# Talk:Noether's theorem

## Initial Discussion

• This is fascinating. They should teach this in undergraduate physics: what a powerful idea!
They do teach it in undergrad physics, usually during the second term of mechanics. —Preceding unsigned comment added by 128.135.100.164 (talk) 23:52, 17 February 2008 (UTC)
Unfortunately, that is not taught in undergrad physics at all schools. And where it is, it is taught in upper division courses, not even necessarily in as general a form as in this article 207.62.246.131 (talk) 04:41, 10 February 2009 (UTC)
• It is actually a misleading idea because it suggests that there is such a thing as a general law of energy conservation in physics. There isn't, because the notion of energy can be strictly defined only in Newtonian Physics. Noether's theorem in fact assumes Newtonian Physics as it uses Lagrangian functions which in turn contain potential energy functions which in turn can only be defined for conservative force fields, i.e. for Newtonian physics (for related aspects see my site http://www.physicsmyths.org.uk/conservation.htm .
"the notion of energy can be strictly defined only in Newtonian Physics". Eh? -- The Anome 18:22, 8 Mar 2004 (UTC)
The Anome is right to question the assertion that it is a "misleading idea". Of course there is a "general law of energy conservation in physics". It it absurd to deny it. Nor is is true that Noether's Theorem assumes conservative force fields only. It assumes only a Lagrangian invariant with respect to the transformations of some group.
Thus, for example, we have Cornelius Lanczos's statement of it as:
Noether considers variational problems having the property that the action integral remains invariant with respect to a group of transformations, applied either to the dependent or the independent variables. She shows that every parameter associated with such transformations leads to a corresponding conservation law (from The Variational Principles of Mechanics, Lanzcos, Cornelius p401, 4th ed. Dover 1970)
Perhaps the confusion comes from confusing the Lagrangian and its fundamental theorems with the Hamiltonian: the latter can be formed only for monogenic forces and constraints, which in turn implies a conservative work function. But there is no such restriction on Noether's theorem. 207.62.246.131 (talk) 04:41, 10 February 2009 (UTC)
You might also want to read this post from John Baez -- The Anome 13:52, 2 May 2004 (UTC)
• Question: Noether or Nöther? -- Anon.
It's Noether. 199.17.230.76 18:31, 23 Oct 2004 (UTC)
Heh. Noether, or neither? :-) zowie 00:17, 27 May 2006 (UTC)
The reason is that the umlaut in German has long been considered an abbreviated form for the same vowel followed by the latter 'e'. But since English has no umlauts, we do not adopt them in the English spelling of German names 207.62.246.131 (talk) 04:41, 10 February 2009 (UTC)
So they're both correct, but Noether is standard. — LlywelynII 10:42, 22 May 2011 (UTC)
No. Only Noether is correct. She never used an umlaut in her own name, as Goethe never used an umlaut in his. Colin McLarty (talk) 18:09, 1 April 2012 (UTC)

## Please state the theorem

The article should cleanly state the theorem before starting to give several proofs. As it stands, the intro contains some general intuitive hints, but the statement of the theorem is nowhere to be found. 199.17.230.76 18:31, 23 Oct 2004 (UTC)

To every symmetry group transformation, there corresponds a conserved current.
How's this formulation? Ancheta Wis 07:46, 24 Oct 2004 (UTC)
Well, that's a slogan but not a theorem. It omits the assumptions, and doesn't refer to cleanly defined concepts. 199.17.230.81 18:50, 24 Oct 2004 (UTC)
The concepts are meaningful to a physicist, but not to a mathematician, I see. I suggest reading invariant, conservation law, law of physics etc. If those are insufficient for you, then there are mathematical reviews in the literature which should meet your viewpoint. I should warn you that even John von Neumann's mathematical reviews of quantum mechanics etc did not survive close scrutiny by others, so you may wish to point out where you see deficiencies, and then we can make this the basis of a to-do list whose objective is to rectify the deficiencies, point by point. Are you willing to concede the concept of an observer, or do we have to go farther back than that? Ancheta Wis 23:19, 24 Oct 2004 (UTC)
I share the concerns of 199.17.230.81. The problem is not (at least not for me) that the concept "symmetry group transformation" and "conserved current" are not clear, but that it is simply not true that to every symmetry corresponds a conservation law. As the first sentence states, the model needs to be based on an action principle, but I did not find an explanation on what this exactly means. -- Jitse Niesen 10:25, 25 Oct 2004 (UTC)

Well, one could try to state the theorem that Emmy Noether actually proved, rather than discussing what people assume it says.

Charles Matthews 12:12, 25 Oct 2004 (UTC)

Great idea! Perhaps we can have three sections: statement, proof, controversy about physical implications. -- The Anome 12:14, 25 Oct 2004 (UTC)

There is even an English translation on the Web:

Charles Matthews 16:08, 25 Oct 2004 (UTC)

Unfortunately, the English translation shows that the theorem in the original paper uses many implicit assumptions given in the introduction to the paper, in particular that one is considering a variational problem. —Preceding unsigned comment added by 129.215.104.121 (talk) 13:43, 16 December 2010 (UTC)

## Textbook style

This article seems to be written for a mathematics textbook, not for an encyclopedia. The following quote illustrates the problem: "But if you think about it, any two conserved currents differ by a divergenceless vector field". Umm, yeah. Maybe I'm being unreasonable expecting that an encyclopedia article should be comprehensible to someone who studied mathematics up to science degree level? Metamatic 16:17, 16 Dec 2004 (UTC)

That's the personal style of one contributor. I deprecate it. Charles Matthews 22:11, 16 Dec 2004 (UTC)
So do I - it may be a relatively unpleasant style even for a math textbook. But it is counterproductive to criticize it if you don't offer a better version than the person who wrote it. Is the beginning acceptable? When we have time, we can add a meaningful and comprehensible treatment of the mathematical issues, too. --Lumidek 22:58, 16 Dec 2004 (UTC)
Criticism may encourage someone else to attempt to translate it into English. And I would offer a better version if I could actually understand it to start with. Metamatic 23:01, 2005 May 27 (UTC)
You might start reading at Divergence#Properties, working backward from the goal of understanding the article, up through the links until an article is comprehensible from your POV. Then start reading forward from that page, drilling down through the links, translating the statements to yourself (not just parroting the content of the article) until you can state something consequential in English (but it is probably best to restrict the statement to the Talk page, until you attain consensus on the content of your statement). When you finally get to something you want you say in Wikipedia, then Be Bold. Ancheta Wis 15:48, 28 May 2005 (UTC)
Indeed, it would be prudent if someone can verify that the proof sections as currently written are not copyright violations of an actual published textbook passage. It reads rather like one. If it really is the case, at the very least we should have a proper citation of the source text to avoid plagiarism concerns. 24.16.32.174 10:31, 24 May 2006 (UTC)

