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Conservation laws and variational sequences in gauge-natural theories

Published online by Cambridge University Press:  18 May 2001

L. FATIBENE ,
M. FRANCAVIGLIA  and
M. PALESE
L. FATIBENE
Affiliation:
Department of Mathematics, University of Turin, Via C. Alberto 10, 10123 Turin, Italy. e-mail: fatibene@dm.unito.it, francaviglia@dm.unito.it, palese@dm.unito.it
M. FRANCAVIGLIA
Affiliation:
Department of Mathematics, University of Turin, Via C. Alberto 10, 10123 Turin, Italy. e-mail: fatibene@dm.unito.it, francaviglia@dm.unito.it, palese@dm.unito.it
M. PALESE
Affiliation:
Department of Mathematics, University of Turin, Via C. Alberto 10, 10123 Turin, Italy. e-mail: fatibene@dm.unito.it, francaviglia@dm.unito.it, palese@dm.unito.it
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Abstract

In the classical Lagrangian approach to conservation laws of gauge-natural field theories a suitable vector density is known to generate the so-called conserved Noether currents. It turns out that along any section of the relevant gauge-natural bundle this density is the divergence of a skew-symmetric tensor density, which is called a superpotential for the conserved currents.

We describe gauge-natural superpotentials in the framework of finite order variational sequences according to Krupka. We refer to previous results of ours on variational Lie derivatives concerning abstract versions of Noether's theorems, which are here interpreted in terms of ‘horizontal’ and ‘vertical’ conserved currents. The gauge-natural lift of principal automorphisms implies suitable linearity properties of the Lie derivative operator. Thus abstract results due to Kolář, concerning the integration by parts procedure, can be applied to prove the existence and globality of superpotentials in a very general setting.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 130 , Issue 3 , May 2001 , pp. 555 - 569
Copyright
2001 Cambridge Philosophical Society

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Footnotes

Work partially supported by GNFM of INDAM, MURST, University of Turin.