Talk:Integrable system

Too technical
I'm a physics student, and I'm not familiar with the level of mathematics that is discussed in this article, but my understanding of integrable systems is this: if the phase space is $$2n$$ dimensional, there should be $$n$$ independent conserved quantities which are in involution, ie, their Poisson brackets separately vanish. This means that the phase space is confined to a surface that is homeomorphic to an $$n$$-torus. It also means that their equations of motion can be reduced to quadratures and can in principle be solved by a sequence of integrals. I think this is the basic definition that most physicists would look for, and it is really, really hard to make that out from this article. I don't feel confident editing it, because I don't understand half the math terms involved. But it would be nice if someone who does could tone it down a bit. SrijitaK (talk) 18:23, 21 February 2012 (UTC)


 * I completely agree! This is way too mathematical. wpoely86 (talk) 10:05, 2 July 2012 (UTC)


 * In fact, this page is completely unreadable. I just want to know what it means for a system to be integrable, and this article is completely useless to 99% of people. Dragonfiremalus (talk) 20:11, 14 December 2013 (UTC)


 * Also agree the article is utterly incomprehensible to a layperson. I would forgive its opaqueness to the average reader if it simply began with a single short sentence that says: "An integrable system in mathematics is one in which < finish the sentence by describing the basic concept in ordinary English without using any technical terms nor referring to the names of particular systems >."  Hypothetical example: "...one in which certain equations can be completely solved by conventional algebra." (This is merely an example of the use of easily understood English; it is not a suggestion for the actual wording, which I'm not qualified to offer.) It seems to me that any qualified mathematician, after giving the matter some thought, should be able to write an opening sentence that can be understood by anyone and does not rely on textbook jargon. DonFB (talk) 01:48, 12 June 2018 (UTC)


 * This is an arcane, highly technical and difficult branch of mathematics, requiring years of background study. Why, exactly, are you interested in reading this article, and what do you hope to get out of it? What brought you to this article? What did you hope to learn? (Yes, people want to read about black holes and quantum mechanics, but without the formulas; but is this also the case for integrable systems? Surely not. So then what?) 67.198.37.16 (talk) 23:30, 9 January 2019 (UTC)

A counter-example would be useful
It would be useful to add a counter-example to the article, i.e. a simple example that is *not* integrable. 18.87.1.116 (talk) 16:35, 3 February 2016 (UTC)


 * An automobile driven on a flat surface is non-integrable. You can drive it anywhere you want. Compare and contrast to a roller-coaster at an amusement park, which is integrable.67.198.37.16 (talk) 23:57, 9 January 2019 (UTC)

Assessment comment
Substituted at 18:59, 29 April 2016 (UTC)

Changes to lede
, I'm not sure if your recent edit to the article is what you intended. Your edit summary said: "misleading link; use a more precise link", but your edit actually reverted my explanatory opening two sentences and restored the previous version of the lede, which I believe is too technical for the very beginning of a Wikipedia article. A new link you added, Integrability conditions for differential systems, points to an article which itself has an overly technical lede and does not define "integral" (or "integrable") in terms that can be understood by a general reader. This was the point of my two-sentence lede to this article: to define an essential term in a manner that a general reader can easily understand. I did not disturb the remainder of this highly technical article. In addition, your edit restored the "incomprehensible" tag I added and then removed after my revision of the lede. Also, your reversion removed the link I added to integral (piped from "integration" in the first sentence) and added a link to integrate, which points to a Disambiguation page with many terms, thus losing the direct link to integral. Please let me know if you believe my revision was technically incorrect, or if you only intended to add the new link to Integrability conditions for differential systems, while preserving the other changes I made (including my removal of the tag I added). DonFB (talk) 22:49, 12 June 2018 (UTC)


 * I think that the accessibility of DonFB's version is very promising, but that Taku has a point about the lack of precision. I propose the following intermediate version:

In calculus, "integration" refers to adding together many small (infinitesimal) parts to find the value of areas, volumes and related totals. An integrable system in mathematics or physics is a system of partial differential equations whose behavior can be expressed in some sense in terms of an integral and its initial conditions.

There are various integrability conditions for differential systems. In the general theory ...
 * (I don't think it's perfect, but maybe it can be a step towards consensus?) --JBL (talk) 22:55, 12 June 2018 (UTC)


 * Do other editors also think it is necessary to definite "integration" in the lede instead of just having a link (sorry about a link to a disambig page; that was a mistake). The target audience should be those who have rudimentary knowledge of differential equations and to those audience the explanation of integration is simply not valuable (like defining Frace in the Paris article). The lede should be essentially saying an integral system is a "nice" system of differential equations as opposed to saying integral is a process of ... and an integral system is a related concept; that would be missing the point (since this article is not about integration). An Integrability conditions for differential systems has a somehow technical lede, which needs to be fixed but not having a link to it is a serious omission, since "integral" in "integral system" refers to integrability conditions after all. -- Taku (talk) 23:11, 12 June 2018 (UTC)
 * I have restored the tag since I wasn't sure my version was accessible enough. -- Taku (talk) 23:23, 12 June 2018 (UTC)


