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Noether's theorem is important, both because of the insight it gives into conservation laws, and also as a practical calculational tool. It allows investigators to determine the conserved quantities (invariants) from the observed symmetries of a physical system. Conversely, it allows researchers to consider whole classes of hypothetical Lagrangians with given invariants, to describe a physical system.<ref name=":0" />{{Rp|page=127}} As an illustration, suppose that a physical theory is proposed which conserves a quantity ''X''. A researcher can calculate the types of Lagrangians that conserve ''X'' through a continuous symmetry. Due to Noether's theorem, the properties of these Lagrangians provide further criteria to understand the implications and judge the fitness of the new theory.
There are numerous versions of Noether's theorem, with varying degrees of generality. There are natural quantum counterparts of this theorem, expressed in the [[Ward–Takahashi identity|Ward–Takahashi identities]]. Generalizations of Noether's theorem to [[superspace]]s also exist.<ref>{{Cite journal|last1=De Azcárraga|first1=J.a.|last2=Lukierski|first2=J.|last3=Vindel|first3=P.|date=1986-07-01|title=Superfields and canonical methods in superspace|url=https://www.worldscientific.com/doi/abs/10.1142/S0217732386000385|journal=Modern Physics Letters A|volume=01|issue=4|pages=293–302|doi=10.1142/S0217732386000385|bibcode=1986MPLA....1..293D|issn=0217-7323|url-access=subscription}}</ref>
== Informal statement of the theorem ==
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