TITLE:
Solving Perturbed Kepler Problem by Quaternion Algebra
AUTHORS:
Jan Vrbik
KEYWORDS:
Quaternions, Kustaanheimo-Stiefel Transformation, Kepler Problem, Perturbation Theory, Orbital Elements
JOURNAL NAME:
Applied Mathematics,
Vol.17 No.6,
June
23,
2026
ABSTRACT: In this article, we use quaternions together with Kustaanheimo-Stiefel transformation to solve the Kepler problem, first its unperturbed version and then when adding a small perturbing force. We assume that the perturbing force is autonomous, which enables us to derive a set of first-order differential equations for the corresponding orbital elements, while keeping them fully autonomous as well. This is achieved without compromising the resulting accuracy; there is no need for averaging out fast oscillations or involving a fast-angle variable. As a consequence, the equations can be solved to arbitrary accuracy by iterative and routine application of the new formulas.