THE VARIATIONAL BICOMPLEX by Ian M. Anderson TABLE OF CONTENTS PREFACE ............................ iii ACKNOWLEDGMENTS† ..................... vii INTRODUCTION† ....................... viii chapter one VECTOR FIELDS AND FORMS ON THE INFINITE JET BUNDLE ............................. 1 A. Infinite Jet Bundles . 1 B. Vector Fields and Generalized Vector Fields . 6 C. Differential Forms and the Variational Bicomplex. 12 D. Prolongations of Generalized Vector Fields . 23 chapter two EULER OPERATORS ...................... 30 A.TotalDifferentialOperators.................. 31 B.EulerOperators....................... 39 C.SomeGeometricVariationalProblems............. 50 chapter three FUNCTIONAL FORMS AND COCHAIN MAPS ......... 65 A.FunctionalForms...................... 66 B.CochainMapsontheVariationalBicomplex........... 83 C.ALieDerivativeFormulaforFunctionalForms......... 99 †preliminary draft, not complete. ii chapter four LOCAL PROPERTIES OF THE VARIATIONAL BICOMPLEX . 111 A. Local Exactness and the Homotopy Operators fortheVariationalBicomplex................ 114 B.MinimalWeightForms................... 137 C.TheJacobianSubcomplex.................. 163 chapter five GLOBAL PROPERTIES OF THE VARIATIONAL BICOMPLEX 188 A. Horizontal Cohomology of the Variational Bicomplex. 191 B. Vertical Cohomology of the Variational Bicomplex. 222 C. Generalized Poincar´e -Cartan Forms and Natural Differential Operators on the Variational Bicomplex. 243 D. Invariant Horizontal Homotopy Operators on ManifoldswithSymmetricConnections............ 258 chapter six EQUIVARIANT COHOMOLOGY OF THE VARIATIONAL BICOMPLEX‡ ........................... A.Gelfand-FuksCohomology.................... B. The Variational Bicomplexfor Natural Riemannian Structures.......................... chapter seven THE VARIATIONAL BICOMPLEX FOR DIFFERENTIAL EQUATIONS‡ ........................... A.Preliminaries......................... B.TheTwoLineTheoremanditsGeneralization........... C.Examples........................... D. The Inverse Problem to the Calculus of Variations . REFERENCES ......................... 286 ‡In preparation iii PREFACE The variational bicomplex is a double complex of differential forms defined on the infinite jet bundle of any fibered manifold π : E → M. This double complex of forms is called the variational bicomplex because one of its differentials (or, more precisely, one of the induced differentials in the first term of the first spec- tral sequence) coincides with the classical Euler-Lagrange operator, or variational derivative, for arbitrary order, multiple integral problems in the calculus of varia- tions. Thus, the most immediate application of the variational bicomplex is that of providing a simple, natural, and yet general, differential geometric development of the variational calculus. Indeed, the subject originated within the last fifteen years in the independent efforts of W. M. Tulczyjewand A. M. Vinogradov to resolve the Euler-Lagrange operator and thereby characterize the kernel and the image of the the variational derivative. But the utility of this bicomplex extents well beyond the domain of the calculus of variations. Indeed, it may well be that the more important aspects of our subject are those aspects which pertain either to the general theory of conservation laws for differential equations, as introduced by Vinogradov, or to the theory of characteristic (and secondary characteristic) classes and Gelfand-Fuks cohomology, as suggested by T. Tsujishita. All of these topics are part of what I. M. Gelfand, in his 1970 address to the International Congress in Nice, called formal differential geometry. The variational bicomplex plays the same ubiquitous role in formal differential geometry, that is, in the geometry of the infinite jet bundle for the triple (E,M,π) that the de Rham complex plays in the geometry of a single manifold M. The purpose of this book is to develop the basic general theory of the variational bicomplex and to present a variety of applications of this theory in the areas of dif- ferential geometry and topology, differential equations, and mathematical physics. This book is divided, although not explicitly, into four parts. In part one, which consists of Chapters One, Two, and Three, the differential calculus of the varia- tional bicomplex is presented. Besides the usual operations involving vector fields and forms on manifolds, there are two additional operations upon which much of the general theory rests. The first of these is the operation of prolongation which lifts generalized vector fields on the total space E to vector fields on the infinite jet bundle J ∞(E). The second operation is essentially an invariant “integration by parts” operation which formalizes and extends the familiar process of forming November 26, 1989 iv