Noether's theorem: Difference between revisions

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{{shortShort description|Statement relating differentiable symmetries to conserved quantities}}

{{About|Emmy Noether's first theorem, which derives conserved quantities from symmetries|}}

{{Use American English|date=March 2019}}

[[File:Noether theorem 1st page.png|thumb| First page of [[Emmy Noether]]'s article "Invariante Variationsprobleme" (1918), where she proved her theorem]]

{{calculusCalculus|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\"ullerMüller | first1 = Johanna | last2 = Hermann | first2 = Sophie | last3 = Samm\"ullerSammüller | 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}}

==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|url-access=subscription}}</ref>

== Informal statement of the theorem ==

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Consider any trajectory <math>q(t)</math> (bold on the diagram) that satisfies the system's [[Euler-Lagrange equation|laws of motion]]. That is, the [[Action (physics)|action]] <math>S</math> governing this system is [[stationary point|stationary]] on this trajectory, i.e. does not change under any local [[Calculus of variations|variation]] of the trajectory. In particular it would not change under a variation that applies the symmetry flow <math>\varphi</math> on a time segment {{closed-closed|''t''₀, ''t''₁}} and is motionless outside that segment. To keep the trajectory continuous, we use "buffering" periods of small time <math>\tau</math> to transition between the segments gradually.

The total change in the action <math>S</math> now comprises changes brought by every interval in play. Parts, where variation itself vanishes, i.e outside <math>[t_0,t_1]</math>, bring no <math>\Delta S</math>. The middle part does not change the action either, because its transformation <math>\varphi</math> is a symmetry and thus preserves the Lagrangian <math>L</math> and the action <math display="inline"> S = \int L </math>. The only remaining parts are the "buffering" pieces. In these regions both the coordinate <math>q</math> and velocity <math>\dot{q}</math> change, but <math>\dot{q}</math> changes by <math>\delta q / \tau</math>, and the change <math>\delta q</math> in the coordinate is negligible by comparison since the time span <math>\tau</math> of the buffering is small (taken to the limit of 0), so <math>\delta q / \tau\gg \delta q</math>. So the regions contribute mostly through their "slanting" <math>\dot{q}\rightarrow \dot{q}\pm \delta q / \tau</math>.

That changes the Lagrangian by <math>\Delta L \approx \bigl(\partial L/\partial \dot{q}\bigr)\Delta \dot{q} </math>, which integrates to

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which expresses the idea that the amount of a conserved quantity within a sphere cannot change unless some of it flows out of the sphere. For example, [[electric charge]] is conserved; the amount of charge within a sphere cannot change unless some of the charge leaves the sphere.

==== Examples ====

'''I. The [[stress–energy tensor]]'''

For illustration, consider a physical system of fields that behaves the same under translations in time and space, as considered above; in other words, <math>L \left(\boldsymbol\varphi, \partial_\mu{\boldsymbol\varphi}, x^\mu \right)</math> is constant in its third argument. In that case, ''N''&nbsp;=&nbsp;4, one for each dimension of space and time. An infinitesimal translation in space, <math>x^\mu \mapsto x^\mu + \varepsilon_r \delta^\mu_r</math> (with <math>\delta</math> denoting the [[Kronecker delta]]), affects the fields as <math>\varphi(x^\mu) \mapsto \varphi\left(x^\mu - \varepsilon_r \delta^\mu_r\right)</math>: that is, relabelling the coordinates is equivalent to leaving the coordinates in place while translating the field itself, which in turn is equivalent to transforming the field by replacing its value at each point <math>x^\mu</math> with the value at the point <math>x^\mu - \varepsilon X^\mu</math> "behind" it which would be mapped onto <math>x^\mu</math> by the infinitesimal displacement under consideration. Since this is infinitesimal, we may write this transformation as

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:<math>\Lambda^\mu_r = -\delta^\mu_r \mathcal{L}</math>

and thus Noether's theorem corresponds<ref name="Goldstein1980" />{{rp|592}} to the conservation law for the [[stress–energy tensor]] ''T''_''μ''^''ν'', where we have used <math>\mu</math> in place of <math>r</math>. To wit, by using the expression given earlier, and collecting the four conserved currents (one for each <math>\mu</math>) into a tensor <math>T</math>, Noether's theorem gives

:<math>

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(we relabelled <math>\mu</math> as <math>\sigma</math> at an intermediate step to avoid conflict). (However, the <math>T</math> obtained in this way may differ from the symmetric tensor used as the source term in general relativity; see [[Stress–energy tensor#Canonical stress.E2.80.93energy tensor|Canonical stress–energy tensor]].)

