On the force fields which are homogeneous of degree $-3$

TL;DR

This paper explores the special properties of force fields that are positively homogeneous of degree -3, showing how they enable phase space reduction and relate to classical dynamics on spheres and ellipsoids.

Contribution

It establishes a connection between homogeneous force fields of degree -3, phase space reduction, and classical geometric dynamics, providing a foundation for Appell's projective dynamics.

Findings

Phase space reduces by two dimensions for degree -3 force fields.

Relation between degree -3 homogeneity and phase space reduction.

Connection to Neumann potential and geodesic motion on ellipsoids.

Abstract

The dynamics defined by a force field which is positively homogeneous of degree $-3$ can always be reduced, by simply constraining it. The dimension of the phase space is reduced by two dimensions, while it may only be reduced by one dimension if the degree of homogeneity is different from $-3$. This remark is an elegant foundation of Appell's projective dynamics. We show how it relates to Kn\"orrer's remark on the correspondence between the Neumann potential on a sphere and the geodesic motion on an ellipsoid.