Yet another approach to the inverse square law and to the circular character of the hodograph of Kepler orbits

TL;DR

This paper presents a geometric approach to derive the inverse square law and the circular hodograph of Kepler orbits, avoiding differential equations and based on new theorems about geometric means and off-center circles.

Contribution

It introduces two mathematical theorems that underpin a geometric derivation of Keplerian motion and hodographs, providing a novel foundation independent of differential calculus.

Findings

Derivation of the inverse square law from geometric principles

Proof of the circular character of hodographs for Kepler orbits

Development of a parametric equation for off-center circles

Abstract

The law of centripetal force governing the motion of celestial bodies in eccentric conic sections, has been established and thoroughly investigated by Sir Isaac Newton in his Principia Mathematica. Yet its profound implications on the understanding of such motions is still evolving. In a paper to the royal academy of science, Sir Willian Hamilton demonstrated that this law underlies the circular character of hodographs for Kepler orbits. A fact which was the object of ulterior research and exploration by Richard Feynman and many other authors [1]. In effect, a minute examination of the geometry of elliptic trajectories, reveals interesting geometric properties and relations, altogether, combined with the law of conservation of angular momentum lead eventually, and without any recourse to dealing with differential equations, to the appearance of the equation of the trajectory and to the derivation of the equation of its corresponding hodograph. On this respect, and for the sake of founding the approach on solid basis, I devised two mathematical theorems; one concerning the existence of geometric means, and the other is related to establishing the parametric equation of an off-center circle, altogether compounded with other simple arguments ultimately give rise to the inverse square law of force that governs the motion of bodies in elliptic trajectories, as well as to the equation of their inherent circular hodographs.