Contemporary Mathematics Volume 132, 1992 INTRODUCTION TO THE VARIATIONAL BICOMPLEX Ian M. Anderson Department of Mathematics Utah State University Logan, Utah 84322 §1. Introduction. The variational bicomplex was first introduced in the mid 1970's as a means of studying the inverse problem of the calculus of variations. This is the prob­ lem of characterizing those differential equations which are the Euler-Lagrange equations for a classical, unconstrained variational problem. Since then, the variational bicomplex has emerged as an effective means for studying other for­ mal, differential-geometric aspects of the calculus of variations. Moreover, it has been shown that the basic variational bicomplex constructed to solve the inverse problem can be modified in various ways and that the cohomology groups asso­ ciated with these modified bicomplexes are relevant to many topics in geometry, mathematical physics and differential equations. The purpose of this paper is to review the general construction of the variational bicomplex, to describe some of its the basic properties, and to survey some recent results. We begin by returning to the genesis of our subject - the inverse problem of the calculus of variations. For the purposes of this introduction, we need not formulate this problem in its full generality. Accordingly, let us consider variational problems for a single function u of three independent variables x, y, z. Given a compact region W in R3 with smooth boundary oW and a first order Lagrangian L = L(x, y, z, U, U:c, u y , u z ), we seek those smooth real-valued functions u:W-+R with prescribed values on oW which minimize the functional f au au au F[u] = Jw L(x,y,z,u, ax' oy' oz)dxdydz. (1.1) Printed October 22,1991 and revised January 23,1992. This research is supported, in part, by grant DMS- 9100674 from the National Science Foundation. 1991 Mathematics Subject Classification. Primary 58E30j Secondary 58A15, 58A20, 49N45. This paper is in final form and no version will be submitted for publication elsewhere. © 1992 American Mathematical Society 0271-4132/92 $1.00 + $.25 per page 51 bz 52 IAN M. ANDERSON The derivation of necessary conditions for a minimum u are based upon the first variational formula: if U f is a smooth one' parameter family of smooth functions on W, then ~(L[u,JI,=o = E(L)[uJ v + Div V[uJ. (1.2) Uf In this equation, v = dd I ' the Euler-Lagrange operator E(L) is € f=O and V is the vector field aL ~ aL ~ aL A V = (~V)l + (~V)J + (~v)k. uUx uUy uUz The total derivative operators Dx, Dy and Dz are defined, for example, by a a a a Dx = ax + U x au + Uxx aux + Uxy aUy +... The total divergence operator Div in (1.2) is defined in terms of the total deriva­ tives Dx, Dy and Dz. In equation (1.2) and in what follows, we shall write L[u], V[u] and so on to indicate that L is a function of the independent variables x, y, z, the dependent variable U and its derivatives to some finite (in this case first) order. With the first variational formula in hand it is not difficult to establish that if U is a local minimum for (1.1), then u is a solution to the Euler-Lagrange equations E(L)[u] = O. The first variational formula plays a central role in our subject and its impor­ tance should be emphasized. Here we note that: (i) the first variational formula (1.2) holds for Lagrangians of any order al­ though the local expressions for the vector field V become rather compli­ cated. (ii) equation (1.2) uniquely characterizes the Euler-Lagrange operator; that is, if T[u] is any differential operator and ~/[U,JI,=o = T[uJ v + Div W[uJ (1.3) then for some vector field W[u], T[u] = E(L)[u]. (iii) As an immediate consequence of (ii), it follows that the Euler-Lagrange operator annihilates divergences. Indeed, if L[u] = Div W[u], then INTRODUCTION TO THE VARIATIONAL BICOMPLEX 53 and hence, by uniqueness, E(L) = O. This proves that E(DivW)[u] = O. (1.4) We can also use the first variational formula to determine when a given dif­ ferential operator T[u] is the Euler-Lagrange operator for some Lagrangian L[u]. This is the simplest version of the inverse problem of the calculus of variations. IfT[u] = E(L)[u], then the first variational formula (1.2) becomes (1.3). To this equation we apply the Euler-Lagrange operator E to deduce, by virtue of (1.4), that (1.5) But it is a straightforward matter to check that and hence we can re-write (1.5) as (~fT[u,t) = E(T[u]v) u where, once again, v = v(x, y, z) = dd £ I . Let us define a linear operator