Cohomology of the variational complex in the class of exterior forms of finite jet order
Let \(Y\to X\) be a smooth fiber bundle, \(J^*_\infty (O^*_\infty\), respectively) be the bigraded differential algebra of exterior forms on \(J^\infty Y\) which are locally (globally) the pullback of exterior forms on finite-order jet manifolds. Cohomologies of the horizontal differential \(d_H\) and of the variational operator \(\delta\) are well-known. The author proves that the graded differential algebra \(O^*_\infty\) has the same cohomology as \(J^*_\infty\). It follows the isomorphism between \(d_H\)- and \(\delta\)-cohomology of the variational complex \[ 0\to \mathbb{R}\to J^0_\infty @>d_H>> J_\infty^{0, 1}\to \cdots\to J_\infty^{0,n} @>\delta>> E_1 @>\delta>> E_1\to \cdots \] and the de Rham cohomology of the fiber bundle \(Y\) (namely \(H^{k<n} (d_H,J^*_\infty)= H^{k<n}(Y)\), \(H^{k-n} (\delta,J^*_\infty) =H^{k\geq n}(Y))\). This provides a solution of the global inverse problem of the classical calculus of variations.
inverse problem of the calculus of variations
scientific article; zbMATH DE number 1779098