Lie-Algebraic Methods in Dynamics and Celestial Mechanics

  • Joachim Baumgarte
Conference paper
Part of the NATO Advanced Study Institutes Series book series (ASIC, volume 82)

Abstract

The connection between the oscillator problem and the Keplerian motion is considered.

First it is shown that the Lie algebra so(3,2) is characteristic for the two-dimensional oscillator problem. After a point transformation (Levi-Civita) this algebra describes the two-dimensional Kepler problem, provided the eccentric anomaly is used as the independent variable.

Secondly, the algebra so(3,2) can be extended to an algebra so (4,2) which yields the guiding principle for reformulating all the Keplerian formulas in three dimensions.

The concretisation of an abstract isomorphism between the two Lie algebras so (4,2) leads on one side to the KS-transformation and on the other side to the transformation into Delaunay similar elements in the eccentric anomaly.

References

  1. Baumgarte, J., (1978), Das Oszillator-Kepler Problem und die Lie-Algebra, Journal fur die reine und angewandte Mathematik, 301, p. 59–76, Walter de Gruyter, Berlin, New York.MathSciNetMATHCrossRefGoogle Scholar
  2. Baumgarte, J., (1980), Eine Lie-Algebra, die Delaunay-similar-Elemente in der exzentrischen Anomalie erzeugt, J. Phys. A: Math. Gen., 13, pp. 1145–1158, printed in Great Britain, The Institute of Physics.MathSciNetADSMATHCrossRefGoogle Scholar
  3. Baugmgarte, J., (1979), Eine kanonische Transformation für die Kepler-Bewegung, welche ohne Dimensionserhöhung die Zeit in die exzentrische Anomalie überführt., Zeitschrift für Angewandte Mathematik un Physik (ZAMP), 30.Google Scholar
  4. Baumgarte, J., (1979), Die Invarianzgruppe des Oszillator-Kepler-Problems., ZAMM, 59, pp. 177–187.MathSciNetADSMATHCrossRefGoogle Scholar

Copyright information

© D. Reidel Publishing Company 1982

Authors and Affiliations

  • Joachim Baumgarte
    • 1
  1. 1.MechanikzentrumTechnischen Universität BraunschweigGermany

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