On the Noether identities for a class of systems with singular Lagrangians

Abstract

A straightforward method for obtaining the generators of Lagrangian gauge transformations within the Lagrangian formalism is presented. This procedure can be carried out completely without the need of developing the Dirac–Hamilton formalism. Singular Lagrangians which may generate Lagrangian constraints are considered. It is assumed that the consistency conditions on the Lagrangian constraints do not generate new independent equations for the accelerations. From the structure of these consistency conditions, the Noether identities are explicitly constructed and the Lagrangian generators of gauge transformations are obtained. It is shown that in the generic case of systems having chains of Lagrangian constraints up to a level k (k≥2), the variations δq under which the action is invariant depend on q, dq/dt, and higher time derivatives up to dkq/dtk. The relation between this Lagrangian approach and the standard Dirac–Hamiltonian approach is discussed.