I doubt the contributions by User:Phys from 2004, which are still probably extant in the article and have the chatty flavour, are anything to worry about. Charles Matthews 12:01, 24 May 2006 (UTC)

For what it's worth, here's a proposed revision which I think greatly minimize the "chattiness", eliminates pointless weasel words, unnecessary fluff, and improves organization (eg. previously the "Application" and "Explanation" sections are all jumbled together). I'm still concern about plagiarism though (due to lack of citation to original sources of the presented proof), as well as whether there might be a proof that might be more accessible. (I personally have no hope of understanding it even with my college engineering background, but fortunately I'm just copyediting here.) 24.16.32.174 12:13, 24 May 2006 (UTC)

I am sorry to re-open the debate on this page. I studied Noether's Theorem as part of a course on Calculus of Variations last year for my MSc in Maths. It seems to me that the main problem with this article (which has great potential) is that it is not clear that Noether's Theorem is part of Calculus of variations and can only be understood as part of that subject. Noether's Theorem is just not comprehensible (or applicable) if problems are not formulated in terms of a Variational principle. So I would strongly recommend that the introduction be re-written to make this clear. If everyone else doesn't have any problems with this I will take on the task of the re-write of the Intro. Wilmot1 13:48, 18 October 2007 (UTC)

## Nina Byers' article

This is my reading of Nina Byers' article, whose reference I added to the page yesterday; I started with the lead sentence of the variational principle article, and augmented it based on the Byers article: Ancheta Wis 20:20, 30 Jan 2005 (UTC)

A variational principle is a principle in physics which is expressed in terms of the calculus of variations. According to Cornelius Lanczos, any physical law which can be expressed as a variational principle describes an expression which is self-adjoint1. These expressions are also called Hermitian. Thus such an expression describes an invariant under a Hermitian transformation. Felix Klein's Erlangen program attempted to identify such invariants under a group of transformations. On July 16, 1918, before a scientific organization in Goettingen, Klein read a paper written by Emmy Noether, because she was not allowed to present the paper before the scientific organization herself. In particular, in what is referred to in physics as Noether's theorem, this paper identified the conditions under which the Poincaré group of transformations (what is now called a gauge group) for General Relativity define conservation laws. (The relationship of these invariants (the symmetries under a group of transformations) and what are now called conserved currents, depends on a variational principle, or action principle.) Noether's papers made the requirements for the conservation laws precise.

Hilbert had derived the same equation as the Einstein equation for General Relativity within a period of the same few weeks as Einstein, in November 1915. The chief difficulty, which concerned David Hilbert, was that the conservation of energy does not hold for a region subject to a gravitational field. (Byers' commentary2 notes that sometimes the objects which are needed to define conserved quantities are not tensors, but pseudotensors.3) Hilbert's unified theory remained uncelebrated because of this difficulty. Noether's theorem remains right in line with current developments in physics to this day.

### Notes

I propose adding this to the Variational principle article, with links to Noether's theorem. Ancheta Wis 09:55, 1 Feb 2005 (UTC)

Well, I would hope to have this clarified. I don't understand what it means for an expression to be self-adjoint, for example: that terminology usually applies to an operator. I also don't understand what is being said about pseudotensors, though that might be my ignorance. The comment about Hilbert's work seems excessive. Charles Matthews 19:13, 28 May 2005 (UTC)
1) Based on my background, to me, a matrix, an operator, or other form (including some expression) can be Hermitian. 2) I refer to Byers' last paragraph of part III in her article. 3) I will strike the Hilbert statement in the Variational principle article, based on your comment. Ancheta Wis 21:13, 28 May 2005 (UTC)

## Conformal transformation

How is the ${\displaystyle \phi ^{4}\,}$ example a conformal transformation? It would seem that it is merely a scale transformation. A general conformal transformation admits special conformal transformations as well. Even the non-interacting case is only conformally invariant in 2D. Furthermore, couple it to gravity, find the stress energy tensor ${\displaystyle T^{\mu \nu }={\frac {1}{\sqrt {g}}}{\frac {\delta S}{\delta g_{\mu \nu }}}\,}$ and find that it is not traceless. Is there an improvement term?--Lionelbrits 03:47, 25 March 2007 (UTC)

## Phase and electric charge

The Noether current associated with phase is a charge carrying EM current, not charge itself -- and I don't think either can be thought of a conjugate variables. The article needs amending to reduce the over-emphasis on conjugate variables. --Michael C. Price talk 13:42, 2 June 2007 (UTC)

## Conserved Current

Under "Mathematical statement of the theorem" I followed the link to "Conserved current" and there it appears to be a thing which is conjugate to a variable which has a differentiable symmetry, or something that fits this theorem. This seems to be a bit tautologous. Is it a current which is non-divergent? I will be adding an appeal to the conserved current page. Please forgive any clumsiness in my editing; this is the first time I have felt obliged to add anything to this very fine project so please, experienced users, if I have erred please let me know gently... Toospaice 04:55, 22 June 2007 (UTC)

## making it more comprehensible

Is there a good reason why this article throws around a lot of terminology, gives a hyper-vague 'mathematical statement', and then spends most of the energy on the highly technical proof? As Jitse Niesen remarked in rating the article, it would have been more logical and more comprehensible to start with a simple finite-dimensional Hamiltonian case, stating the theorem precisely, illustrate it by examples, and then discuss a variational formulation. Also, I know that the following approach was tried at other articles where proof distracts too much from the subject: move the proof into a separate (more technical) article. If there is a strong agreement on keeping the present structure, I would feel inclined to write a separate, gentler article of a type Noether theorem (symplectic geometry). Opinions? Arcfrk 04:14, 26 June 2007 (UTC)

## Example 2 from article

I've removed the following example:

### Example 2: Conservation of linear momentum

Still considering 1-dimensional time, let

i.e. N Newtonian particles where the potential only depends pairwise upon the relative displacement.

