The KS-transformation revisited

Abstract

We show that the Kustaanheimo-Stiefel (KS) regularizing transformation for the perturbed Kepler motion is so deeply rooted to the Keplerian orbital elements as to yield the position vector of a particle on the osculating orbit as the effect of a peculiar roto-dilatation in the physical Euclidean space. Adopting the conventional vector formulation, quaternions and spinors are also involved. A key role is played by (i): a simple hodographical approach to the integrals of the Kepler motion (angular momentum vector, Runge-Lenz vector); (ii) a polarized outlook on the attitude frame of the Kepler orbit; (iii) a simple kinematical expression for the orbital elements. The mechanical energy, the bilinear relation, the gauge transformation — fundamental in the KS-theory — are naturally arrived at, acquiring interesting kinematical interpretations.