'''I. The [[electric charge]]'''

The conservation of [[electric charge]], by contrast, can be derived by considering ''Ψ'' linear in the fields ''φ'' rather than in the derivatives.<ref name="Goldstein1980"/>{{rp|593–594}} In [[quantum mechanics]], the [[probability amplitude]] ''ψ''('''x''') of finding a particle at a point '''x''' is a complex field ''φ'', because it ascribes a [[complex number]] to every point in space and time. The probability amplitude itself is physically unmeasurable; only the probability ''p'' = |''ψ''|² can be inferred from a set of measurements. Therefore, the system is invariant under transformations of the ''ψ'' field and its [[complex conjugate]] field ''ψ''^* that leave |''ψ''|² unchanged, such as

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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}}

</ref>

for a function

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\end{align}</math>

The first term in the brackets is the [[kinetic energy]] of the particle, while the second is its [[potential energy]]. Consider the generator of [[time translation]]s ''<math>Q'' = ''\frac{d''/''}{dt''}</math>. In other words, <math>Q[x(t)] = \dot{x}(t)</math>. The coordinate ''x'' has an explicit dependence on time, whilst ''V'' does not; consequently:

:<math>Q[L] =

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== Applications ==

Application of Noether's theorem allows physicists to gain powerful insights into any general theory in physics, by just analyzing the various transformations that would make the form of the laws involved invariant. For example:

* Invariance of an isolated system with respect to spatial [[translation (physics)|translation]] (in other words, that the laws of physics are the same at all locations in space) gives the law of conservation of [[linear momentum]] (which states that the total linear momentum of an isolated system is constant)

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==See also==

{{Portal|Mathematics|Physics}}

* [[Conservation law]]

{{cols}}

* [[ConservationCharge law(physics)]]

* [[ChargeGauge (physics)symmetry]]

* [[Gauge symmetry (mathematics)]]

* [[Gauge symmetryInvariant (mathematicsphysics)]]

* [[InvariantGoldstone (physics)boson]]

* [[GoldstoneSymmetry boson(physics)]]

*[[Symmetry (physics)]]

{{colend}}

== References ==

{{reflist|37emReflist}}

==Further reading==

* {{citeCite book| isbn=978-3-319-59694-5 |last1=Badin |first1=Gualtiero |last2=Crisciani |first2=Fulvio| |title=Variational Formulation of Fluid and Geophysical Fluid Dynamics –: Mechanics, Symmetries and Conservation Laws | publisher=Springer | year=2018 | pages=218 | doi= 10.1007/978-3-319-59695-2 |bibcode=2018vffg.book.....B |isbn=978-3-319-59694-5 |s2cid=125902566}}

* {{citeCite journalweb |last1=JohnsonBaez |first1=TristanJohn |author-link1=John Baez |title=Noether's Theorem: Symmetryin anda Conservation |journal=Honors Theses |date=2016Nutshell |url=httpshttp://digitalworksmath.unionucr.edu/theseshome/163baez/noether.html |website=math.ucr.edu |access-date=28 August 2020 |publisherdate=[[Union College]]2002}}

* {{citeCite arXiv |eprintlast=physics/9807044Byers |first=Nina |last=Byers|title=E. Noether's Discovery of the Deep Connection Between Symmetries and Conservation Laws |year=1998 |eprint=physics/9807044}}

*{{Cite book | last = Kosmann-Schwarzbach | first = Yvette | author-link = Yvette Kosmann-Schwarzbach | title = The Noether theorems: Invariance and conservation laws in the twentieth century | publisher = [[Springer Science+Business Media|Springer-Verlag]] | series = Sources and Studies in the History of Mathematics and Physical Sciences | year = 2010 | isbn = 978-0-387-87867-6}} [http://www.math.cornell.edu/~templier/junior/The-Noether-theorems.pdf Online copy].

* {{citeCite journal |author1last1=Vladimir Cuesta |first1=Vladimir |author2=Merced Montesinos |author3=José David Vergara |title=Gauge invariance of the action principle for gauge systems with noncanonical symplectic structures |journal=Physical Review D |volume=76 |pages=025025 |year=2007 |issue=2 |doi=10.1103/PhysRevD.76.025025 |bibcode = 2007PhRvD..76b5025C }}

* {{cite journal |last1=Moser |first1=Seth |title=Understanding Noether's Theorem by Visualizing the Lagrangian |journal=Physics Capstone Projects |date=21 April 2020 |pages=1–12 |url=https://digitalcommons.usu.edu/phys_capstoneproject/86/ |access-date=28 August 2020}}

