An explicit integrator for the 3D Kepler problem with exact time–true anomaly relation and exact preservation of integrals of motion
- 1Faculty of Physics, University of Bialystok , ul. Ciołkowskiego, Białystok PL-15-245 ,
Abstract
We develop and present a numerical scheme for the Kepler problem that preserves exactly all integrals of motion; angular momentum, total energy, and the Laplace–Runge–Lenz vector. This property ensures that orbital trajectories retain their precise shape and orientation over long times, avoiding the spurious precession typical of many standard methods. The scheme uses an adaptive time-step derived from a constant angular increment that allows determining time as a function of the true anomaly up to round-off errors. It means that the induced discretization of the Kepler problem is exact. Analytical considerations and numerical experiments demonstrate that the algorithm combines high accuracy with long-term stability, even for orbits with extreme eccentricity, where other methods fail.
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BibTeX
@article{cieslinski2026explicit,
title = {{An explicit integrator for the 3D Kepler problem with exact time–true anomaly relation and exact preservation of integrals of motion}},
author = {Jan L Cieśliński and Maciej Jurgielewicz},
journal = {Monthly Notices of the Royal Astronomical Society},
year = {2026},
month = oct,
doi = {10.1093/mnras/stag1928},
url = {https://academic.oup.com/mnras/article/doi/10.1093/mnras/stag1928/8916627},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - An explicit integrator for the 3D Kepler problem with exact time–true anomaly relation and exact preservation of integrals of motion
AU - Jan L Cieśliński
AU - Maciej Jurgielewicz
PY - 2026
DA - 2026/10/14
JO - Monthly Notices of the Royal Astronomical Society
DO - 10.1093/mnras/stag1928
UR - https://academic.oup.com/mnras/article/doi/10.1093/mnras/stag1928/8916627
AB - We develop and present a numerical scheme for the Kepler problem that preserves exactly all integrals of motion; angular momentum, total energy, and the Laplace–Runge–Lenz vector. This property ensures that orbital trajectories retain their precise shape and orientation over long times, avoiding the spurious precession typical of many standard methods. The scheme uses an adaptive time-step derived from a constant angular increment that allows determining time as a function of the true anomaly up to round-off errors. It means that the induced discretization of the Kepler problem is exact. Analytical considerations and numerical experiments demonstrate that the algorithm combines high accuracy with long-term stability, even for orbits with extreme eccentricity, where other methods fail.
SN - 0035-8711
SN - 1365-2966
ER -