Laplace-Runge-Lenz symmetry in general rotationally symmetric systems

TL;DR

This paper demonstrates that the Laplace-Runge-Lenz symmetry is universally present in all rotationally symmetric systems, regardless of interaction type, and extends its applicability to relativistic and post-Newtonian systems.

Contribution

It proves the universality of the Laplace-Runge-Lenz symmetry using generic Poisson bracket properties and defines generalized vectors that are constant across various systems.

Findings

Laplace-Runge-Lenz vectors are constant in all rotationally symmetric systems.

The symmetry's relativistic origin is supported across all centrally symmetric systems.

Applications include relativistic Coulomb and gravitational systems in post-Newtonian approximation.

Abstract

The universality of the Laplace-Runge-Lenz symmetry in all rotationally symmetric systems is discussed. The independence of the symmetry on the type of interaction is proven using only the most generic properties of the Poisson brackets. Generalized Laplace-Runge-Lenz vectors are definable to be constant (not only piece-wise conserved) for all cases, including systems with open orbits. Applications are included for relativistic Coulomb systems and electromagnetic/gravitational systems in the post-Newtonian approximation. The evidence for the relativistic origin of the symmetry are extended to all centrally symmetric systems.