The relation between the Jacobi morphism and the Hessian in gauge-natural field theories

We generalize a classic result, due to Goldschmidt and Sternberg, relating the Jacobi morphism and the Hessian for first-order field theories to higher-order gauge-natural field theories. In particular, we define a generalized gauge-natural Jacobi morphism where the variation vector fields are Lie derivatives of sections of the gauge-natural bundle with respect to gauge-natural lifts of infinitesimal principal automorphisms, and we relate it to the Hessian. The Hessian is also very simply related to the generalized Bergmann–Bianchi morphism, whose kernel provides necessary and sufficient conditions for the existence of global canonical superpotentials. We find that the Hamilton equations for the Hamiltonian connection associated with a suitably defined covariant strongly conserved current are satisfied identically and can be interpreted as generalized Bergmann–Bianchi identities and thus characterized in terms of the Hessian vanishing.