Variational sequences, representation sequences and applications in physics. (English) Zbl 1347.70043
The aim of the paper is to provide the reader a comprehensive source of the current theory of finite order variational sequences as it stands 25 years after its beginnings. However, the aim of the paper is not merely to provide a revised exposition of known results. Apart from putting known results into a unified treatment, the paper also contains a number of original results in order to develop the theory and make it complete. The aim of the paper is also to stimulate the use and applications of the variational sequences in physics. As a motivation the authors mention a few applications, some of which are interesting and not widely known or even are original here.
Section 2 of the paper introduces the variational sequence as a quotient sequence of the de Rham sequence, developed by Krupka. Section 3 is the core of the paper, and deals with the problem of representation of the variational sequence by differential forms. In general, the authors present two different representation sequences: the Takens representation sequence based on the so called Euler operator, and the Lepage representation sequence (i.e. the sequence of Lepage forms). As a new result concerning Lepage \(n\)-forms, the authors present an intrinsic formula for the Cartan form.
Section 4 deals with the Lie derivative in the variational sequence. It is proved that the Lie derivative of differential forms with respect to prolongations of projectable vector fields preserves the contact structure. Finally, the last section of the paper is devoted to selected applications of the variational sequence theory.
Section 2 of the paper introduces the variational sequence as a quotient sequence of the de Rham sequence, developed by Krupka. Section 3 is the core of the paper, and deals with the problem of representation of the variational sequence by differential forms. In general, the authors present two different representation sequences: the Takens representation sequence based on the so called Euler operator, and the Lepage representation sequence (i.e. the sequence of Lepage forms). As a new result concerning Lepage \(n\)-forms, the authors present an intrinsic formula for the Cartan form.
Section 4 deals with the Lie derivative in the variational sequence. It is proved that the Lie derivative of differential forms with respect to prolongations of projectable vector fields preserves the contact structure. Finally, the last section of the paper is devoted to selected applications of the variational sequence theory.
Reviewer: Miroslav Doupovec (Brno)
MSC:
| 70S10 | Symmetries and conservation laws in mechanics of particles and systems |
| 55N30 | Sheaf cohomology in algebraic topology |
| 55R10 | Fiber bundles in algebraic topology |
| 58A12 | de Rham theory in global analysis |
| 58A20 | Jets in global analysis |
| 58E30 | Variational principles in infinite-dimensional spaces |
Keywords:
fibered manifold; jet space; Lagrangian formalism; variational sequence; variational derivative; cohomology; symmetry; conservation lawReferences:
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