Duality properties of Gorringe-Leach equations

TL;DR

This paper explores the duality relationships between two classes of differential equations that preserve angular momentum direction and have elliptical orbits, revealing a deeper symmetry involving conserved quantities.

Contribution

It introduces a duality correspondence between Gorringe-Leach equations and extends their classes, linking conserved quantities through a novel duality framework.

Findings

The classes are related by a duality of the Arnold-Vassiliev type.

Conserved quantities are dual reflections of each other.

Extended classes maintain elliptical orbit solutions.

Abstract

In the category of motions preserving the angular momentum's direction, Gorringe and Leach exhibited two classes of differential equations having elliptical orbits. After enlarging slightly these classes, we show that they are related by a duality correspondence of the Arnold-Vassiliev type. The specific associated conserved quantities (Laplace-Runge-Lenz vector and Fradkin-Jauch-Hill tensor) are then dual reflections one of the other