Variational first-order partial differential equations
Let \(\pi_1: J^1 Y\to X\) be the jet prolongation of a fibered manifold \(\pi: Y\to X\). Denoting \((x^i, y^\sigma, y^\sigma_j)\) the local fibered coordinates and \(\omega^\sigma= dy^\sigma- \sum y^\sigma_j dx^j\) the contact forms on \(J^1Y\), the article is devoted to formal algebraic properties of the multiple variational integral \(\int\rho\), where \[ \rho= \sum a^{j_1\cdots j_{n-k}}_{\sigma_1\cdots\sigma_k} dy^{\sigma_1}\wedge\cdots\wedge dy^{\sigma_k}\wedge dx^{j_1}\wedge\cdots\wedge dx^{j_{n-k}} (n=\dim X) \] is a differential form on \(Y\). Then the Lagrange function \(L\) is determined by the congruence \(\rho\sim Ldx^1\wedge\cdots\wedge dx^n (\text{mod all }\omega^\sigma)\) and the Euler-Lagrange expressions \(E^\sigma\) from \[ d\rho\sim dL\wedge dx^1\wedge\cdots\wedge dx^n\sim \sum E^\sigma \omega^\sigma \wedge dx^1\wedge\cdots\wedge dx^n (\text{mod }\omega^\sigma\wedge \omega^\tau). \] It follows that the Euler-Lagrange system is of the first-order (and defined on \(J^1Y\)). \{Reviewer's warning: there exist other first-order variational integrals that lead to first-order equations for extremals.\} The authors introduce a large number of concepts (\(Y\)-pertinence, Lepage mappings, dynamical forms, Krupka differential forms, Lepage classes) and deal with various simple interrelations. For instance, the variationality of certain special first-order equations is equivalent to the existence of a special closed differential form. The more involved problems (field theory, Hamilton-Jacobi equations, optimality conditions) are not involved.
scientific article; zbMATH DE number 1915666
scientific article