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

TL;DR

This paper introduces a numerical integrator for the 3D Kepler problem that exactly preserves all physical invariants, ensuring accurate and stable long-term orbital simulations.

Contribution

It presents a novel explicit integrator that maintains all integrals of motion exactly, unlike traditional methods that often introduce errors over time.

Findings

Preserves angular momentum, energy, and Laplace-Runge-Lenz vector exactly.

Ensures orbital trajectories retain their shape and orientation over long simulations.

Combines high accuracy with long-term stability in numerical experiments.

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.