Noether's theorem: Difference between revisions

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==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.