Editing Noether's theorem (section)

==Basic illustrations and background==
As an illustration, if a physical system behaves the same regardless of how it is oriented in space (that is, it is [[Invariant (mathematics)|invariant]]), its [[Lagrangian mechanics|Lagrangian]] is symmetric under continuous rotation: from this symmetry, Noether's theorem dictates that the [[angular momentum]] of the system be conserved, as a consequence of its laws of motion.<ref name=":0">{{Cite book |last1=José |first1=Jorge V. |url=https://www.worldcat.org/oclc/857769535 |title=Classical Dynamics: A Contemporary Approach |last2=Saletan |first2=Eugene J. |date=1998 |publisher=Cambridge University Press |isbn=978-1-139-64890-5 |location=Cambridge [England] |oclc=857769535}}</ref>{{Rp|page=126}} The physical system itself need not be symmetric; a jagged asteroid tumbling in space conserves angular momentum despite its asymmetry. It is the laws of its motion that are symmetric.

As another example, if a physical process exhibits the same outcomes regardless of place or time, then its Lagrangian is symmetric under continuous translations in space and time respectively: by Noether's theorem, these symmetries account for the [[conservation law]]s of [[momentum|linear momentum]] and [[energy]] within this system, respectively.<ref>{{Cite book |last1=Hand |first1=Louis N. |url=https://www.worldcat.org/oclc/37903527 |title=Analytical Mechanics |last2=Finch |first2=Janet D. |date=1998 |publisher=Cambridge University Press |isbn=0-521-57327-0 |location=Cambridge |oclc=37903527}}</ref>{{Rp|page=23}}<ref>{{Cite book |last1=Thornton |first1=Stephen T. |title=Classical dynamics of particles and systems. |last2=Marion |first2=Jerry B. |date=2004 |publisher=Brooks/Cole, Cengage Learning |isbn=978-0-534-40896-1 |edition=5th |location=Boston, MA |oclc=759172774}}</ref>{{Rp|page=261}}

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}}</ref>