Does Newtonian dynamics need Euclidean space?

TL;DR

This paper derives Newtonian force from Kepler's laws, generalizes Keplerian motion beyond Euclidean space, and explores properties of these generalized orbits and hodographs.

Contribution

It introduces a novel derivation of Newtonian force from Kepler's laws and generalizes Keplerian motion to non-Euclidean contexts.

Findings

Newtonian force can be derived from Kepler's laws.

Generalized Keplerian orbits exhibit convexity properties.

Hodographs are described in the generalized setting.

Abstract

We present an elementary deduction of the Newtonian force from Kepler's laws. We relate it to a generalization by Jacobi of the Keplerian motion, where the Euclidean form in the plane is replaced by some function with the same homogeneity. We show how several convexity properties of the generalized Keplerian orbits appear in this context. We describe the generalized hodographs.