For ${\displaystyle {\vec {Q}}}$, let's consider the generator of Galilean transformations (i.e. a change in the frame of reference). In other words,

${\displaystyle Q_{i}[x_{\alpha }^{j}(t)]=t\delta _{i}^{j}.}$

Note that

${\displaystyle Q_{i}[{\mathcal {L}}]=\sum _{\alpha }m_{\alpha }{\dot {x}}_{\alpha }^{i}-\sum _{\alpha <\beta }\partial _{i}V_{\alpha \beta }({\vec {x}}_{\beta }-{\vec {x}}_{\alpha })(t-t)}$
${\displaystyle =\sum _{\alpha }m_{\alpha }{\dot {x}}_{\alpha }^{i}.}$

This has the form of ${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\sum _{\alpha }m_{\alpha }x_{\alpha }^{i}}$ so we can set

${\displaystyle {\vec {f}}=\sum _{\alpha }m_{\alpha }{\vec {x}}_{\alpha }.}$

Then,

${\displaystyle {\vec {j}}=\sum _{\alpha }\left({\frac {\partial }{\partial {\dot {\vec {x}}}_{\alpha }}}{\mathcal {L}}\right)\cdot {\vec {Q}}[{\vec {x}}_{\alpha }]-{\vec {f}}}$
${\displaystyle =\sum _{\alpha }(m_{\alpha }{\dot {\vec {x}}}_{\alpha }t-m_{\alpha }{\vec {x}})}$
${\displaystyle ={\vec {P}}t-M{\vec {x}}_{CM}}$

where ${\displaystyle {\vec {P}}}$ is the total momentum, M is the total mass and ${\displaystyle {\vec {x}}_{CM}}$ is the center of mass. Noether's theorem states:

${\displaystyle {\dot {\vec {j}}}=0\Rightarrow {\vec {P}}-M{\dot {\vec {x}}}_{CM}=0}$.

This example is confused. The conservation of momentum comes from spatial translation invariance. This example uses momentum translation invariance. A boost (change of reference frame) is a translation in momentum and the conserved quantity is the displacement of the center of mass. I may fix this if I find the time but I more than welcome someone else to do this. Alfred Centauri 00:22, 30 July 2007 (UTC)

Hi, I've restored the example, unchanged, to the article on the grounds that all the examples need a drastic rewrite and I don't see that example #2 is any worse than the rest. Look at example #1, for example: Modeling a one dimensional particle and it starts blathering on about curved Riemannian space and metrics. urgh! I can only think that it was a cut-and-paste job from somewhere else.--Michael C. Price talk 07:39, 30 July 2007 (UTC)
Seeing how long the article has stood without much improvement, I'm inclined to remove it all and start afresh. What do you guys think about that? Unfortunately, I don't know that much of Noether's theorem in a PDE setting, which makes it hard to write about that and impossible to try and fix the examples. Or perhaps you think that it's better if I write a bit about the ODE setting and leave the examples as they are? -- Jitse Niesen (talk) 08:33, 30 July 2007 (UTC)
I favour leaving the examples in until we have better material available. Bad though the examples are, they are of some help. I was intending to read the article completely -- if it is readable -- before deciding on a course of action. In the meantime do add any ODE (ordinary DE?) stuff you know. I like the suggestion mentioned in the previous section, which you've made, of starting from some simple examples before moving on to the more generic proofs. --Michael C. Price talk 09:05, 30 July 2007 (UTC)
I've changed my mind, having reached the point where example 1 has reduced to
${\displaystyle {\frac {\partial L}{\partial t}}={\frac {\mathrm {d} L}{\mathrm {d} t}}}$
which is just plain silly. I've only ever seen Noether's theorem applied non-trivially to the field theory case, anyway. I would say, blast (or tuck away in a more technical article) everything from section 3 (Proof) onwards and crib what we need from the excellent Baez and Byers links in the reference section. They can't copyright the laws of physics, so we should be okay.--Michael C. Price talk 15:45, 30 July 2007 (UTC)

<sigh> Apparently you guys didn't understand... Example 2 is not an example of conservation of momentum. If you are going to put it back in, at least change the title of the example. Alfred Centauri 12:56, 30 July 2007 (UTC)

Okay, I'll leave you to update example 2. --Michael C. Price talk 14:33, 30 July 2007 (UTC)

Sorry, but that example is either wrong or I misunderstand it. Shouldn't it be similar to

${\displaystyle {\vec {j}}=\sum _{\alpha }\left({\frac {\partial }{\partial {\dot {\vec {x}}}_{\alpha }}}{\mathcal {L}}\right)\cdot {\vec {Q}}[{\vec {x}}_{\alpha }]-{\vec {f}}}$
${\displaystyle =\sum _{\alpha }(m_{\alpha }{\dot {\vec {x}}}_{\alpha }t-m_{\alpha }{\vec {x}})}$
${\displaystyle ={\vec {P}}t-M{\vec {x}}_{CM}}$

where ${\displaystyle {\vec {P}}}$ is the total momentum (which is time dependent), M is the total mass and ${\displaystyle {\vec {x}}_{CM}}$ is the center of mass (which is time dependent). So we need some kind of product rule and then Noether's theorem states:

${\displaystyle {\frac {d}{dt}}{\vec {j}}=0\Rightarrow {\vec {P}}+{\dot {\vec {P}}}t-M{\dot {\vec {x}}}_{CM}=0}$.

