A simple derivation of Kepler's laws without solving differential equations

TL;DR

This paper presents an accessible, geometry-based derivation of Kepler's laws using a discrete time approach and polar coordinates, avoiding complex differential equations and making the concepts suitable for beginners.

Contribution

It introduces a straightforward, geometric method to derive Kepler's laws without differential equations, enhancing pedagogical understanding for students.

Findings

Derivation of Kepler's laws using discrete time and geometry

Explicit expression of velocity and trajectory equations

Accessible approach suitable for beginners

Abstract

Proceeding like Newton with a discrete time approach of motion and a geometrical representation of velocity and acceleration, we obtain Kepler's laws without solving differential equations. The difficult part of Newton's work, when it calls for non trivial properties of ellipses, is avoided by the introduction of polar coordinates. Then a simple reconsideration of Newton's figure naturally leads to en explicit expression of the velocity and to the equation of the trajectory. This derivation, which can be fully apprehended by beginners at university (or even before) can be considered as a first application of mechanical concepts to a physical problem of great historical and pedagogical interest.