A new look at the Feynman `hodograph' approach to the Kepler first law

TL;DR

This paper revisits Feynman's geometric approach to deriving Kepler's first law using hodographs, extends it to hyperbolic orbits, and explores related properties and generalizations to curved spaces.

Contribution

It provides a geometric extension of Feynman's method to hyperbolic orbits and discusses properties of conic director circles and their relation to conserved quantities.

Findings

Geometric derivation of hyperbolic orbits without calculus

Analysis of director circles and their relation to hodographs

Extension of the approach to curved spaces like spheres and hyperbolic planes

Abstract

Hodographs for the Kepler problem are circles. This fact, known since almost two centuries ago, still provides the simplest path to derive the Kepler first law. Through Feynman `lost lecture', this derivation has now reached to a wider audience. Here we look again at Feynman's approach to this problem as well as at the recently suggested modification by van Haandel and Heckman (vHH), with two aims in view, both of which extend the scope of the approach. First we review the geometric constructions of the Feynman and vHH approaches (that prove the existence of {\itshape elliptic} orbits without making use of integral calculus or differential equations) and then we extend the geometric approach to cover also the {\itshape hyperbolic} orbits (corresponding to $E>0$). In the second part we analyse the properties of the director circles of the conics, which are used to simplify the approach and we relate with the properties of the hodographs and with the Laplace-Runge-Lenz vector, the constant of motion specific to the Kepler problem. Finally, we briefly discuss the generalisation of the geometric method to the Kepler problem in configuration spaces of constant curvature, i.e. in the sphere and the hyperbolic plane.