* {{citeCite journal |author1=Hanca, J. |author2=Tulejab, S. |author3=Hancova, M. |title=Symmetries and conservation laws: Consequences of Noether's theorem |journal=American Journal of Physics |volume=72 |issue=4 |pages=428–35 |year=2004 |doi= 10.1119/1.1591764 |url=http://www.eftaylor.com/pub/symmetry.html |bibcode = 2004AmJPh..72..428H }}

*{{Cite book | last = Olver | first = Peter |author-link=Peter J. Olver | title = Applications of Lie groups to differential equations | publisher = [[Springer Science+Business Media|Springer-Verlag]] | edition = 2nd | series = [[Graduate Texts in Mathematics]] | volume = 107 | year = 1993 | isbn = 0-387-95000-1 }}

* {{Cite thesis |last1=Johnson |first1=Tristan |date=2016 |title=Noether's Theorem: Symmetry and Conservation |type=Bachelor's (honors) |url=https://arches.union.edu/do/7fd3d251-014f-40f1-ae43-f08dccc7ee0e |publisher=[[Union College]] |access-date=10 August 2025}}

*{{Cite book | last = Sardanashvily | first = G. | author-link=Gennadi Sardanashvily | title = Noether's Theorems. Applications in Mechanics and Field Theory | publisher = [[Springer Science+Business Media|Springer-Verlag]] | year = 2016 | isbn = 978-94-6239-171-0 }}

* {{Cite book | last = Kosmann-Schwarzbach | first = Yvette | author-link = Yvette Kosmann-Schwarzbach | title = The Noether theorems: Invariance and conservation laws in the twentieth century | publisher = [[Springer Science+Business Media |Springer-Verlag]] | series = Sources and Studies in the History of Mathematics and Physical Sciences | year = 2010 | isbn = 978-0-387-87867-6}} [httphttps://www.math.cornell.edu/~templier/junior/The-Noether-theorems.pdf Online copy].

* {{citeCite arXiv |last1=Leone |first1=Raphaël |title=On the wonderfulness of Noether's theorems, 100 years later, and Routh reduction |date=11 April 2018 |class=physics.hist-ph |eprint=1804.01714 }}

* {{citeCite journal |author1last=Merced Montesinos |first=Merced |author2=Ernesto Flores |journal=Revista Mexicana de Física |title=Symmetric energy–momentum tensor in Maxwell, Yang–Mills, and Proca theories obtained using only Noether's theorem |volume=52 |pages=29–36 |year=2006 |issue=1 |url=http://rmf.smf.mx/pdf/rmf/52/1/52_1_29.pdf |arxiv=hep-th/0602190 |bibcode=2006RMxF...52...29M |access-date=2014-11-12 |archive-date=2016-03-04 |archive-url=https://web.archive.org/web/20160304023543/http://rmf.smf.mx/pdf/rmf/52/1/52_1_29.pdf |url-status=dead }}

* {{citeCite journal |last1=Moser |first1=Seth |title=Understanding Noether's Theorem by Visualizing the Lagrangian |journal=Physics Capstone Projects |date=21 April 2020 |pages=1–12 |url=https://digitalcommons.usu.edu/phys_capstoneproject/86/ |access-date=28 August 2020}}

* {{citeCite book | last1 = Neuenschwander | first1 = Dwight E. | title = Emmy Noether's Wonderful Theorem | publisher = Johns Hopkins University Press | year = 2010 | isbn = 978-0-8018-9694-1}}

* {{citeCite web |author1last=Emmy Noether |first=Emmy |year=1918 |title=Invariante Variationsprobleme |language=de |url=http://de.wikisource.org/wiki/Invariante_Variationsprobleme }}

* {{citeCite journal |author1last=Emmy Noether |translatorfirst=Mort TavelEmmy |year=1971 |title=Invariant Variation Problems |translator=Mort Tavel |journal=Transport Theory and Statistical Physics |volume=1 |issue=3 |pages=186–207 |arxiv=physics/0503066 |doi=10.1080/00411457108231446 |bibcode = 1971TTSP....1..186N |s2cid=119019843 }} (Original in ''Gott. Nachr.'' 1918:235–257)

* {{Cite book | last = Olver | first = Peter |author-link=Peter J. Olver | title = Applications of Lie groupsGroups to differentialDifferential equationsEquations | publisher = [[Springer Science+Business Media |Springer-Verlag]] | edition = 2nd | series = [[Graduate Texts in Mathematics]] | volume = 107 | year = 1993 | isbn = 0-387-95000-1 }}

* {{citeCite arXiv |last1=Quigg |first1=Chris |title=Colloquium: A Century of Noether's Theorem |date=9 July 2019 |class=physics.hist-ph |eprint=1902.01989 }}

