COHOMOLOGY OF LAGRANGE COMPLEXES INVARIANT UNDER PSEUDOGROUPS OF LOCAL TRANSFORMATIONS.

The inverse problem of the Calculus of Variations for Lagrangians and Euler–Lagrange equations invariant under a pseudogroup $\mathcal{P}$ of local transformations of the base manifold is considered. Exploiting some ideas of Krupka, a theorem is proved showing that, if the configuration space consists of sections of tensor bundles or of local maps of a manifold into another, then such inverse problem is solvable whenever a certain cohomology class of $\mathcal{P}$-invariant forms on the configuration space is vanishing. In addition, for a few pseudogroups, the cohomology groups considered in the main result are explicitly determined in terms of the de Rham cohomology of the configuration space. [ABSTRACT FROM AUTHOR]