Noether's theorem in classical field theory

Within the lagrangian formalism in classical field theory Noether's theorem is generalized so as to abolish the role of invariance considerations in it. Examples of application of the generalized formulation are presented for comparison with the usual formulation. It is shown that symmetry transformations do not necessarily lead to continuity equations (from which conservation laws are obtained), while transformations associated with continuity equations are not necessarily symmetry transformations. The inverse Noether theorem and families of transformations leading to the same continuity equations are discussed. Possible utility of the generalized formulation even when there is no lagrangian density is suggested.