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\(\mathcal D\)-modules, contact valued calculus and Poincare-Cartan form. (English) Zbl 1011.58011
The author studies systematically the basic structures of the higher order variational calculus on a fibered manifold \(E\rightarrow M\) from the viewpoint of the theory of differential operators. First of all he clarifies that every linear connection on \(M\) induces a decomposition morphism on the \(k\)-th order tangent bundle of \(M\). This is applied to the differential calculus in the contact module of the jet bundles over \(E\). In particular, this approach yields an interesting new point of view to the variational bicomplex over \(E\). Next the author shows that his approach is an efficient tool for constructing the Euler-Lagrange and Helmholtz-Sonin operators. Finally he clarifies how the construction of the Poincaré-Cartan form depends on the connection in question and deduces an explicit coordinate formula for its coefficients.

58E30 Variational principles in infinite-dimensional spaces
58A20 Jets in global analysis
58J10 Differential complexes
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