Laplace–Runge–Lenz vector

In classical mechanics, the Laplace–Runge–Lenz vector (LRL vector) is a vector used chiefly to describe the shape and orientation of the orbit of one astronomical body around another, such as a binary star or a planet revolving around a star. For two bodies interacting by Newtonian gravity, the LRL vector is a constant of motion, meaning that it is the same no matter where it is calculated on the orbit;[1][2] equivalently, the LRL vector is said to be conserved. More generally, the LRL vector is conserved in all problems in which two bodies interact by a central force that varies as the inverse square of the distance between them; such problems are called Kepler problems.[3][4][5][6]

Thus the hydrogen atom is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law of electrostatics, another inverse-square central force. The LRL vector was essential in the first quantum mechanical derivation of the spectrum of the hydrogen atom,[7][8] before the development of the Schrödinger equation. However, this approach is rarely used today.

In classical and quantum mechanics, conserved quantities generally correspond to a symmetry of the system.[9] The conservation of the LRL vector corresponds to an unusual symmetry; the Kepler problem is mathematically equivalent to a particle moving freely on the surface of a four-dimensional (hyper-)sphere,[10] so that the whole problem is symmetric under certain rotations of the four-dimensional space.[11] This higher symmetry results from two properties of the Kepler problem: the velocity vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points.[12]

The Laplace–Runge–Lenz vector is named after Pierre-Simon de Laplace, Carl Runge and Wilhelm Lenz. It is also known as the Laplace vector,[13][14] the Runge–Lenz vector[15] and the Lenz vector.[8] Ironically, none of those scientists discovered it.[15] The LRL vector has been re-discovered and re-formulated several times;[15] for example, it is equivalent to the dimensionless eccentricity vector of celestial mechanics.[2][14][16] Various generalizations of the LRL vector have been defined, which incorporate the effects of special relativity, electromagnetic fields and even different types of central forces.[17][18][19]

  1. ^ Goldstein, H. (1980). Classical Mechanics (2nd ed.). Addison Wesley. pp. 102–105, 421–422.
  2. ^ a b Taff, L. G. (1985). Celestial Mechanics: A Computational Guide for the Practitioner. New York: John Wiley and Sons. pp. 42–43.
  3. ^ Goldstein, H. (1980). Classical Mechanics (2nd ed.). Addison Wesley. pp. 94–102.
  4. ^ Arnold, V. I. (1989). Mathematical Methods of Classical Mechanics (2nd ed.). New York: Springer-Verlag. p. 38. ISBN 0-387-96890-3.
  5. ^ Sommerfeld, A. (1964). Mechanics. Lectures on Theoretical Physics. Vol. 1. Translated by Martin O. Stern (4th ed.). New York: Academic Press. pp. 38–45.
  6. ^ Lanczos, C. (1970). The Variational Principles of Mechanics (4th ed.). New York: Dover Publications. pp. 118, 129, 242, 248.
  7. ^ Pauli, W. (1926). "Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik". Zeitschrift für Physik. 36 (5): 336–363. Bibcode:1926ZPhy...36..336P. doi:10.1007/BF01450175. S2CID 128132824.
  8. ^ a b Bohm, A. (1993). Quantum Mechanics: Foundations and Applications (3rd ed.). New York: Springer-Verlag. pp. 205–222.
  9. ^ Hanca, J.; Tulejab, S.; Hancova, M. (2004). "Symmetries and conservation laws: Consequences of Noether's theorem". American Journal of Physics. 72 (4): 428–35. Bibcode:2004AmJPh..72..428H. doi:10.1119/1.1591764.
  10. ^ Fock, V. (1935). "Zur Theorie des Wasserstoffatoms". Zeitschrift für Physik. 98 (3–4): 145–154. Bibcode:1935ZPhy...98..145F. doi:10.1007/BF01336904. S2CID 123112334.
  11. ^ Bargmann, V. (1936). "Zur Theorie des Wasserstoffatoms: Bemerkungen zur gleichnamigen Arbeit von V. Fock". Zeitschrift für Physik. 99 (7–8): 576–582. Bibcode:1936ZPhy...99..576B. doi:10.1007/BF01338811. S2CID 117461194.
  12. ^ Hamilton, W. R. (1847). "The hodograph or a new method of expressing in symbolic language the Newtonian law of attraction". Proceedings of the Royal Irish Academy. 3: 344–353.
  13. ^ Goldstein, H. (1980). Classical Mechanics (2nd ed.). Addison Wesley. p. 421.
  14. ^ a b Arnold, V. I. (1989). Mathematical Methods of Classical Mechanics (2nd ed.). New York: Springer-Verlag. pp. 413–415. ISBN 0-387-96890-3.
  15. ^ a b c Goldstein, H. (1975). "Prehistory of the Runge–Lenz vector". American Journal of Physics. 43 (8): 737–738. Bibcode:1975AmJPh..43..737G. doi:10.1119/1.9745.
    Goldstein, H. (1976). "More on the prehistory of the Runge–Lenz vector". American Journal of Physics. 44 (11): 1123–1124. Bibcode:1976AmJPh..44.1123G. doi:10.1119/1.10202.
  16. ^ Hamilton, W. R. (1847). "Applications of Quaternions to Some Dynamical Questions". Proceedings of the Royal Irish Academy. 3: Appendix III.
  17. ^ Landau, L. D.; Lifshitz E. M. (1976). Mechanics (3rd ed.). Pergamon Press. p. 154. ISBN 0-08-021022-8.
  18. ^ Fradkin, D. M. (1967). "Existence of the Dynamic Symmetries O4 and SU3 for All Classical Central Potential Problems". Progress of Theoretical Physics. 37 (5): 798–812. Bibcode:1967PThPh..37..798F. doi:10.1143/PTP.37.798.
  19. ^ Yoshida, T. (1987). "Two methods of generalisation of the Laplace–Runge–Lenz vector". European Journal of Physics. 8 (4): 258–259. Bibcode:1987EJPh....8..258Y. doi:10.1088/0143-0807/8/4/005. S2CID 250843588.
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