Noether's theorem: Difference between revisions

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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]]) proven by mathematician [[Emmy Noether]] in 1915<ref name="DickNoetherBio1981">{{clarifyCite book |last=Dick |first=Auguste |url=http://link.springer.com/10.1007/978-1-4684-0535-4 |title=Emmy Noether 1882–1935 |date=September1981 2024|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}}