  We introduce an generalized action functional describing the equations of
motion and the variational equations for any Lagrangian system. Using this
novel scheme we are able to generalize Noether's theorem in such a way that to
any $n$-parameter continuous symmetry group of the Lagrangian there exist 1)
the usual $n$ constants of motion and 2) $n$ extra constants valid in the
variational equations. The new constants are related to the infinitesimal
generators of the symmetry transformation by relations similar to the ones that
stem from the `nonextended' Noether theorem.
