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

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.