Recall that an invariant of a group G acting on a manifold M is a function I: M → ℝ which is not affected by the group transformations. A differential invariant is merely an invariant, in the standard sense, for a prolonged group of transformations (or, more generally, a group of contact transformations) acting on the jet space Jn. Just as the ordinary invariants of a group action serve to characterize invariant equations, so differential invariants will completely characterize invariant systems of differential equations for the group, as well as invariant variational principles. As such they form the basic building blocks of many physical theories, where one begins by postulating the invariance of the differential equations, or the variational problem, under a prescribed symmetry group. Historically, the subject was initiated by Halphen, [108], and then developed in great detail, with numerous applications, by Lie, [156], and Tresse, [214]. In this chapter, we discuss the basic theory of differential invariants, and some fundamental methods for constructing them. Applications will appear here and in the following two chapters. The explicit examples of group actions that are presented to illustrate the general theory will be fairly elementary, in part because, rather surprisingly, the complete classification of differential invariants for many of the groups of physical importance, including the general linear, affine, Poincaré, and conformal groups, does not yet seem to be known!

Differential Invariants

The basic problem to be addressed in the first part of this chapter is the classification of the differential invariants of a given group action.