Kepler's Differential Equations

TL;DR

This paper explores Kepler's pioneering use of differential equations to describe planetary motion, highlighting his early insights into force-based orbital dynamics before Newton.

Contribution

It presents Kepler's original results in modern language and provides translations of key chapters from Astronomia Nova, emphasizing his early differential approach.

Findings

Kepler demonstrated planetary orbits as solutions to differential equations.

He attributed planetary motion to a force from the sun.

Kepler's work predates and foreshadows Newton's calculus-based physics.

Abstract

Although the differential calculus was invented by Newton, Kepler established his famous laws 70 years earlier by using the same idea, namely to find a path in a nonuniform field of force by small steps. It is generally not known that Kepler demonstrated the elliptic orbit to be composed of intelligeable differential pieces, in modern language, to result from a differential equation. Kepler was first to attribute planetary orbits to a force from the sun, rather than giving them a predetermined geometric shape. Even though neither the force was known nor its relation to motion, he could determine the differential equations of motion from observation. This is one of the most important achievements in the history of physics. In contrast to Newton's Principia and Galilei's Dialogo Kepler's text is not easy to read, for various reasons. Therefore, in the present article, his results -- most of them well known -- are first presented in modern language. Then, in order to justify the claim, the full text of some relevant chapters of Astronomia Nova is presented. The translation from latin is by the present author, with apologies for its shortcomings.