Generalisations of the Laplace-Runge-Lenz vector

TL;DR

This paper explores various problems related to the Kepler problem that possess conserved vectors similar to the Laplace-Runge-Lenz vector, broadening understanding of orbital dynamics and symmetries.

Contribution

It identifies and analyzes classes of problems sharing conserved vectors akin to the Laplace-Runge-Lenz vector, extending the concept beyond the classical Kepler problem.

Findings

Multiple classes of problems share conserved vectors similar to the Laplace-Runge-Lenz vector.

Conserved vectors play a significant role in analyzing these generalized problems.

The study broadens the understanding of symmetries in orbital mechanics.

Abstract

The characteristic feature of the Kepler Problem is the existence of the so-called Laplace--Runge--Lenz vector which enables a very simple discussion of the properties of the orbit for the problem. It is found that there are many classes of problems, some closely related to the Kepler Problem and others somewhat remote, which share the possession of a conserved vector which plays a significant r\^ole in the analysis of these problems.