a Dipartimento di Fisica, Università degli Studi di Milano and I.N.F.N., Sezione di Milano, Via Celoria 16, 20133, Milano, Italy
b Dipartimento di Fisica, Università degli Studi di Milano, Via Celoria 16, 20133, Milano, Italy
Abstract:
By means of the jet-bundle formalism, the Second Noether Theorem is formulated for a general first-order Lagrangian field theory with infinitesimal local symmetries. These symmetries are implemented by a linear differential operator acting between the sections of a vector bundle and vector fields on the configuration bundle. The problem of the degeneration of the Lagrangian system is examined from a covariant and an instantaneous (i.e. space+time split) viewpoint. It is shown that in the instantaneous approach the presence of infinitesimal local symmetries leads to degeneration of the theory. Vertical local symmetries are shown to imply degeneration also in the covariant formalism. These results can be extended to higher-order Lagrangians as well.