What is a Kustaanheimo-Stiefel transformation? What is a Kustaanheimo-Stiefel transformation? Which applications has it in physics? Can you point me to a reference, where this transformation is explained?
 A: http://arxiv.org/abs/0803.4441

The Kustaanheimo-Stiefel transform turns a gravitational two-body problem into a harmonic oscillator, by going to four dimensions. In addition to the mathematical-physics interest, the KS transform has proved very useful in N-body simulations, where it helps handle close encounters. Yet the formalism remains somewhat arcane, with the role of the extra dimension being especially mysterious. This paper shows how the basic transformation can be interpreted as a rotation in three dimensions

A: Concerning applications to quantum mechanics, the K-S transformation has been used to calculate the Green's function of the hydrogen atom via path integrals. See, for example, Ho & Inomata, Phys. Rev. Lett. 48, 231–234 (1982).
A: The Kustaanheimo-Stiefel transformation is a special case of the Hurwitz transformation. 

In classical mechanics, the Kustaanheimo–Stiefel transformation is used for the regularization of the Kepler problem. In quantum mechanics, the latter transformation makes it possible to transform the Schrödinger equation for the three-dimensional hydrogen atom (in an electromagnetic field) into a Schrödinger equation for a four-dimensional isotropic harmonic oscillator.

It can also be derived by introducing parabolic coordinates, see Yoshida (1982): A New Derivation of the Kustaanheimo-Stiefel Variables.
