Vector constants of the motion and orbits in the Coulomb/Kepler problem

American Journal of Physics

Volumn 71, Issue 12, 2003, Pages 1292-1293

  Muñoz, Gerardo ^a  

^a CALIFORNIA STATE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0344287520     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.1596174     Document Type: Article

References (18)

1. P. S. Laplace, Celestial Mechanics (Chelsea, New York, 1969). Vol. 1, p. 344, Eqs. [572].
2. C. Runge, Vektoranalysis (Hirzel, Leipzig, 1919), Vol. 1, p. 70.
3. W. Lenz, "On the course of the motion and the quantum states of the disturbed Kepler motion," Z. Phys. 24, 197-207 (1924).
4. H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, MA, 1980). 2nd ed.
5. H. Goldstein, "More on the prehistory of the Laplace or Runge-Lenz vector," Am. J. Phys. 44, 1123-1124 (1976).
6. 2/2m + V(r,θ,φ), the only potential possessing a LapIace-Runge-Lenz-type constant of the motion is V(r, θ φ) = - k/r. See V. M. Gorringe and P. G. L. Leach, "The first integrals and their Lie algebra of the most general autonomous Hamiltonian of the form H = T + V possessing a Laplace-Runge-Lenz vector." J. Aust. Math. Soc. B, Appl. Math. 34, 511-522 (1993).
7. See. for example, J. B. Marion and S. T. Thornton, Classical Dynamics of Particles and Systems (Saunders College Publishing, Philadelphia, 1995), 4th ed.; K. R. Symon, Mechanics (Cummings, 1971), 3rd ed. The method presented in these textbooks is due to Johann Bernoulli. For an interesting historical account, see E. J. Aiton, "The contributions of Isaac Newton, Johann Bernoulli and Jakob Hermann to the inverse problem of central forces," Studia Leibnitiana, Sonderheft, edited by H.-J. Hess and F. Nagel (Franz Steiner Verlag Wiesbaden GMBH, Stuttgart, 1989), Vol. 17, pp. 48-58.
8. See. for example, J. B. Marion and S. T. Thornton, Classical Dynamics of Particles and Systems (Saunders College Publishing, Philadelphia, 1995), 4th ed.; K. R. Symon, Mechanics (Cummings, 1971), 3rd ed. The method presented in these textbooks is due to Johann Bernoulli. For an interesting historical account, see E. J. Aiton, "The contributions of Isaac Newton, Johann Bernoulli and Jakob Hermann to the inverse problem of central forces," Studia Leibnitiana, Sonderheft, edited by H.-J. Hess and F. Nagel (Franz Steiner Verlag Wiesbaden GMBH, Stuttgart, 1989), Vol. 17, pp. 48-58.
9. See. for example, J. B. Marion and S. T. Thornton, Classical Dynamics of Particles and Systems (Saunders College Publishing, Philadelphia, 1995), 4th ed.; K. R. Symon, Mechanics (Cummings, 1971), 3rd ed. The method presented in these textbooks is due to Johann Bernoulli. For an interesting historical account, see E. J. Aiton, "The contributions of Isaac Newton, Johann Bernoulli and Jakob Hermann to the inverse problem of central forces," Studia Leibnitiana, Sonderheft, edited by H.-J. Hess and F. Nagel (Franz Steiner Verlag Wiesbaden GMBH, Stuttgart, 1989), Vol. 17, pp. 48-58.
10. 3 for all classical potential problems," Prog. Theor. Phys. 37. 798-812 (1967); T. Yoshida, "Determination of the generalized Laplace-Runge-Lenz vector by an inverse matrix method," Am. J. Phys. 57, 376-377 (1989); C. C. Yan, "Determination of vector constant of motion for a particle moving in a conservative force field," J. Phys. A 24, 4731-4738 (1991). However, these constants of the motion do not entail a degeneracy unless they are single-valued or at most finitely-multivalued. See P. Stehle and M. Y. Han, "Symmetry and degeneracy in classical mechanics," Phys. Rev. 159, 1076-1082(1967).
