A new look at the Feynman ‘hodograph’ approach to the Kepler first law

Abstract

Hodographs for the Kepler problem are circles. This fact, known for almost two centuries, still provides the simplest path to derive the Kepler first law. Through Feynman's 'lost lecture', this derivation has now reached a wider audience. Here we look again at Feynman's approach to this problem, as well as the recently suggested modification by van Haandel and Heckman (vHH), with two aims in mind, 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 elliptic orbits without making use of integral calculus or differential equations) and then extend the geometric approach to also cover the 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 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.

Velocity-imaging the rapidly precessing planetary disc around the white dwarf HE 1349–2305 using Doppler tomography

“…One curious property of eccentric orbits in velocity space is that they are represented as offset circles, which was identified independently by both Möbius (1843) and Hamilton (1847), and has been discussed in numerous other works since (e.g., Cariñena et al 2016 and references therein), where velocity diagrams are usually referred to as 'hodographs'. We re-illustrate here that eccentric orbits can be represented as offset circles in velocity space using the formalism described in Appendix A of Manser et al (2016b).…”

Hamilton symmetry in relativistic Coulomb systems

“…The Hamilton symmetry is therefore an extension of central symmetry. Further information and discussions regarding the classical KC analytic hodograph solution may be found in various publications [4,5,6,7,8,9,10,11].…”

“…another consequence of (11), allows transition from spatial dependencies to velocity dependencies. In particular, using ( 16) the energy integral (12) becomes…”

“…The hodograph method -studying the dynamics of a system in velocity space -was originally invented by Hamilton [1] and successfully applied [2,3] to prove geometrically the relation between Kepler's laws and Newton's law of universal attraction. Although largely unfamiliar with, it is a method much simpler and more elegant to study Newtonian Kepler/Coulomb (KC) systems than the familiar analytic solutions in ordinary space [4,5,6,7,8,9,10,11].…”

Back to epicycles – relativistic Coulomb systems in velocity space

“…Although hardly known, the virtue of the hodograph method is that its application to classical systems with 1/r potential provides, in a very simple and elegant way, the full solution -all the necessary information regarding the dynamics of the system, including spatial trajectories -just from the discussion in velocity space. Its merits have been discusses on several occasions in the last decades [4,5,6,7,8,9,10].…”