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Learning about Differential Equations from Their Symmetries

Introduction

Generally, we teach our sophomore engineering students a course in techniques for solving differential equations. It often appears to the students that we are offering a disjoint collection of methods that we expect them to apply to a sequence of contrived examples. Later in their education many of these students are introduced to partial differential equations and again they are besieged by a multitude of techniques and "special" solutions. Almost all of the methods that we teach can be derived from one basic idea: the existence of symmetries of differential equations.

In the late 1870s Sophus Lie, working at the University of Christiania (now Oslo), applied his theories of transformation groups to differential equations. It was well known at the time that Abel's theory on the roots of polynomials was best understood in terms of the theories of Galois regarding symmetries of polynomial equations. Lie believed that his theories of continuous transformation groups would lead to a similar description of solutions to differential equations. He soon realized that many of the standard techniques for solving differential equations could be subordinated to a general method. In [1] Lie wrote that "the foundation of this method is the concept of an infinitesimal transformation and closely related to it the concept of a one-parameter group" (translation by F. Schwartz in [2]).

The transformations that Lie was writing about are usually called symmetries. They are found by posing the question: "What are the transformations of the variables in a differential equation that map solutions of the equation to other solutions of the equation?" This is the fundamental question that we should always keep in mind when discussing the symmetries of a differential equation.

A symmetry of an ordinary differential equation can be used to reduce the order of the equation. My first example shows how this may be done. Also, Alexei Bocharov discusses Mathematica's implementation of this technique in [3]. For partial differential equations, I will show how knowledge of a symmetry can be used to reduce the number of independent variables by one. So, for example, an equation in two independent variables can be converted to an ordinary differential equation.

Answering the question of what are the possible symmetries of any given differential equation leads to a massive computational problem. For this reason Lie's technique was rarely used until recently. The advent of powerful computer algebra systems has made it feasible to carry out such long calculations. MathSym is a Mathematica package that performs many of the computations necessary to apply Lie's technique. In addition, it incorporates ideas from Gröbner bases to reduce the equations, which must be solved in order to compute the symmetries of an equation.

I will not give a full description of the method of symmetry reduction of differential equations. Rather, I suggest the following references for someone interested in learning more about these techniques. The mathematical details of Lie's technique may be found in [4, 5, 6, 7]. A description of the use of symmetries in Mathematica appears in [3]. For a discussion of Gröbner bases, see [8] and [9]. Application of symmetry reduction to two equations from fluid mechanics may be found in [10].