 * About "do other editors think it is necessary", the right question is not whether it is necessary but whether it is a good idea. I found it jarring at first, because I mostly read mathematics written for other professional mathematicians. But after thinking about it for a few minutes, I decided that it made it a realistic possibility for one paragraph of this article to be understandable by someone who does not know beyond high school mathematics. And that seems like a good thing to me. (I imagine that once a few more WT:WPM editors weigh in, it will be made incomprehensible again.) --JBL (talk) 23:23, 12 June 2018 (UTC)


 * I like the second sentence in JBL's proposed intro above much better than the second sentence in the current version ("An integrable system in mathematics or physics is one which uses this process"). XOR&#39;easter (talk) 23:39, 12 June 2018 (UTC)

I agree that my edit created a lede that "went from very accessible to heavily technical", as JBL said on the Project Talk page. In fact, I do think the remainder of the lede needs more work to improve its accessibility, but I'm not a mathematician, so I've tread lightly. Regarding JBL's comment above: because his suggested wording uses "integral" near the end of the 2nd sentence, it might be more logical to link that word to the "Integral" article and remove the piped link that I added from "integration" in the first sentence. I understand Takuya's point about the audience, but I remain of the view that, for the uninitiated, the only easily understood clue to the actual meaning of this article is the first sentence as currently worded. Speaking as a layperson, I think the remainder of the article remains all but "incomprehensible", but that's an extremely harsh tag and I'm willing to remove it again--if the opening sentence remains in or very close to its present form as a kindly gesture to non-mathematicians who, for whatever reason, happen to stumble across this article. Also, editors in general should work to fulfill the guidelines I linked to in the summary for my original edit: Jargon and ExplainLead. DonFB (talk) 23:44, 12 June 2018 (UTC)


 * For the sake of discussion, here is the version I proposed (it's not original and is a variant of D. Lazard's):
 * In mathematics and physics, an integrable systems is a system modeled by partial differential equations that is integral in some sense; i.e., roughly speaking, its behavior is determined by initial conditions and one can integrate from the initial conditions.
 * I still maintain that not having a link to "integrability condition" article is a serious omission, since integral refers specifically to that not integration.
 * I also think this version has more precision/accuracy (e.g., "in terms of an integral and its initial conditions" is too vague).
 * -- Taku (talk) 23:53, 12 June 2018 (UTC)
 * I'm not opposed to including that link. But I would want to preserve my efforts at accessibility of the lede while including it. A question: In reference to differential equations, does "integrable" refer to the act of "integration"? DonFB (talk) 00:04, 13 June 2018 (UTC)
 * No "integeal" comes from the integrality condition" of a differential system; an integral system is a fancy version of a differential equation that is integrable; i.e., one can solve it by integration. -- Taku (talk) 00:08, 13 June 2018 (UTC)
 * For example, in the above, when the poster is saying "... ie, their Poisson brackets separately vanish.", they is referring to the integrality condition; i.e., the vanishing of the bracket. This is why the link to "integrablity" is more precise (thus??? preferable). -- Taku (talk) 00:40, 13 June 2018 (UTC)
 * I think "the remainder of the article remains all but "incomprehensible"," is a correct assessment and putting the tag is correct; Wikipedia articles are not perfect! The article need some to discuss a sort of the basic case mentioned in the above thread. Incidentally the article also lacks some (moderately) advanced stuff like Hitch's integrable system; this is an omission that is not acceptable, given its importance in algebraic geometry. -- Taku (talk) 01:01, 13 June 2018 (UTC)

Point of order. The subject of this article is not "integration". It is "integrable system". These are special types of dynamical systems. Packing the lead with irrelevant things about calculus and computing areas and volumes has nothing to do with the subject of this article. To improve the lead of the article, go and get a textbook on integrable systems and see what it says. Sławomir Biały (talk) 01:22, 13 June 2018 (UTC)
 * That's basically what I was trying to say, but don't you think having a link to integrability conditions for differential systems is necessary, since "integrable" really comes from there. I don't have a strong opinion on the other wordings. -- Taku (talk) 02:08, 13 June 2018 (UTC)


 * Taku, you keep banging on about that link; have you noticed that it is included in my proposal? (In a direct way, unlike the easter egg in the present version.) --JBL (talk) 02:11, 13 June 2018 (UTC)
 * Yes in your proposal but not in the version put by DonFB or the old one. -- Taku (talk) 02:12, 13 June 2018 (UTC)
 * On a second reading of JBL's proposal, I still like the "An integrable system in mathematics or physics" sentence, but the jump from areas and volumes to that is jarring. I don't think I'd make the mental connection without already knowing the subject matter. XOR&#39;easter (talk) 03:03, 13 June 2018 (UTC)
 * I think that proposed sentence isn't quite right; "in terms of an integral and initial conditions" is potentially misleading; an integrable system isn't like say Cauchy's integral formula that expresses a complex function in terms of an integral. An integrable system is a differentiable system satisfying some integrability conditions (that's what I remember from school) and saying anything else is misleading in my opinion, if accessible. If the readers misunderstand the concept, that's not making the article accessible (but making it inaccurate). -- Taku (talk) 03:58, 13 June 2018 (UTC)

