Quaternions for Regularizing Celestial Mechanics - the Right Way

by Jörg Waldvogel
Seminar for Applied Mathematics
Swiss Federal Institute of Technology ETH
CH-8092 Zürich, Switzerland


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.

Download a preliminary version of the paper (19 pages) "Quaternions for regularizing celestial mechanics - the right way": spolpaper.pdf
Accepted by Celestial Mechanics and Dynamical Astronomy, February 8, 2008. Published on-line March 18, 2008. The original publication is available at http://www.springerlink.com/content/c82l8x187t5w6086/fulltext.pdf

View the presentation (27 frames), "Quaternions for regularizing celestial mechanics - the right way",
given at the meeting "Theory and Applications of Dynamical Systems" in honor of Claude Froeschlé
Spoleto, Italy, June 24 - 28, 2007 spoleto.pdf

View the presentation (18 frames), "Theory of Kepler Motion by Regularization",
given at the meeting CELMEC 5, San Martino al Cimino, Viterbo, Italy, September 6 - 12, 2009 viter09.pdf

A forerunner: Quaternions and the perturbed Kepler problem, 2006

Quaternions, introduced by W. R. Hamilton (1844) as a generalization of complex numbers, lead to a remarkably simple representation of the regularization of the spatial case of binary collisions in celestial mechanics. The transformation suggested by Kustaanheimo and Stiefel (KS) in 1964 may be handled in complete formal agreement with the planar case regularized by Levi-Civita (1920) by means of a conformal squaring.

Download the complete paper (14 pages), appeared in "Celestial Mechanics and Dynamical Astronomy" 95 (2006), 201 - 212: viterbo.pdf