This is a totally different result (regarding interpretation) compared to the article. — Preceding unsigned comment added by 92.75.133.242 (talk) 00:16, 8 September 2011 (UTC)

## WikiProject class rating

This article was automatically assessed because at least one WikiProject had rated the article as start, and the rating on other projects was brought up to start class. BetacommandBot 10:00, 10 November 2007 (UTC)

## Simpler special-case proof?

What do people think of prefacing the proof section by a simple proof of a special case amenable to techniques of freshman-level calculus? The proof for time-invariant symmetries of configuration spaces built on R^n, or even just R, for example. The proof for the very general case currently presented is accessible only to specialists, I fear. PerVognsen (talk) 09:35, 23 November 2007 (UTC)

## Disagrees with Emmy Noether

According to the article on Emmy Noether, Noether's theorem provides a one-to-one correspondence between conserved currents and symmetries. This article doesn't show how to get from a conserved quantity to a symmetry. Also, it doesn't assign a unique conserved current to each symmetry, since there's so much latitude in choosing f, which is introduced without explanation during the proof. It's not even clear that such an f can always be chosen, but when it can, any divergence free vector field can be added to f, which will give a complete different conserved current. Frankly, this article doesn't make any sense, especially the proof. Maybe we should explain what a "functional derivation" is, and why it can be applied to numbers, functions from M to T, and functions from M to R. I'm not sure whether it even makes sense to think of Q[] as a derivation since it's being applied to functions like phi which take values in an arbitrary manifold, not a ring. 128.208.87.93 (talk) 02:29, 6 March 2008 (UTC)

I am surprised. A symmetry characterizes that which remains unchanged under a transformation. A conserved quantity remains unchanged by definition. What is it that you do not see? --Ancheta Wis (talk) 11:18, 6 March 2008 (UTC)
Historically, this arose from the Erlangen program. In fact, Felix Klein presented her paper in Göttingen because she was not allowed to do so, July 16th, 1918. The Emmy Noether article needs work, by the way. --Ancheta Wis (talk) 11:33, 6 March 2008 (UTC)
The Erlangen program article has a table which lists 3 columns of transformations (dilation, reflection, translation) but which is missing a fourth column, the Inversion transformation, whose invariants are an example of a hidden symmetry. (These sorts of transformations are studied in physics right now) Roger Penrose has publicized some aspects of this fourth column, for example in his lectures about the Weyl curvature hypothesis. --Ancheta Wis (talk) 02:45, 7 March 2008 (UTC)
The whole point is that there isn't a unique conserved current associated with each symmetry. Take translations for example; we have so many equally valid conserved stress-energy tensors out there, each only differing by a surface term. However, the integral of any conserved current from this family over a spatial cross section will give the same Noether charge, which is really what matters. AnonyScientist (talk) 08:49, 19 April 2008 (UTC)

## Converse of Noether's theorem true?

The Goldstein reference (p. 594) states that the converse of Noether's theorem is not true, that some conservation laws cannot be derived from Noether's theorem. He cites the conserved quantities associated with soliton solutions, as appear in the Sine-Gordon equation and the Korteweg-de Vries equation. I'm not sure if this is a fair example, but please see the discussion at the Laplace-Runge-Lenz vector page as well. There is a way to derive its conservation from the Noether theorem, but some physicists seem to view the required "symmetry" transformation as cheating. ;) I'm willing to believe that Goldstein was mistaken, but I think we need a reference to the scientific literature before we can include it in the article. Willow (talk) 21:04, 18 April 2008 (UTC)

About solitonic superselection sectors, in quantum physics, there is a symmetry which multiplies each sector with a phase which is proportional to the topological number. However, classically, the Noether charge, which corresponds to the topological number only generates the trivial transformation despite the fact that it's not constant. The problem is, the topological sectors are disjoint and the topological charge is constant for each sector, even though it's not constant between sectors. So you're right, there is a classical counterexample in a weak sense. AnonyScientist (talk) 07:08, 19 April 2008 (UTC)
The treatment of this issue in the literature is pretty confused, in the sense that people use "Noether's theorem" to refer to different mathematical statements. The best summary I've found so far is in this paper, http://www.jstor.org/stable/2029651?origin=JSTOR-pdf, which of course isn't on the open web. Emmy Noether, in her original paper, used the invariance of the action functional, but some other texts I've looked at only treat the invariance of the Lagrangian itself. Because different Lagrangians can lead to the same equations of motion, knowing a conserved quantity and the Lagrangian for a system doesn't lead to a unique invariance, since there may be another Lagrangian with a different symmetry that creates the same equations of motion. I don't have that version of Goldstein, so I don't know what's going on in his examples. AnonyScientist, you'd have to explain what that means to me—even with a background as far as grad school, I don't know enough to interpret your statement :). Iainuki (talk) 13:17, 3 October 2008 (UTC)
I don't have Goldstein's version as well, but according to the examples you listed, his statement is probably correct. In Ch. 10 of L.H. Ryder's QUANTUM FIELD THEORY (2nd ed.), he shows how various conserved quantities arise from topological aspects of the theory, rather than from some symmetry of the action. For ex., in pg. 395 he discusses a conserved current and clearly states that it "...does not follow from the invariance of [the Lagrangian density] under any symmetry transformation. It is therefore not a Noether current." So I guess both Goldstein and Ryder probably know what they're talking about... NN22 (talk) 18:06, 10 April 2010 (UTC)
V.I. Arnold, Mathematical Methods in Classical Mechanics, claims that many references incorrectly state the converse is true. 129.79.110.70 (talk) 15:35, 7 April 2011 (UTC)

## Dali picture

While a very famous painting n'all, why on earth is there one of Dali's paintings on this article? What has "warping clocks" got to do with invariance under transformation? Deamon138 (talk) 10:03, 3 May 2008 (UTC)

Somebody removed it: good and thanks whoever it was! Deamon138 (talk) 16:20, 4 May 2008 (UTC)
I didn't notice this comment before I removed it; I was working from a list of nonfree images in math articles, which have to be cleaned up every few months. There's a policy, WP:NFCC, which says among other things we can't use nonfree images when a free image could serve the same purpose. In this case, we could definitely make a free image that illustrates a coordinate transformation. — Carl (CBM · talk) 16:43, 4 May 2008 (UTC)

## Confusing nature?

Somebody came and removed the confusing tag. Is this correct? I myself find it confusing, but then no more than a lot of Physics/Mathematics topics and though they're two subjects I know most about. Deamon138 (talk) 19:40, 4 May 2008 (UTC)

## This can't be right

As another example, if a physical experiment has the same outcome regardless of place or time (having the same outcome, say, in Cleveland on Tuesday and Samaria on Wednesday), then its Lagrangian is symmetric under continuous translations in space and time; by Noether's theorem, these symmetries account for the conservation laws of linear momentum and energy within this system, respectively.

