Diierential Invariants

This paper summarizes recent results on the number and characterization of di erential invariants of transformation groups Generalizations of theorems due to Ovsiannikov and to M Green are presented as well as a new approach to nding bounds on the number of independent di erential invariants Consider a group of transformations acting on a jet space coordinatized by the inde pendent variables the dependent variables and their derivatives Scalar functions which are not a ected by the group transformations are known as di erential invariants Their importance was emphasized by Sophus Lie who showed that every invariant system of di erential equations and every invariant variational problem could be directly expressed in terms of the di erential invariants As such they form the basic building blocks of many physical theories where one begins by postulating the invariance of the equations or the variational principle under a prescribed symmetry group Lie also demonstrated how di erential invariants could be used to integrate invariant ordinary di erential equations and succeeded in completely classifying all the di erential invariants for all pos sible nite dimensional Lie groups of point transformations in the case of one independent and one dependent variable Lie s results were pursued by Tresse and much later Ovsiannikov In this paper I will summarize some recent new results extending these earlier classi cation theorems which were discovered in the course of writing the forthcoming book Space considerations preclude the inclusion of proofs and signi cant examples here It is worth remarking that surprisingly the complete classi cation of di erential invariants for many of the groups of physical importance including the general linear a ne conformal and Poincar e groups does not yet seem to be known y Supported in part by NSF Grant DMS Consider the spacey M X U R R whose coordinates represent our indepen dent variables x x x X and dependent variables u u u U Let J denote the associated jet bundle of order n whose coordinates x u n represent the independent variables and the derivatives u I u x xk q i p of the dependent variables of orders k I n Thus dimJ p q n where q n q p n n The number of derivative coordinates of order exactly n is denoted by qn dimJ n dimJ q n q n qpn q p n