Nightingale, “A Short Course in General Relativity,” Springer, New York, 1995, p. 147.

[1] C. G. Jacobi, “Sur le Movement d’un Point et sur un cas Particulier du Probleme des trios Corps,” Compte Rendus de l’Académie des Sciences, Vol. 3, 1836, p. 59.

[2] G. W. Hill, “Researches in the Lunar Theory,” American Journal of Mathematics, Vol. 1, No. 1, 1878, pp. 5-26. doi:10.2307/2369430

[3] H. Poincaré, “Les Methodes Mouvelles de la Mecanique Celeste,” Vol. 3, Gauthier-Villars, Paris, 1892.

[4] G. D. Birkhoff, “The Restricted Problem of Three Bodies,” Rendiconti del Circolo Matematico di Palermo, Vol. 39, 1915, pp. 265-334. doi:10.1007/BF03015982

[5] L. Euler, “Theoria Motuum Lumae,” Typis Academiae Imperialis Scientiarum, Petropoli, 1772.

[6] E. Krefetz, “Restricted Three-Body Problem in the Post-Newtonian Approximation,” Astronomical Journal, Vol. 72, 1967, p. 471. doi:10.1086/110252

[7] G. Contopoulos, “In Memoriam D. Eginitis,” D. Kotsakis, Ed., Athens, 1976, p. 159.

[8] S. Weinberg, “Gravitation and Cosmology Principles and Applications of the General Theory of Relativity,” Chapter 9, John Wiley & Sons, New York, 1972.

[9] M. H. Soffel, “Relativity in Astrometry, Celestial Mechanics and Geodesy,” Springer-Verlag, Berlin, 1989, p. 173.

[10] V. A. Brumberg, “Relativistic Celestial Mechanics,” Nauka Press (Science), Moscow, 1972.

[11] V. A. Brumberg, “Essential Relativistic Celestial Mechanics,” Adam Hilger, Ltd., New York, 1991.

[12] K. B. Bhatnagar and P. P. Hallan, “Existence and Stability of L4,5 in the Relativistic Restricted Three Body Problem,” Celestial Mechanics, Vol. 69, No. 3, 1998, pp. 271-281.

[13] F. W. Lucas, “Chaotic Amplification in the Relativistic Restricted Three-Body Problem,” Verlag der Zeitschrift für Naturforschung, Tübingen, Vol. 58a, No. 1, 2003, pp. 13-22.

[14] C. W. Misner, K. S. Thorne and J. A. Wheeler, “Gravitation,” Freeman and Company, San Francisco, 1973, p. 840.

[15] J. Foster and J. D. Nightingale, “A Short Course in General Relativity,” Springer, New York, 1995, p. 147.