Literature review (i)

This is the first part of the literature review, as submitted in part for the annual review and which will eventually form the introduction to my thesis. The reproduction of it here is provided for the interested reader, as a reference for others involved in the area of symmetry analysis and as an easily locatable reference for myself. There are nine sections (all of which will be linked to in each section) and a bibliography will be made available at a later point.

Towards the end of the 19th century, the Norwegian mathematician Sophus Lie developed an astonishing theory that provided a fruitful mechanism for solving differential equations. Inspired by Galois theory and the work of his compatriot Abel in the group theory of polynomial equations, Lie's great achievement was to show that the various and seemingly unrelated ad hoc techniques which existed for solving particular differential equations of a certain type were all, in fact, special cases of a general integration procedure. The effect of this discovery was to significantly unify and extend the known integration methods, at once bringing together the integrating factor, reduction of order, separation of variables, conservation laws, invariant solutions, invertible linear transformations, homogenous equations and linear superposition of solutions amongst others under the umbrella of consequences of the continuous group theory.

Although originally interested in geometry, Lie devoted the rest of his mathematical career to developing his far-reaching theory of continuous groups, with his formulation of the theory and background providing for a wide range of applications of his continuous groups — now known as Lie groups — in many different pure and applied areas and disciplines. These include algebraic topology, differential geometry, bifurcation theory and numerical analysis in mathematics as well as advances in other mathematics-based sciences.

Surprisingly, Lie's methods and the immediate developments that resulted did not find a wide mathematical audience, a possible result — as Olver suggests — of the "inelegance" of the Lie groups that arose through the study of differential equations and their local action. Instead, the global, abstract version of Lie group theory began to dominate efforts and led to little attention or research in the direction of local Lie groups. Fittingly, however, the subject returned to prominence thanks to its application to problems in fluid dynamics; it is thus within the field of differential equations that Lie groups are most readily applied and perhaps most successfully employed.