Now why can't there be a reproducible experiment that doesn't conserve energy? Such alternate laws of physics have been discussed. —Preceding unsigned comment added by 75.45.2.84 (talk) 03:18, 16 June 2008 (UTC)
There is another condition — the laws of physics in question must arise the principle of least action. Can you give an example of a hypothetical action which is symmetrical with respect to translation through time and yet violates conservation of energy? Remember that you are not free to define energy arbitrarily — it must be the concept of energy indicated by Noether's theorem. JRSpriggs (talk) 14:53, 16 June 2008 (UTC)
I haven't yet reached a high enough level to understand a good deal of this yet, however, is Noether's theorem applicable to Relativistic physics, or does it only work with classical newtonian physics? I'm asking because above you mentioned conservation of energy (which can only be using classical physics) but using relativity we would say "conservation of energy-mass content" or whatever more technical name there is for that. So can Noether's theorem be used to show the conservation of that instead using Einstein etc? Deamon138 (talk) 20:43, 16 June 2008 (UTC)
Yes. It applies to relativity as well as classical physics. In relativity, the energy of an object includes the energy equivalent of the object's mass. We just say "mass-energy" (i.e. energy) for the benefit of those who are not used to thinking in terms of relativity. I am not aware that Einstein ever doubted the conservation of energy or felt it necessary to prove that it is conserved. However, he did many things, with most of which I am not specifically familiar. JRSpriggs (talk) 21:08, 16 June 2008 (UTC)
Try, for example, this Alfred Centauri (talk) 00:22, 18 June 2008 (UTC)
Thanks for that link. Following a quick scan, it looks detailed yet easier to follow than this article! Should be a good read cheers. Deamon138 (talk) 22:28, 18 June 2008 (UTC)

## Redundancy and derivations

I'd like to begin reducing the redundancy in the article, especially the multiple repetitions of the conservation laws of energy, and linear/angular momenta, but I thought it'd be best to open it up for discussion first.

That said, I'd also like to add a simpler derivation for people who've had only calculus and never heard of a fiber bundle defined on a manifold. I don't mean to eliminate the latter derivations, just augment them to broaden the article's accessibility. Willow (talk) 21:10, 26 June 2008 (UTC)

I don't know about the redundancy that you mentioned, but this article certainly could be more accessible, and if you can accomplish that aspect, that would be fantastic. It's always seemed to me that the Physics and Mathematics articles are orientated too much towards experts, and other subjects (say Philosophy, History or Politics) that I don't profess to know a great deal about, seem to be easily accessible to me, so much so in fact, that a lot of what forms my philosophical and political beliefs has stemmed from using Wikipedia as a first point of call. I would almost certainly recommend Wikipedia for someone wanting to learn about such subjects, but I wouldn't necessarily recommend it for physics or maths articles at the moment, especially maths heavy articles like this. Deamon138 (talk) 21:26, 26 June 2008 (UTC)

Thanks, Deamon! We try to do our best; I totally sympathize, since it doesn't come easily to me, either, I think partly because things are rarely explained well. Unfortunately, even if someone's heart is in the right place, they don't always succeed. :( This article is probably going to be one of the thorniest of all. :P

Two other physics-y articles I've been working on are universe and Newton's theorem of revolving orbits; the latter is a little math-y but it has some nifty animations. :) If you had any suggestions for those articles, I'd be very grateful. :) Willow (talk) 12:05, 28 June 2008 (UTC)

Universe seems to be a pretty decent article. I had a skim rad of it and can't see anything wrong with it, it seems pretty thorough, but without going into too much detail. I would say that the Newton's theorem of revolving orbits article seemed okay, the general jist of the it is easy to follow, but I would say that it (the maths particularly) is just a little above me right now, but it's not on the whole bad. That's the thing really: that I like learning about Physics concepts, but I want to learn them while seeing mathematical explanations for these concepts, and that's the hardest part to explain. I'm just impatient really, if I wait until I start a Physics degree in October, I will see all the mathematical derivations I will ever need! (Oh and the media for those two articles was particularly good, I especially liked the Big Bang-->Expansion-->WMAP image showing space-time expansion.) Deamon138 (talk) 19:34, 28 June 2008 (UTC)

## Length as a conserved quantity

Isn't length itself a conserved quantity due to the formula x2 + y2 = z2? If so then what kind of symmetry is that? If the formula was x3 + y3 = z3 then the length of an object would depend on (the orientation of) your coordinate system. just-emery (talk) 04:15, 16 June 2009 (UTC)

You'd have to write length as an action, that is, as the integral of a Lagrangian, and this would also give you some rather odd equations of motion. It seems unlikely to me.
A bigger problem (from the physical perspective) is that length is *not* a conserved quantity in relativity. Length is only conserved in Newtonian mechanics. Ozob (talk) 23:18, 16 June 2009 (UTC)
I cant help but feel that the idea of symmetry and conserved quantities and more fundamental than the idea of 'action'. the length of an object is not conserved in relativity but the length does not depend on the orientation of your coordinate system. just-emery (talk) 19:36, 17 June 2009 (UTC)
Actually, Special Relativity says that length is not conserved, but depends on coordinate system. ds^2=dx^2+dy^2+dz^2+dt^2 is conserved in Special Relativity. Length varies according to Velocity according to the Lorentz Transformation. Also remember that distance always indicates a change in a quantity, so distance isn't really a symmetry at all. In Euclidian geometry we have the Galilean transformation Hope this helps Quodfui (talk) 10:49, 24 February 2010 (UTC)

To Quodfui: You need a minus sign instead of a plus sign before the dt^2 (also measure time in light-meters instead of seconds). To just-emery: The distance in the co-moving frame between two points on a rigid body remains constant, but no body is perfectly rigid. Also this is not a fundamental physical law, rather a result of the complex array of forces within a solid object. It is true that the approximate conservation of distance is related to the symmetry of the laws of physics under rotations and translations, but as indicated above that has nothing to do with this theorem since it says nothing about the action. JRSpriggs (talk) 04:33, 25 February 2010 (UTC)

