Runge-Lenz Vector as a 3d Projection of SO(4) Moment Map in $\mathbb{R}^{4}\times\mathbb{R}^{4}$ Phase Space

TL;DR

This paper demonstrates that the Runge-Lenz vector in the Kepler problem can be understood as a 3D projection of the SO(4) moment map within a 4D phase space, revealing its geometric symmetry origin.

Contribution

It introduces a geometric algebra approach to relate the Runge-Lenz vector to the SO(4) symmetry in a 4D phase space, providing a new perspective on its origin.

Findings

Runge-Lenz vector is a projection of SO(4) moment map.

Geometric symmetry explains the Runge-Lenz vector.

Uses geometric algebra to analyze phase space symmetries.

Abstract

We show, using the methods of geometric algebra, that Runge-Lenz vector in the Kepler problem is a 3-dimensional projection of SO(4) moment map that acts on the phase space of 4-dimensional particle motion. Thus, RL vector is a consequence of geometric symmetry of $\mathbb{R}^4\times \mathbb{R}^4$ phase space.