Exterior Differential Systems and Euler-Lagrange Partial Differential Equations

Preface During the 1996-97 academic year, Phillip Griffiths and Robert Bryant conducted a seminar at the Institute for Advanced Study in Princeton, NJ, outlining their recent work (with Lucas Hsu) on a geometric approach to the calculus of variations in several variables. The present work is an outgrowth of that project; it includes all of the material presented in the seminar, with numerous additional details and a few extra topics of interest. The material can be viewed as a chapter in the ongoing development of a theory of the geometry of differential equations. The relative importance among PDEs of second-order Euler-Lagrange equations suggests that their geometry should be particularly rich, as does the geometric character of their conservation laws, which we discuss at length. A second purpose for the present work is to give an exposition of certain aspects of the theory of exterior differential systems, which provides the language and the techniques for the entire study. Special emphasis is placed on the method of equivalence, which plays a central role in uncovering geometric properties of differential equations. The Euler-Lagrange PDEs of the calculus of variations have turned out to provide excellent illustrations of the general theory.