Global Generalized Bianchi Identities for Invariant Variational Problems on Gauge-natural Bundles

Abstract

We derive both {\em local} and {\em global} generalized {\em Bianchi identities} for classical Lagrangian field theories on gauge-natural bundles. We show that globally defined generalized Bianchi identities can be found without the {\em a priori} introduction of a connection. The proof is based on a {\em global} decomposition of the {\em variational Lie derivative} of the generalized Euler--Lagrange morphism and the representation of the corresponding generalized Jacobi morphism on gauge-natural bundles. In particular, we show that {\em within} a gauge-natural invariant Lagrangian variational principle, the gauge-natural lift of infinitesimal principal automorphism {\em is not} intrinsically arbitrary. As a consequence the existence of {\em canonical} global superpotentials for gauge-natural Noether conserved currents is proved without resorting to additional structures.

24 pages, minor changes, misprints corrected, a misprint in the coordinate expression of the Jacobi morphism corrected; final version to appear in Arch. Math. (Brno)

References in corpus (2)

• Conserved Quantities from the Equations of Motion (with applications to natural and gauge natural theories of gravitation)
• The Hessian and Jacobi Morphisms for Higher Order Calculus of Variations

Cited by in corpus (10)

• Covariant gauge-natural conservation laws
• The Hessian and Jacobi Morphisms for Higher Order Calculus of Variations
• Lagrangian reductive structures on gauge-natural bundles
• Generalized Bianchi identities in gauge-natural field theories and the curvature of variational principles
• Variational Sequences, Representation Sequences and Applications in Physics
• Cohomological obstructions in locally variational field theories
• Symmetries of Helmholtz forms and globally variational dynamical forms
• Second variational derivative of gauge-natural invariant Lagrangians and conservation laws
• Gauge-natural field theories and Noether Theorems: canonical covariant conserved currents
• Symmetry transformations of extremals and higher conserved quantities: invariant Yang--Mills connections