Variational Sequences

Abstract

We introduced in Chap. 4 the Euler–Lagrange mapping of the calculus of variations as an ${\mathbf{R}}$-linear mapping, assigning to a Lagrangian $\lambda$, defined on the r-jet prolongation $J^{r} Y$ of a fibered manifold Y, its Euler–Lagrange form $E_{\lambda }$. Local properties of this mapping are determined by the components of the Euler–Lagrange form, the Euler–Lagrange expressions of the Lagrangian $\lambda$. In this chapter, we construct an exact sequence of Abelian sheaves, the variational sequence, such that one of its sheaf morphisms coincides with the Euler–Lagrange mapping. Existence of the sequence provides a possibility to study basic global characteristics of the Euler–Lagrange mapping in terms of the cohomology groups of the corresponding complex of global sections and the underlying manifold Y. In particular, for variational purposes, the structure of the kernel and the image of the Euler–Lagrange mapping $\lambda \to E_{\lambda }$ is considered.