W. G. Unruh
Newton in the Principia used the inverse squared force law (and Galileo’s idea of compound motion) to derive Kepler’s laws. As usual for him, the proof is a purely geometric proof, using no calculus. Maxwell  gave a very different proof. In the 1960’s, Feynman gave a geometric proof very similar to Maxwell’s. Finally, Vogt in the American Journal of Physics also carried out a derivation which started out with the energy conservation equation and the angular momentum conservation to again present a geometric proof. In all of these cases, the derivation that the orbit actually is an ellipse was somewhat torturous, difficult to follow, and non obvious. In the following, following initially the Maxwell-Feynman’s approach, the derivation that the orbit is an ellipse is simplified about as much as possible.
Read more at https://arxiv.org/pdf/1803.06770.pdf