11. 3 for all classical potential problems," Prog. Theor. Phys. 37. 798-812 (1967); T. Yoshida, "Determination of the generalized Laplace-Runge-Lenz vector by an inverse matrix method," Am. J. Phys. 57, 376-377 (1989); C. C. Yan, "Determination of vector constant of motion for a particle moving in a conservative force field," J. Phys. A 24, 4731-4738 (1991). However, these constants of the motion do not entail a degeneracy unless they are single-valued or at most finitely-multivalued. See P. Stehle and M. Y. Han, "Symmetry and degeneracy in classical mechanics," Phys. Rev. 159, 1076-1082(1967).
12. 3 for all classical potential problems," Prog. Theor. Phys. 37. 798-812 (1967); T. Yoshida, "Determination of the generalized Laplace-Runge-Lenz vector by an inverse matrix method," Am. J. Phys. 57, 376-377 (1989); C. C. Yan, "Determination of vector constant of motion for a particle moving in a conservative force field," J. Phys. A 24, 4731-4738 (1991). However, these constants of the motion do not entail a degeneracy unless they are single-valued or at most finitely-multivalued. See P. Stehle and M. Y. Han, "Symmetry and degeneracy in classical mechanics," Phys. Rev. 159, 1076-1082(1967).
13. 3 for all classical potential problems," Prog. Theor. Phys. 37. 798-812 (1967); T. Yoshida, "Determination of the generalized Laplace-Runge-Lenz vector by an inverse matrix method," Am. J. Phys. 57, 376-377 (1989); C. C. Yan, "Determination of vector constant of motion for a particle moving in a conservative force field," J. Phys. A 24, 4731-4738 (1991). However, these constants of the motion do not entail a degeneracy unless they are single-valued or at most finitely-multivalued. See P. Stehle and M. Y. Han, "Symmetry and degeneracy in classical mechanics," Phys. Rev. 159, 1076-1082(1967).
14. W. R. Hamilton, "On the applications of the method of quaternions to some dynamical questions," in The Mathematical Papers of Sir William Rowan Hamilton, edited by H. Halberstam and R. E. Ingram (Cambridge U.P., Cambridge, 1967), Vol. III, pp. 441-448.
15. A number of alternative formulations of Eq. (4) have appeared in the pages of this journal, however. See H. Abelson, A. diSessa, and L. Rudolph, "Velocity space and the geometry of planetary orbits," Am. J. Phys. 43, 579-589 (1975); R. P. Patera, "Momentum-space derivation of the Runge-Lenz vector," ibid. 49, 593-594 (1981); D. Derbes, "Reinventing the wheel: Hodographic solutions to the Kepler problems." ibid. 69, 481-489 (2001). The present note should be regarded as a brief complement to these interesting papers.
16. A number of alternative formulations of Eq. (4) have appeared in the pages of this journal, however. See H. Abelson, A. diSessa, and L. Rudolph, "Velocity space and the geometry of planetary orbits," Am. J. Phys. 43, 579-589 (1975); R. P. Patera, "Momentum-space derivation of the Runge-Lenz vector," ibid. 49, 593-594 (1981); D. Derbes, "Reinventing the wheel: Hodographic solutions to the Kepler problems." ibid. 69, 481-489 (2001). The present note should be regarded as a brief complement to these interesting papers.
17. A number of alternative formulations of Eq. (4) have appeared in the pages of this journal, however. See H. Abelson, A. diSessa, and L. Rudolph, "Velocity space and the geometry of planetary orbits," Am. J. Phys. 43, 579-589 (1975); R. P. Patera, "Momentum-space derivation of the Runge-Lenz vector," ibid. 49, 593-594 (1981); D. Derbes, "Reinventing the wheel: Hodographic solutions to the Kepler problems." ibid. 69, 481-489 (2001). The present note should be regarded as a brief complement to these interesting papers.
18. A derivation of the Laplace-Runge-Lenz vector from first principles may be found in H. Kaplan, "The Runge-Lenz vector as an 'extra' constant of the motion," Am. J. Phys. 54, 157-161 (1986). The method Kaplan attributes to E. T. Whittaker was in effect used by Laplace in Ref. 1 over a century earlier.