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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]]) published by the mathematician [[Emmy Noether]] 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 applies to continuous and smooth [[Symmetry (physics) |symmetries of physical space]]. Noether's formulation is quite general and has been applied across classical mechanics, high energy physics, and recently statistical mechanics.<ref>{{cite journal | title = Gauge Invariance of Equilibrium Statistical Mechanics | journal = Physical Review Letters | year = 2024 | volume = 133 | issue = 21 | doi = 10.1103/PhysRevLett.133.217101 | last1 = M\"uller | first1 = Johanna | last2 = Hermann | first2 = Sophie | last3 = Samm\"uller | first3 = Florian | last4 = Schmidt | first4 = Matthias | page = 217101 | arxiv = 2406.19235 | bibcode = 2024PhRvL.133u7101M }}</ref>
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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=== Geometric derivation ===
The Noether’s theorem can be seen as a consequence of the [[fundamental theorem of Calculus#Generalizations|fundamental theorem of calculus]] (known by various names in physics such as the [[Generalized Stokes theorem]] or the [[Gradient theorem]]):<ref>{{cite journal | last= Houchmandzadeh |first=B. | year = 2025 | title = A geometric derivation of Noether's theorem | journal = European Journal of Physics | volume = 46 |issue=2 | pages = 025003 |doi=10.1088/1361-6404/adb546 |arxiv=2502.19438 |bibcode=2025EJPh...46b5003H |url= https://hal.science/hal-04682603v3/document}}
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for a function
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