Deriving Kepler's Laws Using Quaternions

TL;DR

This paper introduces a quaternion-based approach to derive Kepler's laws, offering a more intuitive and elegant method compared to traditional techniques, by leveraging quaternionic representations of planetary motion.

Contribution

It presents a novel quaternionic framework for deriving Kepler's laws, simplifying the understanding of planetary orbits and motion.

Findings

Quaternionic formulation accurately derives Kepler's laws

Provides an intuitive geometric interpretation of planetary motion

Validates the quaternionic approach through standard celestial mechanics methods

Abstract

In the past, Kepler painstakingly derived laws of planetary motion using difficult to understand and hard to follow techniques. In 1843 William Hamilton created and described the quaternions, which extend the complex numbers and can easily describe rotations in three dimensional space. In this article, we will harness this system to provide a new and intuitive way to derive Kepler's laws. This will include using a quaternionic version of the spatial Kepler problem differential equation, and using the general solution to describe the motion of planets orbiting a central body. We use the standard method for regularizing celestial mechanics, but this article will be solely focused on showing the validity of Kepler's laws.