We have derived the generalized Noether's first theorem in canonical formalism for field theory with singular Lagrangian and generalized Noether's identities in canonical formalism for variant Lagrangian. The application to the Dirac's constraint theory was discussed. We point out that some variance system may has a Dirac constraint. A counter-example to a conjecture of Dirac was given which indicates that Dirac's conjecture is invalid in this example. The application of generalized Noether's identities to the chromodynamics is also discussed.
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