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Celestial Mechanics and Dynamical Astronomy

, Volume 102, Issue 1–3, pp 149–162 | Cite as

Quaternions for regularizing Celestial Mechanics: the right way

  • Jörg Waldvogel
Original Article

Abstract

Quaternions have been found to be the ideal tool for describing and developing the theory of spatial regularization in Celestial Mechanics. This article corroborates the above statement. Beginning with a summary of quaternion algebra, we will describe the regularization procedure and its consequences in an elegant way. Also, an alternative derivation of the theory of Kepler motion based on regularization will be given. Furthermore, we will consider the regularization of the spatial restricted three-body problem, i.e. the spatial generalization of the Birkhoff transformation. Finally, the perturbed Kepler motion will be described in terms of regularized variables.

Keywords

Quaternions Regularization Kustaanheimo–Stiefel transformation Kepler formulas Birkhoff transformation Perturbed Kepler problem Joukowsky-Birkhoff mapping Quaternion algebra 

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References

  1. Arribas, M., Elipe, A., Palacios, M.: Quaternions and the rotation of a rigid body. Celest. Mech. Dyn. Astron. 96, 239–251 (2006)zbMATHCrossRefADSMathSciNetGoogle Scholar
  2. 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)Google Scholar
  3. Benest, D., Frœschlé, C. (eds): Singularities in Gravitational Systems, 215 pp. Lecture Notes in Physics. Springer-Verlag, Berlin (2002)Google Scholar
  4. Chelnokov, Y.N.: On regularization of the equations of the three-dimensional two-body problem. Izv. Akad. Nauk SSSR, Ser. Mekh. Tverd. Tela [Mechanics of Solids], 12–21 (1981); 151–158 (1984)Google Scholar
  5. Chelnokov, Y.N.: Application of quaternions in the theory of orbital motion of a satellite. Kosm. Issled. [Cosmic Res.] 30, 759–770 (1992); 31, 3–15 (1993)Google Scholar
  6. Chelnokov, Y.N.: Application of quaternions in the mechanics of space flight. Gyroscopy Navigation 4(27), 47–66 (1999)Google Scholar
  7. Hamilton, W.R.: On quaternions, or a new system of imaginaries in algebra. Philos. Mag. 25, 489–495 (1844)Google Scholar
  8. Hopf, H.: Über die Abbildung der dreidimensionalen Sphäre auf die Kugelfläche. Math. Ann. 104 (1931). Reprinted in Selecta Heinz Hopf, pp. 38–63. Springer-Verlag, Berlin (1964)Google Scholar
  9. Kustaanheimo, P.: Spinor regularisation of the Kepler motion. Ann. Univers. Turkuensis, Ser. A, 73, 1–7 (1964); Publ. Astr. Obs. Helsinki 102 Google Scholar
  10. Kustaanheimo, P., Stiefel, E.L.: Perturbation theory of Kepler motion based on spinor regularization. J. Reine Angew. Math. 218, 204–219 (1965)zbMATHMathSciNetGoogle Scholar
  11. Levi-Civita, T.: Sur la régularisation du problème des trois corps. Acta Math. 42, 99–144 (1920)CrossRefMathSciNetGoogle Scholar
  12. Stiefel, E.L., Scheifele, G.: Linear and Regular Celestial Mechanics, 301 pp. Springer-Verlag, Berlin (1971)Google Scholar
  13. Stiefel, E.L., Waldvogel, J.: Généralisation de la régularisation de Birkhoff pour le mouvement du mobile dans l’espace àtrois dimensions. C.R. Acad. Sci. Paris 260, 805 (1965)MathSciNetGoogle Scholar
  14. Sundman, K.F.: Recherches sur le problème des trois corps. Acta Societatis Scientificae Fennicae 34, 6 (1907)Google Scholar
  15. Vivarelli, M.D.: The KS transformation in hypercomplex form. Celest. Mech. Dyn. Astron. 29, 45–50 (1983)zbMATHMathSciNetGoogle Scholar
  16. Vrbik, J.: Celestial mechanics via quaternions. Can. J. Phys. 72, 141–146 (1994)ADSGoogle Scholar
  17. Vrbik, J.: Perturbed Kepler problem in quaternionic form. J. Phys. A 28, 193–198 (1995)CrossRefMathSciNetGoogle Scholar
  18. Waldvogel, J.: Die Verallgemeinerung der Birkhoff-Regularisierung für das räumliche Dreikörperproblem. Bulletin Astronomique, Série 3, Tome II, Fasc. 2, 295–341 (1967a)Google Scholar
  19. Waldvogel, J.: The restricted elliptic three-body problem. In: Stiefel E. et al. (eds.) Methods of Regularization for Computing Orbits in Celestial Mechanics. NASA Contractor Report NASA CR, vol. 769, pp. 88–115 (1967b)Google Scholar
  20. Waldvogel, J.: Order and chaos in satellite encounters. In: Steves, B.A. et al. (eds) Chaotic Worlds: From Order to Disorder in Gravitational N-Body Dynamical Systems, pp. 233–254. Springer, Dordrecht (2006)Google Scholar
  21. Waldvogel, J.: Quaternions and the perturbed Kepler problem. Celest. Mech. Dyn. Astron. 95, 201–212 (2006b)zbMATHCrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Swiss Federal Institute of Technology ETHZurichSwitzerland

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