Global generalized Bianchi identities for invariant variational problems on gauge-natural bundles

Marcella Palese and Ekkehart Winterroth

Department of Mathematics, University of Torino, Via C. Alberto 10, 10123 Torino, Italy


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.

AMSclassification. 58A20, 58A32, 58E30, 58E40, 58J10, 58J70.

Keywords. Jets, gauge-natural bundles, variational principles, generalized Bianchi identities, Jacobi morphisms, invariance and symmetry properties.