Jets and Differential Invariants

Transformation groups figure prominently in Lie’s theory of symmetry groups of differential equations, which we discuss in Chapter 4. They will act on the basic space coordinatized by the independent and dependent variables relevant to the system of differential equations under consideration. Since we are dealing with differential equations we must be able to handle the derivatives of the dependent variables on the same footing as the independent and dependent variables themselves. This chapter is devoted to a detailed study of the proper geometric context for these purposes —the so-called “jet spaces” or “jet bundles”, well known to nineteenth century practitioners, but first formally defined by Ehresmann, [16]. After presenting a simplified version of the basic construction, we then discuss how group transformations are “prolonged” so that the derivative coordinates are appropriately acted upon, and, in the case of infinitesimal generators, deduce the explicit prolongation formula. A differential invariant is merely an invariant, in the standard sense, for a prolonged group of transformations acting on the jet space J. 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, [21], and then developed in great detail, with numerous applications, by Lie, [33], and Tresse, [53]. In this chapter, we discuss the basic theory of differential invariants, and some fundamental methods for constructing them. Applications of these results to the study of differential equations and variational problems will be discussed in the following chapters.