* {{citeCite journal |author1=Sardanashvily |first=G. |year=2009 |title=Gauge Conservation Laws in a General Setting: Superpotential |journal=[[International Journal of Geometric Methods in Modern Physics]] |titlevolume=Gauge conservation laws in a general setting. Superpotential6 |volumeissue=6 |pages=1047–1056 |year=2009 |arxiv=0906.1732 |bibcode = 2009arXiv0906.1732S |doi=10.1142/S0219887809003862|issue=6 }}

* {{Cite book | last = Sardanashvily | first = G. | author-link=Gennadi Sardanashvily |year=2016 |title = Noether's Theorems.: Applications in Mechanics and Field Theory | publisher = [[Springer Science+Business Media |Springer-Verlag]] | year = 2016 | isbn = 978-94-6239-171-0 }}

==External links==

* [http://www.mathpages.com/home/kmath564/kmath564.htm Noether's Theorem] at MathPages.<!-- Previously a referenced note; reference is lost, but we can assume this is still a valid citation -->

* {{cite web |author1=Emmy Noether |year=1918 |title=Invariante Variationsprobleme |language=de |url=http://de.wikisource.org/wiki/Invariante_Variationsprobleme }}

* {{cite journal |author1=Emmy Noether |translator=Mort Tavel |year=1971 |title=Invariant Variation Problems |journal=Transport Theory and Statistical Physics |volume=1 |issue=3 |pages=186–207 |arxiv=physics/0503066 |doi=10.1080/00411457108231446 |bibcode = 1971TTSP....1..186N |s2cid=119019843 }} (Original in ''Gott. Nachr.'' 1918:235–257)

*{{cite arXiv |eprint=physics/9807044 |first=Nina |last=Byers|title=E. Noether's Discovery of the Deep Connection Between Symmetries and Conservation Laws |year=1998}}

* {{cite web |last1=Baez |first1=John |author-link1=John Baez |title=Noether's Theorem in a Nutshell |url=http://math.ucr.edu/home/baez/noether.html |website=math.ucr.edu |access-date=28 August 2020 |date=2002}}

*{{cite journal |author1=Vladimir Cuesta |author2=Merced Montesinos |author3=José David Vergara |title=Gauge invariance of the action principle for gauge systems with noncanonical symplectic structures |journal=Physical Review D |volume=76 |pages=025025 |year=2007 |issue=2 |doi=10.1103/PhysRevD.76.025025 |bibcode = 2007PhRvD..76b5025C }}

*{{cite journal |author1=Hanca, J. |author2=Tulejab, S. |author3=Hancova, M. |title=Symmetries and conservation laws: Consequences of Noether's theorem |journal=American Journal of Physics |volume=72 |issue=4 |pages=428–35 |year=2004 |doi= 10.1119/1.1591764|url=http://www.eftaylor.com/pub/symmetry.html|bibcode = 2004AmJPh..72..428H }}

* {{cite arXiv |last1=Leone |first1=Raphaël |title=On the wonderfulness of Noether's theorems, 100 years later, and Routh reduction |date=11 April 2018|class=physics.hist-ph |eprint=1804.01714 }}

*[http://www.mathpages.com/home/kmath564/kmath564.htm Noether's Theorem] at MathPages.<!-- Previously a referenced note; reference is lost, but we can assume this is still a valid citation -->

*{{cite journal |author1=Merced Montesinos |author2=Ernesto Flores |journal=Revista Mexicana de Física |title=Symmetric energy–momentum tensor in Maxwell, Yang–Mills, and Proca theories obtained using only Noether's theorem |volume=52 |pages=29–36 |year=2006 |issue=1 |url=http://rmf.smf.mx/pdf/rmf/52/1/52_1_29.pdf |arxiv=hep-th/0602190 |bibcode=2006RMxF...52...29M |access-date=2014-11-12 |archive-date=2016-03-04 |archive-url=https://web.archive.org/web/20160304023543/http://rmf.smf.mx/pdf/rmf/52/1/52_1_29.pdf |url-status=dead }}

* {{cite book | last1 = Neuenschwander | first1 = Dwight E. | title = Emmy Noether's Wonderful Theorem | publisher = Johns Hopkins University Press | year = 2010 | isbn = 978-0-8018-9694-1}}

* {{cite arXiv |last1=Quigg |first1=Chris |title=Colloquium: A Century of Noether's Theorem |date=9 July 2019|class=physics.hist-ph |eprint=1902.01989 }}

*{{cite journal|author1=Sardanashvily|journal=[[International Journal of Geometric Methods in Modern Physics]]|title=Gauge conservation laws in a general setting. Superpotential |volume=6 |pages=1047–1056 |year=2009 |arxiv=0906.1732|bibcode = 2009arXiv0906.1732S|doi=10.1142/S0219887809003862|issue=6 }}

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