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Variational sequences on finite order jet spaces. (English) Zbl 0813.58014
Janyška, Josef (ed.) et al., Differential geometry and its applications. International conference, Brno, Czechoslovakia, 27 Aug. - 2 Sept. 1989. Singapore: World Scientific. 236-254 (1990).
The variational sequence of order $$r$$ is introduced in the following way. Let $$Y$$ be a fibered manifold with base $$X$$ and $$J^{r}Y$$ the $$r$$-jet prolongation of $$Y$$; further $$\Omega^ k_ r$$ denotes the sheaves of $$k$$-forms over $$J^ rY$$ while $$\Omega^ k_{r(c)}$$ stands for their subsheaves of contact. We set $$\theta^ k_ r = d\Omega^{k - 1}_{r(c)} + \Omega^ k_{r(c)}$$ and consider the quotient sheaves $$\Omega^ k_ r/ \theta^ k_ r$$ jointly with the quotient morphisms $$E_ k : \Omega^ k_ r/\theta^ k_ r \to \Omega^{k + 1}_ r/\theta^{k + 1}_ r$$ induced by the exterior derivatives $$d_ k : \Omega^ k_ r \to \Omega^{k + 1}_ r$$.
The quotient sequence: $0 \to R \to \Omega^ 0_ r \to \Omega^ 1_ r/\theta^ 1_ r \to \Omega^ 2_ r/\theta^ 2_ r \to \cdots$ is called the variational sequence of order $$r$$ over $$Y$$. One of the main theorems of the paper states that such a sequence is an acyclic resolution of the constant sheaf $$R$$ over $$Y$$. Explicit expression for $$E_{n + 1}$$ is provided and it is shown that $$E_ n$$ is the Euler- Lagrange mapping in the sense introduced earlier by the same author.
The paper reads well and constitutes a valuable contribution to the calculus of variations on manifolds.
For the entire collection see [Zbl 0777.00040].

MSC:
 5.8e+31 Variational principles in infinite-dimensional spaces