On the Gravitational Two-Body System and an Infinite Set of Laplace-Runge-Lenz Vectors


The current approach of a system of two bodies that interact through a gravitational force goes beyond the familiar expositions [1-3] and derives some interesting features and laws that are overlooked. A new expression for the angular momentum of a system in terms of the angular momenta of its parts is deduced. It is shown that the characteristics of the relative motion depend on the system’s total mass, whereas the characteristics of the individual motions depend on the masses of the two bodies. The reduced energy and angular momentum densities are constants of motion that do not depend on the distribution of the total mass between the two bodies; whereas the energy may vary in absolute value from an infinitesimal to a maximum value which occurs when the two bodies are of equal masses. In correspondence with infinite possible ways to describe the absolute rotational positioning of a two body system, an infinite set of Laplace-Runge-Lenz vectors (LRL) are constructed, all fixing a unique orientation of the orbit relative to the fixed stars. The common expression of LRV vector is an approximation of the actual one. The conditions for nested and intersecting individual orbits of the two bodies are specified. As far as we know, and apart from the law of periods, the laws of equivalent orbits concerning their associated periods, areal velocities, angular velocities, velocities, energies, as well as, the law of total angular momentum, were never considered before.

Share and Cite:

C. Viazminsky and P. Vizminiska, "On the Gravitational Two-Body System and an Infinite Set of Laplace-Runge-Lenz Vectors," Applied Mathematics, Vol. 4 No. 5, 2013, pp. 774-784. doi: 10.4236/am.2013.45106.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] H. Goldstein, C. P. Poole and J. L. Safko, “Classical Mechanics,” Addison Wesley, Boston, 2001.
[2] S. W. Groesberg, “Advanced Mechanics,” John Wiley & Sons, Inc., Hoboken, 1998.
[3] S. R. Spiegel, “Theoretical Mechanics,” In: Schaum Out line Series, Mc Graw Hill Book Company, New York, 1967.
[4] H. Goldstein, “More on the Prehistory of the Runge-Lenz Vector,” American Journal of Physics, Vol. 44, No. 11, 1976, pp. 1123-1124. doi:10.1119/1.10202
[5] A. Alemi, “Laplace-Runge-Lenz Vector,” 2009. www.cds.Caltech.edu/Wiki/Alemicds205final.pdf
[6] “Laplace-Runge-Lenz Vector,” Wikipedia, 2013. http://en.wikipedia.org/wiki/Laplace%E2%80%93Runge%E2%80%93Lenz_vector
[7] P. G. L. Leach and G. P. Flessas, “Generalizations of the Laplace-Runge-Lenz Vector,” Journal of Nonlinear Mathematical Physics, Vol. 10, No. 3, 2003, p. 340.
[8] J. Moser, “Regularization of Kepler’s Problem and the Averaging Method on a Manifold,” Communications on Pure and Applied Mathematics, Vol. 23, No. 4, 1970, pp. 609-636. doi:10.1002/cpa.3160230406
[9] H. H. Rogers, “Symmetry Transformations of the Classical Kepler Problem,” Journal of Mathematical Physics, Vol. 14, No. 8, 1973, p. 1125. doi:10.1063/1.1666448
[10] W. R. Hamilton, “The Hodograph or a New Method of Expressing in Symbolic Language the Newtonian Law of Attraction,” Proceedings of the Royal Irish Academy, Vol. 3, 1847, pp. 344-353.
[11] W. R. Hamilton, “Applications of Quaternions to Some Dynamical Questions,” Proceedings of the Royal Irish Academy, Vol. 3, 1847, Appendix III.
[12] J. Hermann, “Unknown Title,” Giornale de Letterati D’ Italia, Vol. 2, 1710, pp. 447-467.
[13] J. Hermann, “Extrait d’une lettre de M. Herman a’ M datée de Padoüe le 12, Juillet 1710,” Histoire Toire de l’Academie Royale des Sciences (Paris), 1732, pp. 519-521.
[14] J. Bernoulli, “Extrait de la Response de M. Bernoulli a M. Herman date de Basle le 7. October 1710,” Histoire de l’Academie Royale des Sciences (Paris), 1732, pp 521-544.
[15] P. S. Laplace, “Traite de Mecanique Celeste,” Tome I, Premiere Partie, Liver II, 1799, p. 165.
[16] J. W. Gibbs, J. W. Gibbs and E. B. Wilson, “Vector Analysis,” Scribners, New York, 1910, p. 135.
[17] C. Runge, “Vektoranalysis,” Verlag Von S. Hirzel, Leipzig, 1919.
[18] W. Lenz, “Uberden Bewegungsverlauf und Quantenzust Ande der Gestorten Keplerbewegung,” Zeitschrift fur Physik, Vol. 24, No. 1, 1924, pp. 197-207. doi:10.1007/BF01327245
[19] W. Rindler, “Essential Relaivity,” Springr-Verlag, Berlin, 2006.
[20] N. W. Evans, “Superintegrability in Classical Mechanics,” Physical Review A, Vol. 41, No. 10, 1990, pp. 5666-5676. doi:10.1103/PhysRevA.41.5666
[21] F. D. Lawden, “Tensor Calculus and Relativity,” Chap man and Hall, London, 1975.

Copyright © 2024 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.