Preface the formal adjoint of a linear differential operator. Chapters Four and Five form the second part of the book. Here, the local and global cohomological properties of the variational bicomplex are studied in some detail. While the issues, methods and applications found in these two chapters differ considerable from one another, they both pertain to what may best be called the theory of the “free” variational bicomplex — no restrictions are imposed on the domain of definition of the dif- ferential forms in the variational bicomplex nor do we restrict our attention to sub-bicomplexes of invariant forms. In short, part two of this book does for the variational bicomplex what, by analogy, the Poincar´e Lemma does for the de Rham complex. In Chapter Six, the third part of the book, we let G be a group of fiber preserving transformations on E. The action of G on E lifts to a group action G on J ∞(E). Because G respects the structure of the variational bicomplex, we can address the problem of computing its G equivariant cohomology. The Gelfand- Fuks cohomology of formal vector fields is computed anewfrom this viewpoint. We also showhowcharacteristic and secondary characteristic classes arise as equivari- ant cohomology classes on the variational bicomplex for the bundle of Riemannian structures. In the final part, Chapter Seven, we look at systems of differential equa- tions R on E as subbundles R∞ of J ∞(E) and investigate the cohomology of the variational bicomplex restricted to R∞. Cohomology classes nowrepresent vari- ous deformation invariants of the given system of equations — first integrals and conservation laws, integral invariants, and variational principles. Vinogradov’s Two Line Theorem is extended to give a sharp lower bound for the dimension of the first nonzero cohomology groups in the variational bicomplex for R. A newperspective is given to J. Douglas’ solution to the inverse problem to the calculus of variations for ordinary differential equations. A major emphasis throughout the entire book is placed on specific examples, problems and applications. These are test cases against which the usefulness of this machinery can, at least for now, be judged. Through different choices of the bundle E, the group G and the differential relations R, these examples also illustrate how the variational bicomplex can be adapted to model diverse phenomena in differential geometry and topology, differential equations, and mathematical physics. They suggest possible avenues for future research. The general prequisites for this book include the usual topics from the calculus on manifolds and a modest familiarity with the classical variational calculus and its role in mechanics, classical field theory, and differential geometry. In the early chapters, some acquaintance with symmetry group methods in differential equations would surely be helpful. Indeed, we shall find ourselves referring often to Applica- v The Variational Bicomplex tions of Lie Groups to Differential Equations, P. Olver’s fine text on this subject. In Chapter Five, the global properties of the variational bicomplex are developed using the generalized Mayer-Vietoris argument, as explained in the wonderful book, Differential Forms in Algebraic Topology, by R. Bott and L. Tu. The classical in- variant theory of the general linear and orthogonal groups play a central role in our calculations of equivariant cohomology in Chapter Six. Some of the material presented here represents new, previously unpublished re- search by the author. This includes the results in §3B on cochain maps between bicomplexes, the entire theory of minimal weight forms developed in §4B, the analy- sis of locally variational operators in §4C, the existence of global homotopy operators in §5D, and the calculation of the equivariant cohomology of the variational bicom- plex on Riemannian structures in Chapter Six. Newproofs of some of the basic properties of the variational bicomplex are given in §4A, §5A, and §5B and §7B. Also, many of the specific examples and applications of the variational bicomplex are presented here for the first time. vi ACKNOWLEDGMENTS It has taken a long time to research and to write this book. It is now my plea- sure to thank those who helped to initiate this project and to bring this work to completion. Its genesis occurred, back in the fall of 1983, with a seemingly innocu- ous suggestion by Peter Olver concerning the use of the higher Euler operators to simplify the derivation of the horizontal homotopy operators for the variational bi- complex. Over time this idea developed into the material for sections §2A, §2B and §5D. The general outline of the book slowly emerged from series of lectures given at the University of Utah in the fall of 1984, at the University of Waterloo in Jan-