## $$\mathcal D$$-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.

### MSC:

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

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