Comment on `conservative discretizations of the Kepler motion

We show that the exact integrator for the classical Kepler motion, recently found by Kozlov ({\it J. Phys. A: Math. Theor.} {\bf 40} (2007) 4529-4539), can be derived in a simple natural way (using well known exact discretization of the harmonic oscillator). We also turn attention on important earlier references, where the exact discretization of the 4-dimensional isotropic harmonic oscillator has been applied to the perturbed Kepler problem.

💡 Research Summary

The paper revisits the “exact integrator” for the classical Kepler problem that was recently introduced by Kozlov (J. Phys. A 40, 2007). The author shows that this integrator is not a novel invention but follows directly from the well‑known exact discretization of the harmonic oscillator when the Kepler problem is regularized by the Levi‑Civita (or Kustaanheimo‑Stiefel) transformation.

First, the author recalls that the Kepler equations can be mapped to a four‑dimensional isotropic harmonic oscillator by introducing a complex two‑vector (Q) and a fictitious time (s) defined through (dt = r,ds), where (r) is the radial distance. In the new variables the equations of motion become the linear system
\