Scientific Research

An Academic Publisher

**Hodographs of the Gravitational Two-Body System and Discrepancies between Newtonian Laws of Equivalent Kepler Orbits and General Relativity** ()

We show first that an orbit, which is naturally characterized by its eccentricity and semi-latus rectum, can equally be characterized by other sets of parameters, and proceed to determine mass-independent characterizations. The latter is employed to obtain the laws of equivalent orbits, which by definition have the same eccentricity and orbit’s parameter [1]. These laws relate the values of the same physical observables on two equivalent orbits to the corresponding total mass; they include the laws of velocity, angular velocity, radial velocity, areal velocity, acceleration, period, energy and angular momentum. Regardless of the share of the two bodies of a fixed total mass, the same relative orbit occurs for the same initial conditions. Moreover, the same orbit can be traced by different total masses but with different relative velocities. The concept of a gravitational field generated by a set of masses is shown to be meaningful only when the center of mass is not changed by the test mass. The associated concept of the “nothing”, which is an infinitesimal mass that allows for the property just mentioned to be fulfilled, is introduced and its orbits are determined. The perturbation of the nothing orbits due to its replacement by a finite mass is determined. It is proved that such a replacement can have a qualitative effect resulting in a “phase transition” of an orbit from unbound to bound, and that the nothing’s circular orbits cannot be occupied by any material body. The Galileo law of free fall, on which the equivalence principle hinges and which is exact only for “nothing-like” falling objects, is revised to determine the duration of free fall of a body of an arbitrary mass. The wholeness of Newton’s laws and the associated concept of force as an interaction are highlighted, and some contradictions between the Newtonian laws of equivalent Kepler’s orbits and the general relativistic predictions are discussed. It is demonstrated that Newton’s law of gravitation is not an approximation of Einstein field Equations even in the case of a static weak field. However, both theories have a common limit corresponding to the case in which the alien concept of a field can be incorporated in the Newtonian theory. We also show that the relative velocity’s hodograph [2-4], the alternative Laplace-Runge-Lenz (LRL) vector derived by Hamilton [4-6], as well as an infinite set of LRL vectors, result all from one vector. The hodograph is a proper circular arc for hyperbolic motion, a circle less a point for parabolic motion, and a full circle for bound motion.

Share and Cite:

*Applied Mathematics*, Vol. 5 No. 1, 2014, pp. 55-69. doi: 10.4236/am.2014.51007.

Conflicts of Interest

The authors declare no conflicts of interest.

[1] | C. P. 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. |

[2] | A. Alemi, “Laplace-Runge-Lenz Vector,” 2009. www.cds.Caltech.edu/Wiki/Alemicds205final.pdf |

[3] | E. L. Butikov, “The Velocity Hodograph for Arbitrary Keplerian Motion,” European Journal of Physics, Vol. 21, No. 4, 2000, pp. 1-6. |

[4] |
Wikipedia, “Laplace-Runge-Lenz Vector,” 2013. http://en.wikipedia.org/wiki/Laplace%E2%80%93Runge%E2%80%93Lenz_vector |

[5] | 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. |

[6] | W. R. Hamilton, “Applications of Quaternions to Some Dynamical Questions,” Proceedings of the Royal Irish Academy, Vol. 3, Appendix III, 1847, p. xxxvi-1. |

[7] | S. W. Groesberg, “Advanced Mechanics,” John Wiley & Sons, Inc., Hopoken, 1998. |

[8] | H. Goldstein, C. P. Poole and J. L. Safko, “Classical Mechanics,” Addison Wesley, Boston, 2001. |

[9] | S. R. Spiegel, “Theoretical Mechanics,” Schaum Outline Series, McGraw Hill Book Company, New York, 1967. |

[10] | W. Rindler, “Essential Relativity,” Springr-Verlag, Berlin, 2006. |

[11] | F. D. Lawden, “Tensor Calculus and Relativity,” Chapman and Hall, London, 1975. |

[12] | L. D. Landau and E. M. Lifshitz, “The Classical Theory of Fields,” Pergamon International Library, Pergamon, 1980. |

[13] | http://en.wikipedia.org/wiki/Equivalence_principle |

[14] | A. Einstein, “Relativity, the Special and General Theory,” Henry Holt and Company, New York, 1920. |

[15] | C. Pollock. http://www.math.toronto.edu/~colliand/426_03/Papers03/C_Pollock.pdf |

[16] | L. P. Eisenhart, “Riemannian Geometry,” Princeton University Press, Princeton, 1968. |

Copyright © 2020 by authors and Scientific Research Publishing Inc.

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