Symmetries in finite order variational sequences. (English) Zbl 1006.58014
Summary: We refer to Krupka’s variational sequence, i.e. the quotient of the de Rham sequence on a finite order jet space with respect to a ‘variationally trivial’ subsequence. Among the morphisms of the variational sequence there are the Euler-Lagrange operator and the Helmholtz operator.
In this note we show that the Lie derivative operator passes to the quotient in the variational sequence. Then we define the variational Lie derivative as an operator on the sheaves of the variational sequence. Explicit representations of this operator give us some abstract versions of Noether’s theorems, which can be interpreted in terms of conserved currents for Lagrangians and Euler-Lagrange morphisms.
In this note we show that the Lie derivative operator passes to the quotient in the variational sequence. Then we define the variational Lie derivative as an operator on the sheaves of the variational sequence. Explicit representations of this operator give us some abstract versions of Noether’s theorems, which can be interpreted in terms of conserved currents for Lagrangians and Euler-Lagrange morphisms.
MSC:
| 58E30 | Variational principles in infinite-dimensional spaces |
| 58A12 | de Rham theory in global analysis |
| 58A20 | Jets in global analysis |
| 58J10 | Differential complexes |
Keywords:
fibered manifold; jet space; variational sequence; symmetries; conservation laws; Euler-Lagrange morphism; Helmholtz morphismReferences:
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