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.
References
[1]
Serre, J.P. (1973) A Course in Arithmetic, Graduate Texts in Mathematics. Springer.
[2]
Chelnokov, Y.N. (2022) Quaternion Methods and Models of Regular Celestial Mechanics and Astrodynamics. AppliedMathematicsandMechanics, 43, 21-80. https://doi.org/10.1007/s10483-021-2797-9
[3]
Goldstein, H. (1980) Classical Mechanics. 2nd Edition, Addison-Wesley.
[4]
Kustaanheimo, P., Schinzel, A., Davenport, H. and Stiefel, E. (1965) Perturbation Theory of Kepler Motion Based on Spinor Regularization. Journal für die reine und angewandte Mathematik, 1965, 204-219. https://doi.org/10.1515/crll.1965.218.204
[5]
Stiefel, E.L. and Scheifele, G. (1971) Linear and Regular Celestial Mechanics. Springer-Verlag.
[6]
Vrbik, J. (2023) New Methods of Celestial Mechanics. Bantham Science Publishers.
[7]
Rudin, W. (1987) Real and Complex Analysis. 3rd Edition, McGraw-Hill.
[8]
Cary, J.R. (1981) Lie Transform Perturbation Theory for Hamiltonian Systems. PhysicsReports, 79, 129-159. https://doi.org/10.1016/0370-1573(81)90175-7
[9]
Boccaletti, D. and Pucacco, G. (1999) Theory of Orbits. Volume 2: Perturbative and Geometrical Methods. Springer-Verlag.
[10]
Vrbik, J. (1997) Oblateness Perturbations to Fourth Order. Monthly Notices of the Royal Astronomical Society, 291, 65-70. https://doi.org/10.1093/mnras/291.1.65
[11]
Vrbik, J. (2009) Second Erratum: Oblateness Perturbations to Fourth Order. Monthly Notices of the Royal Astronomical Society, 399, 1088. https://doi.org/10.1111/j.1365-2966.2009.15412.x
[12]
Vrbik, J. (1995) Perturbed Kepler Problem in Quaternionic Form. Journal of Physics A: Mathematical and General, 28, 6245-6252. https://doi.org/10.1088/0305-4470/28/21/027
[13]
Vrbik, J. (2001) Quaternionic Processor. Celestial Mechanics and Dynamical Astronomy, 80, 111-118. https://doi.org/10.1023/a:1011979701759
[14]
Vrbik, J. (1996) Resonance Formation of Kirkwood Gaps and Asteroid Clusters. Journal of Physics A: Mathematical and General, 29, 3311-3316. https://doi.org/10.1088/0305-4470/29/12/033
