Symmetry transformations of extremals and higher conserved quantities: invariant Yang--Mills connections

We characterize symmetry transformations of Lagrangian extremals generating `on shell' conservation laws. We relate symmetry transformations of extremals to Jacobi fields and study symmetries of higher variations by proving that a pair given by a symmetry of the $l$-th variation of a Lagrangian and by a Jacobi field of the $s$-th variation of the same Lagrangian (with $s<l$) is associated with an `off shell' conserved current. The conserved current associated with two symmetry transformations is constructed and, as a case of study, its expression for invariant Yang--Mills connections on Minkowski space-times is obtained.