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
Download a preliminary version of the paper (19 pages) "Quaternions for
regularizing celestial mechanics - the right way":
Accepted by Celestial Mechanics and Dynamical Astronomy, February 8, 2008.
Published on-line March 18, 2008. The original publication is available at
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
View the presentation (18 frames), "Theory of Kepler Motion by
given at the meeting CELMEC 5,
San Martino al Cimino, Viterbo, Italy, September 6 - 12, 2009
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: