Symmetries in finite order variational sequences

Mauro Francaviglia; Marcella Palese; Raffaele Vitolo

Czechoslovak Mathematical Journal (2002)

  • Volume: 52, Issue: 1, page 197-213
  • ISSN: 0011-4642

Abstract

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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.

How to cite

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Francaviglia, Mauro, Palese, Marcella, and Vitolo, Raffaele. "Symmetries in finite order variational sequences." Czechoslovak Mathematical Journal 52.1 (2002): 197-213. <http://eudml.org/doc/30693>.

@article{Francaviglia2002,
abstract = {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.},
author = {Francaviglia, Mauro, Palese, Marcella, Vitolo, Raffaele},
journal = {Czechoslovak Mathematical Journal},
keywords = {fibered manifold; jet space; variational sequence; symmetries; conservation laws; Euler-Lagrange morphism; Helmholtz morphism; fibered manifold; jet space; variational sequence; symmetries; conservation laws; Euler-Lagrange morphism; Helmholtz morphism},
language = {eng},
number = {1},
pages = {197-213},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Symmetries in finite order variational sequences},
url = {http://eudml.org/doc/30693},
volume = {52},
year = {2002},
}

TY - JOUR
AU - Francaviglia, Mauro
AU - Palese, Marcella
AU - Vitolo, Raffaele
TI - Symmetries in finite order variational sequences
JO - Czechoslovak Mathematical Journal
PY - 2002
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 52
IS - 1
SP - 197
EP - 213
AB - 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.
LA - eng
KW - fibered manifold; jet space; variational sequence; symmetries; conservation laws; Euler-Lagrange morphism; Helmholtz morphism; fibered manifold; jet space; variational sequence; symmetries; conservation laws; Euler-Lagrange morphism; Helmholtz morphism
UR - http://eudml.org/doc/30693
ER -

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Citations in EuDML Documents

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  1. Marcella Palese, Ekkehart Winterroth, E. Garrone, Second variational derivative of local variational problems and conservation laws
  2. Mauro Francaviglia, M. Palese, E. Winterroth, Locally variational invariant field equations and global currents: Chern-Simons theories
  3. Marcella Palese, Variations by generalized symmetries of local Noether strong currents equivalent to global canonical Noether currents
  4. Marcella Palese, Ekkehart Winterroth, Global generalized Bianchi identities for invariant variational problems on gauge-natural bundles

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