Quaternions and the perturbed Kepler problem

  • Jörg Waldvogel
Conference paper


Quaternions, introduced by Hamilton (Philos. Mag. 25, 489–495, 1844) as a generalization of complex numbers, lead to a remarkably simple representation of the perturbed three-dimensional Kepler problem as a perturbed harmonic oscillator. The paper gives an overview of this technique, including an outlook to applications in perturbation theories.


Kustaanheimo-Stiefel regularization Quaternions Perturbed Kepler problem Birkhoff transformation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Birkhoff, G.D.: The restricted problem of three bodies. Rendiconti del Circolo Matematico di Palermo 39, 1 (1915); Reprinted in Collected Mathematical Papers, vol. 1. Dover Publications, New York (1968)CrossRefGoogle Scholar
  2. Celletti, A.: The Levi-Civita, KS and radial-inversion regularizing transformations. In: Benest, D., Frœschlé, C. (eds.), Singularities in Gravitational Systems, Lecture Notes in Physics, pp. 25–48. Springer-Verlag, Berlin, Heidelberg, New York (2002)Google Scholar
  3. Hamilton, W.R.: On quaternions, or a new system of imaginaries in algebra. Philos. Mag. 25, 489–495 (1844)Google Scholar
  4. Hopf, H.: Über die Abbildung der dreidimensionalen Sphäre auf die Kugelfläche. Math. Ann. 104 (1931); Reprinted in Selecta Heinz Hopf, 38—63. Springer-Verlag, Berlin, Heidelberg, New York (1964)Google Scholar
  5. Kustaanheimo, P.: Spinor regularization of the Keplermotion. Ann. Univ. Turku, Ser. A 73, 1–7 (1964); Publ. Astr. Obs. Helsinki 102 MathSciNetGoogle Scholar
  6. Kustaanheimo, P., Stiefel, E.L.: Perturbation theory of Kepler motion based on spinor regularization. J. Reine Angew. Math. 218, 204–219 (1965)MathSciNetzbMATHGoogle Scholar
  7. Levi-Civita, T.: Sur la régularisation du problème des trois corps. Acta Math. 42, 99–144 (1920)CrossRefMathSciNetGoogle Scholar
  8. Siegel, C.L., Moser, J.K.: Lectures on Celestial Mechanics, 290 pp. Springer-Verlag, Berlin, Heidelberg, New York (1971)zbMATHGoogle Scholar
  9. Stiefel, E.L., Scheifele, G.: Linear and Regular Celestial Mechanics, 301 pp. Springer-Verlag, Berlin, Heidelberg, New York (1971)zbMATHGoogle Scholar
  10. Stiefel, E.L., Waldvogel, J.: Généralisation de la régularisation de Birkhoff pour le mouvement du mobile dans l’espace à trois dimensions. C.R. Acad. Sc. Paris 260, 805 (1965)MathSciNetzbMATHGoogle Scholar
  11. Vivarelli, M.D.: The KS transformation revisited. Meccanica 29, 15–26 (1994)CrossRefzbMATHGoogle Scholar
  12. Vrbik, J.: Celestial mechanics via quaternions. Can. J. Phys. 72, 141–146 (1994)ADSGoogle Scholar
  13. Vrbik, J.: Perturbed Kepler problem in quaternionic form. J. Phys. A 28, 193–198 (1995)CrossRefMathSciNetGoogle Scholar
  14. Waldvogel, J.: Die Verallgemeinerung der Birkhoff-Regularisierung für das räumliche Dreikörperproblem. Bull. Astronomique, Série 3, Tome II, Fasc. 2, 295–341 (1967a)Google Scholar
  15. Waldvogel, J.: The restricted elliptic three-body problem. In: Stiefel, E., Rössler, M., Waldvogel, J., Burdet, C.A. (eds.), Methods of Regularization for Computing Orbits in Celestial Mechanics. NASA Contractor Report NASA CR 769, pp. 88–115 (1967b)Google Scholar
  16. Waldvogel, J.: Order and chaos in satellite encounters. In: Steves, B.A., Maciejewski, A.J., Hendry, M. (eds.), Chaotic Worlds: From Order to Disorder in Gravitational N-Body Dynamical Systems, pp. 233–254. Springer, Dordrecht (2006)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2006

Authors and Affiliations

  • Jörg Waldvogel
    • 1
  1. 1.Seminar for Applied MathematicsSwiss Federal Institute of Technology ETHZurichSwitzerland

Personalised recommendations