From Kepler's Laws to Newtonian Motion and the Direction Angle of Hamilton's Hodograph

TL;DR

This paper demonstrates a concise derivation from Kepler's laws to Newtonian motion, including a straightforward calculation of Hamilton's Hodograph's direction angle and an elegant expression of speed using elliptic functions.

Contribution

It provides a simplified pathway connecting Kepler's laws to Newtonian mechanics and introduces an elegant elliptic function approach to analyze Hamilton's Hodograph.

Findings

Short derivation from Kepler's laws to Newtonian motion

Explicit computation of Hamilton's Hodograph direction angle

Elliptic functions used to express and invert speed as a function of direction angle

Abstract

In this contribution it is shown that the path from Kepler's results to Newtonian motion can be remarkably short and simple. Following this path we also give a straight forward computation of the direction angle of Hamilton's Hodograph. Then we show how the speed as function of the direction angle can be expressed and inverted elegantly using elliptic functions.