On 20/10/2026 at 1:15 PM, in room 2011 of the Faculty of Physics, University of Białystok, prof. dr hab. Jan Cieśliński, Faculty of Physics, University of Bialystok will give a lecture entitled:
„Geometric discretizations of the classical Kepler problem: integrators preserving trajectories and all integrals of motion.”
Tomasz Karpiuk
Andrzej Maziewski
„Geometric discretizations of the classical Kepler problem: integrators preserving trajectories and all integrals of motion.”
prof. dr hab. Jan Cieśliński
Faculty of Physics, University of Bialystok
The Kepler problem is a maximally superintegrable dynamical system, possessing five functionally independent integrals of motion: the energy, the angular momentum vector, and the Laplace-Runge-Lenz vector. We recall a classical numerical scheme based on the Kustaanheimo-Stiefel regularization, which transforms Kepler motion into a harmonic oscillator in higher dimensions and provides a natural framework for conservative discretizations. Then, we present an explicit integrator uniform in the true anomaly, recently developed in joint work with Maciej Jurgielewicz (Monthly Notices of the Royal Astronomical Society, in press), which preserves exactly all classical integrals of motion and additionally provides an explicit exact time discretization. Both numerical schemes can be considered as special cases of a huge family of discretizations inspired by Nambu mechanics, where evolution is generated by multiple integrals of motion treated as ``Hamiltonians'' in the Nambu bracket. To the best of our knowledge, such Nambu-type integrators have not yet been systematically applied to the Kepler problem. This work is in progress. We emphasize that preserving all first integrals ensures the numerical trajectory reproduces the exact orbital shape without any distortion or artificial precession.
