Interpreting the Kustaanheimo-Stiefel transform in gravitational dynamics

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.

To appear in MNRAS

References in corpus (3)

• The Gravitational Million-Body Problem
• A class of symplectic integrators with adaptive timestep for separable Hamiltonian systems
• The Kustaanheimo-Stiefel transformation in geometric algebra

Cited by in corpus (12)

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• Kustaanheimo-Stiefel Regularization and the Quadrupolar Conjugacy
• Kustaanheimo-Stiefel transformation with an arbitrary defining vector
• The Stark problem in the Weierstrassian formalism
• Stability and chaos in Kustaanheimo-Stiefel space induced by the Hopf fibration
• The Lissajous-Kustaanheimo-Stiefel transformation
• A complete and explicit solution to the three-dimensional problem of two fixed centres
• Integrability of close encounters in the spatial restricted three-body problem
• An analytic solution for weak-field Schwarzschild geodesics
• Symplectic structure of equilibrium thermodynamics
• Runge-Lenz Vector as a 3d Projection of SO(4) Moment Map in ${\mathbb{R}}^4\times {\mathbb{R}}^4$ Phase Space
• Perturbed Kepler-problem in general relativity with quaternions