ABSTRACT
In this chapter, the authors attempt to sketch the crucial ideas and key applications that led to a theory of Lie-Bäcklund transformation groups. They reflect a chronological ordering. The chapter also points out the importance of the order of influence, which starts with key applications, on the development of a theory. However, the main emphasis is on results. In order to explain the work of Lie and Backlund, it is important to distinguish between those requirements placed on transformations that arise from tangency conditions alone and those that follow from the group properties. The considerations of Lie and Bäcklund demonstrated that the only generalization of contact transformation groups lies with the possible existence of infinite-order tangent transformation groups, acting on an intrinsically infinite-dimensional space. They themselves did not pursue this possibility.