## Difficult Description

I have an objection to the difficulty level of the article. I am a theoretical Physicist for me its bread and butter stuff. But this does not serve to explain a O-level student, which I have been trying to do with the help of internet and skype. Its a profound but in a sense rather simple result (cont symmetry = conservation law). One of the goals of wikipedia apart from being accurate is to be accessible. I wish there was a section with a very informal but reasonably correct explanation of the same added of what it just means rather than detailed explainations. I am at loss how to explain symmetry in a simple language (conservation is understood at high school level) and would like other contributors to consider this in future revisions. —Preceding unsigned comment added by 213.100.252.242 (talkcontribs) 18:11:14, 2009-09-20

If you want something added to Wikipedia, the thing to do is to add it yourself. Very few editors understand this subject well enough to edit this article constructively, and (if what you say is true) apparently none of us has both the interest and the skill to create a correct but simplified explanation. JRSpriggs (talk) 16:45, 22 September 2009 (UTC)

## Erlangen program

See the Byers citation in the article -- Nina Byers (1998) "E. Noether's Discovery of the Deep Connection Between Symmetries and Conservation Laws." in Proceedings of a Symposium on the Heritage of Emmy Noether, held on 2-4 December, 1996, at the Bar-Ilan University, Israel. Noether's Paper was read to the Society by none other than Felix Klein, who started the Erlangen Program. Obviously, Klein saw the importance of Noether's theorem for physics. --Ancheta Wis (talk) 22:20, 18 January 2010 (UTC)

Here is Felix Klein's statement of Noether's theorem, from Nina Byer's talk on Noether:
--Ancheta Wis   (talk | contribs) 02:29, 26 March 2012 (UTC)
I propose to insert Klein's statement of Noether's theorem into the article, with the Byers citation, after a suitable waiting period. --Ancheta Wis   (talk | contribs) 15:03, 8 April 2012 (UTC)
This seems notable enough. I guess you've had little response because most editors are interested but feel a little out of their depth, as I do. WP:Be boldQuondum 15:15, 8 April 2012 (UTC)
For this statement to make sense, there should be some relationship between the integral I and the Lagrangian. Do you mean that I is the action, i.e. the integral of the Lagrangian itself? Also which Lagrangian "expressions" are you talking about? JRSpriggs (talk) 19:44, 8 April 2012 (UTC)
I can only quote from the AIP Handbook of Physics p.900: I is the action. Let me dig up the citation. --Ancheta Wis   (talk | contribs) 19:52, 8 April 2012 (UTC)
My point is that if you fleshed out the Klein statement so that it could be clearly understood, it would become essentially the same as the statements which were previously in the article. JRSpriggs (talk) 19:58, 8 April 2012 (UTC)
Yes, I agree that Klein's statement is the the same as that in the article, as seen retrospectively from 97 years later. For example, Benenson, W.; Harris, J; Stocker, H.; Lutz, H. eds (2001). AIP Handbook of Physics ISBN 978-0-387-95269-7 p.900, which is section 26.3 "Symmetries and conservation laws", part of Chapter 26, Elementary Particle Physics - standard model. I paraphrase their statement of the theorem, which I agree says the same thing as Klein's statement of Noether's theorem: "... The invariance of the field-theoretical action integral w.r.t. an n-parametric continuous transformation group ⇒ there are n conservation laws." --Ancheta Wis   (talk | contribs) 22:16, 8 April 2012 (UTC)

## curly L

Seems that the article is switching randomly between capital and curly styles of the L symbol. Isn't it conventional for capital L to represent the Lagrangian (which is integrated along the path of motion), and curly L the Lagrangian density (which is integrated over space and time)? Cesiumfrog (talk) 22:50, 8 March 2010 (UTC)

Unfortunately, the curly L is also used to indicate the Lie derivative which is used in the article. So using ordinary L for the Lagrangian density in some places may be the lesser of two evils. JRSpriggs (talk) 08:43, 10 March 2010 (UTC)

## Change of derivatives to the usual notation

In the Preview section, I changed the expression for the conserved current ${\displaystyle j_{r}^{\nu }}$, so as to include the usual notation of the derivatives, i.e. ${\displaystyle \partial _{\nu }\phi }$ instead of ${\displaystyle \phi _{,\nu }}$ which I believe I encountered for the first time here (after 10 years of studying physics). Actually, this was so obtrusive that I created a wikipedia acount just to fix it!

If this change is approved, than I would appreciate it if someone could help me apply it to the rest of the article, 'cause I'd hate to leave it half-baked, but I really don't have the time to do it properly right now (considering I'm a bit of a novice here). Thanks NN22 (talk) 18:30, 10 April 2010 (UTC)

Use of a comma to indicate partial differentiation and also use of a semicolon to indicate covariant differentiation are the norm in general relativity. JRSpriggs (talk) 20:29, 10 April 2010 (UTC)

## Action

This article begins by denoting action as I, rather than ${\displaystyle {\mathcal {S}}}$ and then uses ${\displaystyle {\mathcal {S}}}$ towards the end. Is there a reason for the inconsistency? If not, I suggest changing all the actions to ${\displaystyle {\mathcal {S}}}$. 86.44.129.212 (talk) 23:11, 4 May 2010 (UTC)

## Why Noethers Theorem is a Tautology

Found this very interesting article by Miles Mathis that says this theorem is a tautology.

This should be included in the main Article I think. —Preceding unsigned comment added by 86.46.35.131 (talk) 08:19, 13 August 2010 (UTC)

Careful there, the expression of a logical Tautology is not a triviality, as Ludwig Wittgenstein has noted. See Rewriting system. Apparently a deeper truth is uncovered in the rewriting process. This was part of Imre Lakatos' research in informal mathematics before his untimely death in 1974. See Imre Lakatos (1976) Proofs and refutations in which 'trivial' theorems are found to have counterexamples, which cause researchers to drive toward deeper, dominant theory. --Ancheta Wis (talk) 11:40, 13 August 2010 (UTC)
I agree, this article is irrelevant and misleading to the entry on this theorem. It is trivial to argue that her theorem is a tautology, since Newtonian, Hamiltonian, and Lagrangian mechanics are equivalent formulations for basic mechanics (this means that any statement made by one system necessarily has a corresponding one in the other). And also, as Ancheta astutely points out, a statement being a tautology does not make it arbitrary. Lagrangian and Hamiltonian mechanics become useful when delving into quantum theory and consequently Noether's theorem proves to be a very valuable one. Besides, the article's main argument, that Noether's theorem does not show that invariance "causes" conservation, is based on a silly philosophical misunderstanding: laws of physics do not "cause" observable regularities in our world, they are models we use to understand physical systems which can give us a priori (known in advance) knowledge. Newton's formulation does not "cause" physical events such as energy conservation any more than Noether's. --Karl — Preceding unsigned comment added by 140.103.111.155 (talk) 07:36, 9 March 2012 (UTC)

Um, yes, it's a tautology? It's a theorem - that's the point! Twin Bird (talk) 05:26, 14 March 2012 (UTC)

## Conserved in the ordinary sense vs. in the covariant sense.

The article says that a current is conserved in the sense of the ordinary derivative

${\displaystyle \partial _{\nu }j^{\nu }=0\,,}$

and then says that ${\displaystyle {T_{\mu }}^{\nu }}$ is an example.

