Interpreting the Kustaanheimo-Stiefel transform in gravitational dynamics

TL;DR

This paper provides a geometric interpretation of the Kustaanheimo-Stiefel transform in gravitational dynamics, revealing it as a rotation in three dimensions using quaternion algebra, which clarifies its role in N-body simulations.

Contribution

It offers a novel geometric interpretation of the KS transform as a rotation in three dimensions, simplifying understanding and derivation of regularized equations.

Findings

KS transform can be viewed as a rotation in 3D space.

Quaternion form simplifies the mathematical expressions.

The extra dimension corresponds to the rotation axis ambiguity.

Abstract

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. For example, if we slew a telescope from zenith to a chosen star in one rotation, we can think of the rotation axis and angle as the KS transform of the star. The non-uniqueness of the rotation axis encodes the extra dimension. This geometrical interpretation becomes evident on writing KS transforms in quaternion form, which also helps derive concise expressions for regularized equations of motion.