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

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