Euler operator as part of a cochain complex

Question

I am studying chapter 4 of Olver's "Applications of Lie groups to differential equations", about symmetries in differential equations coming from a variational principle.

The Euler operator is defined by $$ \mathbf{E}_\alpha=\sum_J(-D)_J \frac{\partial}{\partial u_J^\alpha} $$ being $D_{x_j}$ the total derivative operators and $J$ the usual multiindex notation. It plays the role of associating to a variational problem its corresponding variational derivative. That is, if $L$ is the Lagrangian of the variational problem then $\mathbf{E}_\alpha(L)$ are the corresponding Euler-Lagrange equations.

In this book (Theorem 4.7) it is proven that $\mathbf{E}_\alpha(L)\equiv 0$ if and only if $L$ is the total divergence of a $p$-tuple of functions $P=(P_1,\ldots,P_p)$. The total divergence of $P$ is $$ \mbox{Div} P=\sum D_{x_j} P_j. $$

Since we have two operators and the image of one is the kernel of the other, the idea of the de Rham complex comes to my mind. Is there any relation? Is this fact ($\mathbf{E}_\alpha(L)\equiv 0$ if and only if $L=\mbox{Div}(P)$) part of a bigger complex? Can you give me a reference if it is the case?

Answer 1

Yes. The next operator in the sequence is called the Helmholtz operator, followed by higher versions thereof. The main keyword is "variational bicomplex" and a standard reference is

I. M. Anderson, “The variational bicomplex” (1989). [unpublished but easily googlable]

In Olver's book (referenced by the OP) this topic is dealt with in Sec.5.4. In the Notes to Ch.5 he also gives references to the larger literature.