Abstract
The extended canonical Noether identities and canonical first Noethertheorem derived from an extended action in phase space for a system with asingular Lagrangian are formulated. Using these canonical Noetheridentities, it can be shown that the constraint multipliers connected withthe first-class constraints may not be independent, so a query to aconjecture of Dirac is presented. Based on the symmetry properties of theconstrained Hamiltonian system in phase space, a counterexample to aconjecture of Dirac is given to show that Dirac's conjecturefails in such a system. We present here a different way rather thanCawley's examples and other's ones in that there is nolinearization of constraints in the problem. This example has a feature thatneither the primary first-class constraints nor secondary first-classconstraints are generators of the gauge transformation.
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