Noether's theorem: Difference between revisions

Revision as of 17:26, 20 September 2024

Revision as of 17:29, 20 September 2024

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{{calculus|expanded=specialized}}

'''Noether's theorem''' states that every [[continuous symmetry]] of the [[action (physics)|action]] of a physical system with [[conservative force]]s has a corresponding [[conservation law]]. This is the first of two theorems (see [[Noether's second theorem]]) provenpublished by mathematician [[Emmy Noether]] in 1915<ref name="DickNoetherBio1981">{{Cite book |last=Dick |first=Auguste |url=http://link.springer.com/10.1007/978-1-4684-0535-4 |title=Emmy Noether 1882–1935 |date=1981 |publisher=Birkhäuser Boston |isbn=978-1-4684-0537-8 |___location=Boston, MA |language=en |doi=10.1007/978-1-4684-0535-4}}</ref>{{rp|31}} and published in 1918.<ref>{{cite journal | last= Noether |first=E. | year = 1918 | title = Invariante Variationsprobleme | journal = Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen |series=Mathematisch-Physikalische Klasse | volume = 1918 | pages = 235–257 |url= https://eudml.org/doc/59024}}</ref> The action of a physical system is the [[time integral|integral over time]] of a [[Lagrangian mechanics|Lagrangian]] function, from which the system's behavior can be determined by the [[principle of least action]]. This theorem only applies to continuous and smooth [[Symmetry (physics) |symmetries of physical space]].

Noether's theorem is used in [[theoretical physics]] and the [[calculus of variations]]. It reveals the fundamental relation between the symmetries of a physical system and the conservation laws. It also made modern theoretical physicists much more focused on symmetries of physical systems. A generalization of the formulations on [[constants of motion]] in Lagrangian and [[Hamiltonian mechanics]] (developed in 1788 and 1833, respectively), it does not apply to systems that cannot be modeled with a Lagrangian alone (e.g., systems with a [[Rayleigh dissipation function]]). In particular, [[dissipative]] systems with [[Continuous symmetry|continuous symmetries]] need not have a corresponding conservation law.{{Citation needed|reason=The source of this claim would be useful.|date=May 2023}}

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Several alternative methods for finding conserved quantities were developed in the 19th century, especially by [[William Rowan Hamilton]]. For example, he developed a theory of [[canonical transformation]]s which allowed changing coordinates so that some coordinates disappeared from the Lagrangian, as above, resulting in conserved canonical momenta. Another approach, and perhaps the most efficient for finding conserved quantities, is the [[Hamilton–Jacobi equation]].

Emmy Noether's work on the invariance theorem began in 1915 when she was helping [[Felix Klein]] and David Hilbert with their work related to [[Albert Einstein]]'s theory of general relativity<ref name="DickNoetherBio1981">{{Cite book |last=Dick |first=Auguste |url=http://link.springer.com/10.1007/978-1-4684-0535-4 |title=Emmy Noether 1882–1935 |date=1981 |publisher=Birkhäuser Boston |isbn=978-1-4684-0537-8 |___location=Boston, MA |language=en |doi=10.1007/978-1-4684-0535-4}}</ref>{{rp|31}} By March 1918 she had most of the key ideas for the paper which would be published later in the year.<ref>{{Cite book |last=Rowe |first=David E. |url=https://link.springer.com/10.1007/978-3-030-63810-8 |title=Emmy Noether – Mathematician Extraordinaire |date=2021 |publisher=Springer International Publishing |isbn=978-3-030-63809-2 |___location=Cham |language=en |doi=10.1007/978-3-030-63810-8}}</ref>{{rp|81}}

==Mathematical expression==