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The perturbation functor in the calculus of variations. (English) Zbl 1044.58003

In this paper, in the framework of second order calculus of variations on jet bundles, the authors make the observation that the majority of the relevant constructions entering the first variation, the second variation, the Poincare-Cartan form, and the Jacobi morphism can be factorized through a functorial operation which they call the perturbation functor. The perturbation functor, denoted by \({\mathcal P}\), associates to any given Lagrangian \({\mathcal L}\) its first order deformation in such a way that all relevant quantities of the calculus of variation are carried over to the analogous quantities for the new Lagrangian. They develop the basic tools for construction of a reasonable perturbation functor in the physically relevant case of Lagrangian theories of order at most two and they show that the functor commutes with the most of the relevant cohomology functors of the calculus of variations and with the de Rham functor as well.

MSC:

58A12 de Rham theory in global analysis
58A20 Jets in global analysis
58E30 Variational principles in infinite-dimensional spaces
18F15 Abstract manifolds and fiber bundles (category-theoretic aspects)
18G60 Other (co)homology theories (MSC2010)
49K10 Optimality conditions for free problems in two or more independent variables
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