The Jacobi morphism and the Hessian in higher order field theory; with applications to a Yang-Mills theory on a Minkowskian background

Abstract

We characterize the second variation of an higher order Lagrangian by a Jacobi morphism and by currents strictly related to the geometric structure of the variational problem. We discuss the relation between the Jacobi morphism and the Hessian at an arbitrary order. Furthermore, we prove that a pair of Jacobi fields always generates a (weakly) conserved current. An explicit example is provided for a Yang-Mills theory on a Minkowskian background.

30 pages, misprint in the title corrected, presentation includes only second variations (higher variations appear elsewhere)

References in corpus (6)

• Variationally equivalent problems and variations of Noether currents
• Topological obstructions in Lagrangian field theories, with an application to 3D Chern-Simons gauge theory
• Lagrangian reductive structures on gauge-natural bundles
• A variational perspective on classical Higgs fields in gauge-natural theories
• Variational derivatives in locally Lagrangian field theories and Noether--Bessel-Hagen currents
• Higgs fields induced by Yang--Mills type Lagrangians on gauge-natural prolongations of principal bundles

Cited by in corpus (2)

• Symmetry transformations of extremals and higher conserved quantities: invariant Yang--Mills connections
• A variational framework for higher order perturbations