Symmetry group solutions to differential equations

Abstract

In this project we will be looking at Sophus Lie’s desire (his so called idee fixe) to apply Contact Transformations (what would eventually develop into the modern idea of a Lie Algebra) in order to arrive at symmetries of differential equations, and thus certain solutions. Our goal—as well as Lie’s—is to develop a more universal method for solving differential equations than the familiar cook-book methods we learn in an introductory ordinary or partial differential equations class. We answer three questions. What was the historical underpinning of Sophus Lie’s theory? How do we find the symmetry Lie algebras? How do we use the symmetry Lie algebras to find solutions to the differential equation? (In order to answer these questions we will need to fill in some background material and our answers will also result in a novel derivation of the “Fundamental Source Solution.”) Our second objective will be to establish a connection between solvability in Galois Theory and in Differential Equations. We will assume a familiarity with certain concepts from Abstract Algebra.