Abstract. This paper is a review containing new original results on the finite order variational sequence and its different representations with emphasis on applications in the theory of variational symmetries and conservation laws in physics.

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 BergmannBianchi 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.

When a gauge-natural invariant variational principle is assigned, to determine canonical covariant conservation laws, the vertical part of gaugenatural lifts of infinitesimal principal automorphisms -defining infinitesimal variations of sections of gauge-natural bundles -must satisfy generalized Jacobi equations for the gauge-natural invariant Lagrangian. Vice versa all vertical parts of gauge-natural lifts of infinitesimal principal automorphisms which are in the kernel of generalized Jacobi morphisms are generators of canonical covariant currents and superpotentials. In particular, only a few gauge-natural lifts can be considered as canonical generators of covariant gauge-natural physical charges.

We investigate globality properties of conserved currents associated with local variational problems admitting global Euler-Lagrange morphisms. We show that the obstruction to the existence of a global conserved current is the difference of two conceptually independent cohomology classes: one coming from using the symmetries of the EulerLagrange morphism and the other from the system of local Noether currents.

We investigate canonical aspects concerning the relation between symmetries and conservation laws in gauge-natural field theories. In particular, we find that a canonical spinor connection can be selected by the simple requirement of the global existence of canonical superpotentials for the Lagrangian describing the coupling of gravitational and Fermionic fields. In fact, the naturality of a suitably defined variational Lagragian implies the existence of an associated energy-momentum conserved current. Such a current defines a Hamiltonian form in the corresponding phase space; we show that an associated Hamiltonian connection is canonically defined along the kernel of the generalized gauge-natural Jacobi morphism and uniquely characterizes the canonical spinor connection.