Concepts of Continuous Transformation Groups

The foundation of the group — theoretic method is contained in the general theories of continuous transformation groups that were introduced and treated extensively by Lie 1, Lie and Engel 2 and Lie and Scheffers 3 in the latter part of the last century. Subsequently, the books by Cohen 4, Campbell 5, Eisenhart 6, Ovsjannikov 7, Bluman and Cole 8, have contributed greatly to the development and clarification of many of Lie’s theories, particularly its applications to the invariant solutions of differential equations. In the literature of engineering and applied sciences, the works of Birkhoff 9, Morgan 10, Hansen 11, Na, Abbott and Hansen 12, Na and Hansen 13 and Ames 14,15 give quite extensively the general theories involved in the similarity solutions of partial differential equations as applied to engineering problems. It is assumed, however, that the average engineer may not be thoroughly acquainted with the concepts of that branch of modern algebra designated as group theory. For this reason as well as for clarifications of the terms and concepts involved, a brief review of some of the key aspects of the theory of transformation groups will be given in this chapter. Emphasis will be placed on presenting the ideas of Lie groups in a simple and clear manner suitable for an engineer and scientist, instead of the rigorous and mathematically elegant approach used in the books by Eisenhart 6, Ovsjannikov 7 and Blumau and Cole 8.