Abstract
We generalize the first and second Noether theorems (Noether identities) to a constrained system in phase space. As an example, the conservation law deriving from Lagrange's formalism cannot be obtained fromH E via the generalized first Noether theorem (GFNT); Dirac's conjecture regarding secondary first-class constraints (SFCC) is invalid in this example. A preliminary application of the generalized Noether identities (GNI) to nonrelativistic charged particles in an electromagnetic field shows that on the constrained hypersurface in phase space one obtains electric charge conservation. This conservation law is valid whether Dirac's conjecture holds true or not.
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Zi-Ping, L., Xin, L. Generalized Noether theorems and applications. Int J Theor Phys 30, 225–233 (1991). https://doi.org/10.1007/BF00670715
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DOI: https://doi.org/10.1007/BF00670715