Interpreting the Kustaanheimo-Stiefel transform in gravitational dynamics

The Kustaanheimo–Stiefel transformation in geometric algebra

The Kustaanheimo–Stiefel (KS) transformation maps the non-linear and singular equations of motion of the three-dimensional Kepler problem to the linear and regular equations of a four-dimensional harmonic oscillator. It is used extensively in studies of the perturbed Kepler problem in celestial mechanics and atomic physics. In contrast to the conventional matrixbased approach, the formulation o...

Basics of regularization theory

We consider the dynamics of three point masses, where we assume that the mass of the third body is so small that it does not affect the motion of the primaries. In the framework of the restricted three–body problem, we investigate the collisional trajectories, which correspond to a singularity of the equations of motion. We investigate the regularizing techniques known as Levi–Civita, Kustaanhe...

Time-Symmetrized Kustaanheimo-Stiefel Regularization

In this paper we describe a new algorithm for the long-term numerical integration of the two-body problem, in which two particles interact under a Newtonian gravitational potential. Although analytical solutions exist in the unperturbed and weakly perturbed cases, numerical integration is necessary in situations where the perturbation is relatively strong. Kustaanheimo{Stiefel (KS) regularizati...

Generalisation of the Levi-Civita/Kustaanheimo-Stiefel regularisation: Scaling invariance

Efficient Orbit Integration by Kustaanheimo-stiefel Regularization Using Time Element

We have found that the introduction of the time element into the Kustaanheimo-Stiefel (K-S) regularization changes the rate of error growth from quadratic to linear with respect to the physical time, even when a scaling is not applied. It also mostly wipes out the periodic error components in the case of unperturbed orbits, independently of the application of the scaling. Since the increase in ...