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

TL;DR

This paper explores the geometric structure of higher order variational problems, characterizing the second variation via Jacobi morphisms and currents, with applications to Yang-Mills theory on Minkowski space.

Contribution

It introduces a novel geometric framework linking Jacobi morphisms, the Hessian, and conserved currents in higher order field theories, including explicit Yang-Mills examples.

Findings

Jacobi morphism characterizes second variation in higher order Lagrangians

Pairs of Jacobi fields generate conserved currents

Explicit Yang-Mills example on Minkowski 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.