An explicit integrator uniform in the true anomaly and exactly preserving all integrals of motion in the three-dimensional Kepler problem

2025/12/13 by Jan L. Cieśliński, Cieśliński, Jan L., Maciej Jurgielewicz +1
Mathematics · Engineering · Physics and Astronomy · #Numerical methods for differential equations #Spacecraft Dynamics and Control #Pulsars and Gravitational Waves Research

paper · pdf · doi:10.48550/arxiv.2512.12099

Abstract

We develop a numerical scheme for the Kepler problem that preserves exactly all first integrals: 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. Analytical considerations and numerical experiments demonstrate that the algorithm combines high accuracy with long-term stability.

Citations

• An adaptive symplectic integrator for gravitational dynamics
• Improving the accuracy of the AVF method
• Construction of higher order symplectic integrators
• Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations

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• Elementary Solution of Kepler Problem (and a few other problems)
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