CHAPTER VII - Differential Invariants

Abstract

This chapter focuses on the theory and applications of differential invariants of Lie transformation groups, which can be infinite. It discusses the concepts of a differential invariant and special transformation of a differential equation, the so-called group stratification. The central theorem is about the finiteness of bases for differential invariants of arbitrary Lie groups. In the proposed variant Tresse's theorem is completely established in infinitesimal language by the means of invariant differential operators. An equation E is called automorphic relative to the given group G if all its solutions are situated on the orbit of one of them. In other words, from one given solution U, it is possible to find all solutions by applying to U the transformations of the group G. Classical problems of the construction and classification of automorphic systems provide excellent examples of the applications of the general theory, and these play an important role in the problems of invariant transformations for differential equations.