t £=0 H(T)[v] acting on functions v: W -r R by . H(T)[v] = ~fT[u']ld - E(T[u]v). We conclude that if T[u] is an Euler-Lagrange operator, then H(T) = O. (1.6) This is called the Helmholtz equation. For instance, ifT is a second order operator then the condition (1.6) is equivalent to three differential conditions, the first being aT aT 1 aT 1 aT ~ = Dx [-~-] + -2 Dy [-~-] + -2 Dz [-~-] uu x UUxx uUxy UUxz . b···l . r d an d t h e remammg two emg SImI ar equatIOns lor ~aT an ~.aT uUy UUz We can summarize our discussion to this point as follows. Let F[u] denote the space of smooth functions in the variables {x, y, z, u, U x , uy , U z , U xx , ... }, let V[u] be the space of vector fields on R3 with coefficients in F[u] and let V[u] be the space of linear differential operators, also with coefficients in F[u]. Let Grad, 54 IAN M. ANDERSON Curl, and Div be the total gradient, curl and divergence operators on R3 defined in terms of the total derivative operators D:c , Dy and Dz . Then, in summary, the sequence of spaces and maps Grad o ~ R ----. F[u] ~ V[u] ~V[u] ~ F[u] (1.7) ----.E F[u] ~V[u] is a cochain complex - the composition of successive maps is zero. One of the maps in (1.7) is the Euler-Lagrange operator E and for this reason we call this sequence the Euler-Lagrange complex. As we shall see, this complex is exact and this fact solves the simplest version of the inverse problem of the calculus of variations. A differential operator T E F[u] is an Euler-Lagrange operator if and only if the Helmholtz equation H(T) = 0 is satisfied. Some of the primary purposes of the variational bicomplex can now be de­ scribed. First, the variational bicomplex provides us with a straightforward, differential geometric generalization of our model complex (1.7). We begin with a fib er bundle 7r : E ---+ M. For the problem at hand the coordinates on Mare the independent variables and the fiber coordinates are the dependent variables. Then the objects of interest - Lagrangians, partial differential equations, cur­ rents, etc. - are all realized as various types of differential forms on the infinite jet bundle J OO (E) of E . The calculus of vector fields and forms on JOO(E), along with the fundamental operations of vector field prolongation and ((integration by parts" (which we shall make precise in §2) provide a powerful formal variational calculus. This variational calculus also plays an important role in the theory of symmetry group methods for differential equations and in the calculation of conservation laws. vVe sketch the construction of the variational bicomplex in §2. It is not readily apparent how to continue the complex (1.7) by defining an­ other operator whose domain is the space of linear operators V[u] and whose kernel is the image of the Helmholtz operator. The second immediate use of the variational bicomplex is to construct, again in aremarkably straightforward manner, the full continuation of the Euler-Lagrange complex. Thirdly, in situations where the Euler-Lagrange complex is not exact, the variational bicomplex provides us with the powerful algebraic apparatus of spec­ tral sequences as a general approach to computing its cohomology. Depending upon the situation, these homological methods are combined with techniques from algebraic topology and global analysis, invariant theory and exterior differ­ ential systems. Some of the examples and theorems presented in §3, §4 and §5 illustrate why the cohomology of the Euler-Lagrange complex is of such interest. Th~ basic references for our subject are Anderson [2], Krasilschik [15], Olver [20], Saunders [21], Tulczyjew [27], Tsujishita [24], [25] and Vinogradov [29]. It is a pleasure to thank Mark Gotay, Jerry Marsden and Vince Moncrief for their efforts in organizing the Seattle AMS Summer Research Conference on Classical Field Theory. This work is supported in part by National Science Foundation grant DMS- 9100674. INTRODUCTION TO THE VARIATIONAL BICOMPLEX 55 §2. The Variational Bicomplex. The first step towards a complete definition of the variational bicomplex is a description of the mathematical data from which it is constructed. This data varies with the application at hand; however, for most situations the following is prescribed: (i) a fiber bundle 7r: E --+ M; (ii) a transformation group G on E; and (iii) a set of differential equations or even differential inequalities n on the local sections of E. iFrom the fiber bundle 7r: E --+ M we can construct the basic variational bicom­ plex.