2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/96205We consider the second variational derivative of a given gauge-natural invariant Lagrangian taken with respect to (prolongations of) vertical parts of gauge-natural lifts of infinitesimal principal automorphisms. By requiring such a second variational derivative to vanish, {\em via} the Second Noether Theorem we find that a covariant strongly conserved current is canonically associated with the deformed Lagrangian obtained by contracting Euler--Lagrange equations of the original Lagrangian with (prolongations of) vertical parts of gauge-natural lifts of infinitesimal principal automorphisms lying in the kernel of the generalized gauge-natural Jacobi morphism.17 pages; some misprints corrected, few changes, reference list updated, v3 to appear in Proc. IX Int. Conf. on Diff. Geom. and its Appl. (Prague 30/08-03/09/2004)Mathematical PhysicsDifferential Geometry58A20;58A32;58E30;58E40;58J10;58J70text