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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.

MSC:
58E30 Variational principles in infinite-dimensional spaces
58A12 de Rham theory in global analysis
58A20 Jets in global analysis
58J10 Differential complexes
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