The Lie group approach to solving differential equations

Despite their central place in mathematical physics, Lie groups are generally regarded as requiring a long apprenticeship in the theory of differentiable manifolds and topological groups. However, this is not how they arose historically. Sophus Lie was prompted to study these structures in the late nineteenth century. He became convinced that the power of Galois theory in the investigation of the solutions of algebraic equations could be harnessed to the study of differential equations, and could be made to yield equally striking results there. This application has been almost forgotten today, or relegated to the province of specialists, but the purpose of this paper is to argue that an elementary treatment is both possible and enlightening. It is hoped that this treatment will help to motivate students to believe that Lie groups can be not only useful, but natural, objects of study, with an intuitive interpretation in terms of phenomena that can be visualized at a physical level, certainly in the one-parameter case.