But the conservation for ${\displaystyle {T_{\mu }}^{\nu }}$ is covariant; that is,

${\displaystyle \nabla _{\nu }{T_{\mu }}^{\nu }=0\,,}$

and it is frequently the case that ${\displaystyle \partial _{\nu }{T_{\mu }}^{\nu }}$ is non-zero.

What gives? Thanks — Quantling (talk) 20:49, 22 September 2010 (UTC)

To Quantling: I assume that you are referring to subsection Noether's theorem#Field-theory version in the section Noether's theorem#Mathematical expression. In this section we are talking about field theory in either classical physics or special relativity, not general relativity. In this context, there is no difference between partial and covariant derivatives.
If you want to see my application of Noether's theorem to general relativity, see Talk:Stress-energy-momentum pseudotensor#Applying Noether's theorem to general relativity. JRSpriggs (talk) 09:03, 23 September 2010 (UTC)

In non-Cartesian coordinates (i.e., spherical, cylindrical, or other curvilinear coordinates), even classical physics and special relativity will generally not give us ${\displaystyle {T_{\mu }}_{,\nu }^{\nu }=0}$ —which is perhaps why no one ever speaks of conservation of radial momentum, for example. However, I agree that the covariant derivative reduces to the ordinary derivative for classical physics and special relativity in Cartesian coordinates. Thanks for your link to Noether's theorem in the context of general relativity. I'll take a look at some of the articles mentioned there. Quantling (talk) 13:07, 23 September 2010 (UTC)

Of course, you are correct that using non-linear coordinates would require a correction to the partial derivatives. That is why we normally stick to Cartesian coordinates for simplicity in general purpose articles. In general relativity, we do not have that option. JRSpriggs (talk) 14:54, 23 September 2010 (UTC)

## Painfully unclear section

In the Mathematical expression section, what the hell are Q, T and ε???

There is absolute zero explanation of what these are in the full article. It is clear that t is time, q are generalized coordinates and p are generalized momenta. My best guess for ε is a perturbation parameter (clued from Perturbation theory), and that Q is a generalized force (hinted from Appell's equation of motion). Anyway - for that reason that section has been tagged until someone explains them.--Maschen (talk) 13:14, 3 December 2011 (UTC)

I think that the perturbations are being treated as vectors. For each r, (Tr, Qr) is a basis vector generator of the set of perturbation vectors. The εr are the infinitesimal coefficients of the generators in the linear combination which gives the actual perturbation of interest. JRSpriggs (talk) 00:14, 5 December 2011 (UTC)
Right. I don't understand generators. Would you mind re-writng that section to make it clearer and obvious what these symbols mean? Or anyone else who reads this and understands? Thanks,--Maschen (talk) 13:27, 5 December 2011 (UTC)

The informal statement is sickeningly poetic, and the mathematical formulation is focused on consequences, not the theorem itself. Even the derivation only makes it half-clear from the start just what the section is setting out to prove. This is a mathematical theorem, not a piece of art for Physics for Poets students to admire, and the formulation should be what the theorem is, not how it strikes the cargo-cult mystic's intuition. Twin Bird (talk) 05:25, 14 March 2012 (UTC)

You need to be more specific, if you want me to make any change in the article. Vague accusations will not cut it. JRSpriggs (talk) 10:06, 14 March 2012 (UTC)
Twin Bird, see Leyden jar for more information on the history of the concept. Especially note Benjamin Franklin's viewpoint. Next, read faraday cage. Next, lift the charges on the cage to the generalized surfaces in Maxwell's equations. Finally, lift once more (beyond electrical charge, to noetherian charge) to get to Noether's conceptual level. This is the Erlangen program of Felix Klein, who presented Noether's paper, because she was not allowed to, being female. --Ancheta Wis (talk) 17:42, 14 March 2012 (UTC)
(Sorry, my computer's been down.)
The problem's not that I don't understand, but that to frame it in this very philosophical, physical way, as you just did again, does a disservice to Noether as a mathematician. It would be as if the page on Stokes' theorem were primarily built around the equivalence of the differential and integral forms of Maxwell's equations, and the "derivation" was just a version of the intuition pump used to explain the fundamental theorem of calculus in high school. Such a page would have no defenders at all; likewise, this page would have no defenders except that Noether's status as a symbol of adversity has overtaken her cold results, and given pop scientists free rein to bang on about "fine technical points aside," "the essence of," "the flavor of," etc. - with such a shocking self-indictment as "historical context" placed above "mathematical expression"! - making it next to useless to anyone interested in the actual mathematics, which is the audience such an article really ought to be written for. Shortly, I'll be trolling the library for decent secondary sources, but this page needs to be overhauled so badly I'm not sure where to start. Twin Bird (talk) 21:13, 19 March 2012 (UTC)
One starting point might be with Noether charge (which resolves back to this article ;-(), so a generating set might be more to your taste. A physicist would likely be interested in generating sets of Lie groups, i.e., a Root system, but don't let that stop you. There is a ton of infrastructure that you probably require. But I suspect that what you seek is OR. All the better. That might encourage someone to find your hoped-for result in a publication. Good luck. I hope you get a result you want. At the very least, a citation. --Ancheta Wis   (talk | contribs) 16:44, 20 March 2012 (UTC)
Maxwell's equations (plus Lorentz force) are redundant enough to derive conservation of charge.
The analog at the symmetry level (gauge theory) is the statement of redundant degrees of freedom. --Ancheta Wis   (talk | contribs) 20:11, 25 March 2012 (UTC)
Twin Bird, see #Erlangen program above for Felix Klein's statement of Noether's theorem. --Ancheta Wis   (talk | contribs) 02:33, 26 March 2012 (UTC)

## Historical context

The article says that "Modern physics has revealed that the conservation laws of momentum and energy are only approximately true" but that "the conservation of four-momentum in special relativity and the zero covariant divergence of the stress-energy tensor in general relativity" are rigorous.

