arXiv subjects

Maciej Jurgielewicz

Publications and source records attributed to Maciej Jurgielewicz.

2 recordsLinked to original sources

arXiv2025

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

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.

math.NA↗

arXiv2023

On geometric interpretation of Euler's substitutions

We consider a classial case of irrational integrals containing a square root of a quadratic polynomial. It is well known that they can be expressed in terms of elementary functions by one of three Euler's substitutions. It is less known that the Euler substittutions have a beautiful geometric interpretation. In the framework of this interpretation one can see that the number 3 is not the most suitable. We show that it is natural to introduce the fourth Euler substitution. By the way, it is not clear who was the first to attribute these three substitutions to Euler. In his original treatise Leonhard Euler uses two substitutions which are sufficient to cover all cases.

math.HO↗