Symmetry transformation group arising from the Laplace-Runge-Lenz vector

TL;DR

This paper derives the symmetry group associated with the direction of the Laplace-Runge-Lenz vector in the Kepler problem, revealing a semi-direct product structure different from the traditional SO(4) symmetry.

Contribution

It explicitly formulates the infinitesimal dynamical symmetry related to the LRL vector's direction and identifies the resulting symmetry group as a semi-direct product of SO(3) and R^3.

Findings

The symmetry group is the semi-direct product of SO(3) and R^3.

The explicit form of the symmetry transformations is obtained.

The action of these symmetries on conserved quantities is characterized.

Abstract

The Kepler problem in classical mechanics exhibits a rich structure of conserved quantities, highlighted by the Laplace--Runge--Lenz (LRL) vector. Through Noether's theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which is well known in the literature. However, the physically relevant part of the LRL vector is its direction angle in the plane of motion (since its magnitude is just a function of energy and angular momentum). The present work derives the infinitesimal dynamical symmetry corresponding to the direction part of the LRL vector, and obtains the explicit form of the symmetry transformations that it generates. When combined with the rotation symmetries,the resulting symmetry group is shown to be the semi-direct product of $SO(3)$ and $R^3$. This stands in contrast to the $SO(4)$ symmetry group generated by the LRL symmetries and the rotations. As a by-product, the action of the new infinitesimal symmetries on all of the conserved quanties is obtained. The results are given in terms of the physical kinematical variables in the Kepler problem, rather than in an enlarged auxiliary space in which the LRL symmetries are usually stated.