First of all, conservation of 4-momentum also implies conservation of each individual component, which corresponds exactly to conservation of 3-momentum and energy (granted 3-momentum and energy aren't individually conserved under a boost, but they aren't under a change in velocity in Newtonian mechanics either).

Second of all, the stress-energy tensor and its zero divergence is just as much a part of SR (an SR based theories, like quantum field theory) as it is of GR. The presence of Killing vectors in GR and the fact that the covariant derivative of 4-momentum along a geodesic is zero (the definition of a geodesic) seems more analogous to conservation of momentum and energy than the statement about the stress-energy tensor. --Mpalenik (talk) 15:56, 2 April 2012 (UTC)

## Potential rewrite of Noether's theorem#Mathematical expression

given that the tag is still there...

As mentioned by others, this section starts off like rubbish - I have absolutely no clue what it is saying. The rest is not particularly followable. For instance:

"Then a generic perturbation can be written as a linear sum of the individual types of perturbations
${\displaystyle \delta t=\sum _{r}\epsilon _{r}T_{r}\!}$
${\displaystyle \delta \mathbf {q} =\sum _{r}\epsilon _{r}\mathbf {Q} _{r}.}$
Using these definitions, Emmy Noether showed that the N quantities
${\displaystyle \left({\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}\cdot {\dot {\mathbf {q} }}-L\right)T_{r}-{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}\cdot {\mathbf {Q} }_{r}}$
are conserved, i.e., are constants of motion; this is a simple version of Noether's theorem."

It doesn't even say what Tr, Qr actually are ε. Where the does the last equation come from? How does it explain conservation and symmetry? For what I mean see Schrödinger equation#Equation. How do the examples follow from this statement? They just say "invariant under t → t + δt or q → q + δq leads to this expression", but how? why? I can't understand it at all, it just disjointly states variations in time, position, and angle and gives equations which leave the reader clueless where they come from.

I have a reference which actually provides an origin to the theorem, and gives an simple yet infallible way to find the conserved quantities, in my sandbox. I would replace that entire section and the examples subsection, but the rest of the article uses the perturbation method. =(

12:54, 11 June 2012 (UTC)

## Hi

Noether's theorem has been reformulated/sharpened/specialized/generalized and used in different contexts many times over the years so that it is understandable that there are many different views on what it is and what it says. But I have to agree with a poster above that this article sometimes handles the theorem (whatever it is) with more poetry than truth. Here are a few points that could be strengthened in the article:

• Define what a symmetry of the action is.

The opening sentence is pretty good: Noether's (first) theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. In order to really have a symmetry of the action (in whatever version of the theorem), one should explicitly assume that the Euler-Lagrange equations do not necessarily hold. Otherwise, if they hold, any "differentiable" variation satisfying suitable boundary conditions would pass for a symmetry. I think that invariance of the action is better terminology than symmetry of the action in the statement.

• Under various assumptions, the "strength" of the theorem varies. My memory may be fading, but invariance of the action, the Lagrangian, and the Lagrangian density, in turn, gives more and more useful statements of the theorem.
• The explicit expressions for conserved quantities (conserved currents) explicitly do assume that the Euler-Lagrange equations hold. (This point is sort of a mirror image of the first point)
• When (above point) the theorem gives explicit expressions for conserved quantities (conserved currents), then the expression is not unique. In the vicinity of this, a link to e.g. Belinfante–Rosenfeld stress–energy tensor would be appropriate.
• There are converses of the theorem under certain hypotheses. (Simple connectedness is sufficient is sufficient in some versions.)
• There are "Hamiltonian" versions of this theorem, see e.g. John M. Lee, "Smooth Manifolds". Chapter 19. Neat and clean - including a converse.
• The article could go on to explain in what sense the conserved quantities serve as "generators of" the symmetry that gave rise to them in the first place. Here, the Poisson bracket (perhaps generalized to fields) is essential.
• The "prettiest" context in physics may be in classical field theory (including relativistic quantum mechanics with fixed particle numbers, i.e. RQM but not QFT).
• Passing to QFT seems to be non-trivial to me. The action (or even the Lagrangian) are no longer necessarily functions (or fuctionals), but rather "operator valued functionals". At this point the Ward identities article could take over and be beefed up. YohanN7 (talk) 17:58, 28 January 2013 (UTC)

## Pinned down the problem with the examples

(I have strong maths but I'm an outsider to physics, that's why I don't edit the page directly. Please forgive me if the following turns out to be wrong, I've done my best.)

The problem with the examples is not that they are in themselves wrong, but that they apply an idiom of Noether's Theorem that differs significantly from the ones introduced on the page.

The theorems on the page, at least those before the manifold/fiber section, assume that we wiggle the generalized coordinates in such a way that *the action of any trajectory does not change*.

By contrast, the examples apply an idiom of the theorem where the Lagrangian (as opposed to the action) is *not* constant under wiggling, but *changes up to some 'error term' d/dt f*.

For example, the Galilei transformations in Example 2 aren't actually symmetries, as can be easily checked: they make the the Langrangian change by, well, d/dt f, where f is as in the example.

This is serious because: (1) The whole page's narrative crumbles, because the wigglings in the examples are not symmetries! (2) Everyone who is new to the material will be confused by the mismatch between the theorems and the examples.

It took me hours to reverse-engineer the candidate theorem used in the examples before it dawned on me what was wrong. If finally found my findings confirmed here: http://matheplanet.com/default3.html?call=article.php?sid=1264&ref=https%3A%2F%2Fwww.google.de%2F (It's in German, and they use a capital F instead of a small f.)

Ironically, it seems that the sections "Manifold/fiber bundle derivation" and "Generalization to Lie algebras" employ an error term d/dt f. But I guess few people will work through those before studying the examples. — Preceding unsigned comment added by 212.114.237.92 (talk) 10:16, 21 July 2013 (UTC)