In his edit summary at the article, Sławomir Biały said the recent change "does not seem to have been written by anyone with the foggiest idea of the subject of this article." That's basically accurate. I made the edit after struggling mightily to understand the article's meaning by reading it. Not succeeding, because, as a layman, I found it incomprehensible (as have other readers; see above), and rather than "go and get a textbook on integrable systems and see what it says," as Biały so helpfully suggested, I instead did some quick online research and wrote two new sentences in the Introduction (the lede) in an effort to give general readers a very basic understanding of the meaning of the title of the article, and hence, the subject. Unfortunately, the latest reversion has returned the article to what I, again as a layman, regard as a condition of incomprehensibility (as indicated by the Tag at the top). The article is not succeeding in fulfilling the recommendations at WP:EXPLAINLEAD, which says, "It is particularly important for the first section (the "lead" section, above the table of contents) to be understandable to a broad readership." It also says, "In general, the lead should not assume that the reader is well-acquainted with the subject of the article." The Manual of Style says, under the heading Technical Language: "Some topics are intrinsically technical, but editors should try to make them understandable to as many readers as possible. Minimize jargon, or at least explain it..." I linked both guidance pages (which contain ExplainLead and Jargon) in my edit summary. I believe these guidelines are very important to the usefulness of Wikipedia and should be followed as much as possible. That's not happening here. For example, nowhere does the article give a simple definition of the critical word contained in its very title: "Integrable".

My rudimentary understanding that integrable means amenable to "integration", which in turn means summing a series of really small values to produce a number which can represent things like area or volume, may be incorrect. I am certainly open to correction by people who possess serious education in the subject. In that regard, I must respectfully disagree with Taku, who wrote on the Math Project Talk page in reference to this article that "A problem can exist without a solution." That's certainly been true in the history of mathematics, but expressing ourselves in plain English is not an insurmountable problem. I am not campaigning to change the body of the article, which is quite technical. As the guidelines I referenced explain, the technical details belong in the body, but the lede should be comprehensible to everyone. I recognize, of course, that this is hardly the only article on Wikipedia that suffers from the problem I'm describing. A final question: in your own words (whoever wishes to respond), and without resorting to any technical terminology or repeating the very word to be defined, or using a word similar to it (which the lede now does twice), what is your definition of Integrable? DonFB (talk) 05:20, 13 June 2018 (UTC)


 * I think that the effort to bridge the gap between precalculus and integrable systems by starting with the summation of infinitesimally small chunks is vain. I suggest to explain that integration is not only a term used in calculus, but also refers to solving differential equations, which is of course related, but already quite distant from the term as introduced in calculus. Mentioning the fact that systems of partial differential equation enormously add to these distance, might allow to justify the introduction of the term integrable as qualifying these systems as such, if their solution is theoretically accessible under specific (boundary) conditions.


 * I arrogate to understand DonFB's wishes, and support JBL's target of accessibility, but I also think that Taku has a point when warning from fake explanations (as I see the summing of infinitesimals). Perhaps I can make my point of view clearer by suggesting for the intro a not too big number of roughly equidistant steps from "integration in calculus" to "integrable systems". To my perception stating that "DEQs are integrated" is a central but missing statement for beginners. Relying solely on textbooks (Sławomir Biały) might be in vain for this purpose, too. Purgy (talk) 09:15, 13 June 2018 (UTC)


 * Changes to the lead aimed at making the subject clearer, actually should clarify the subject. The subject is not integral, but is integrable system, which is only distantly connected with the notion of "integral" from calculus (see below).  It would be something like beginning the article integral with a sentence like "Addition is something that one can do with numbers."  That's true, and is distantly connected to integration, but is not really helpful in conveying what integration is.  Especially not for the first sentence, where one normally expects a statement of what the topic is.  The essence of integrable systems is that they are determined from initial conditions, not that they involve infinitesimal calculus.  (In fact, all continuous dynamical systems involve calculus.)  I don't wish to defend the current article, since it actually is incomprehensible.  But reliable sources should dictate the focus and content.  Writing an encyclopedia article without understanding the topic, and without using any external sources is not a good idea, for reasons that are hopefully obvious.  That was why I suggested getting a textbook on the subject.  That should more or less dictate how we approach the subject.   Sławomir Biały  (talk) 10:40, 13 June 2018 (UTC)
 * I want to make explicitely clear that my remark about textbooks was in no way doubting the principle of reliable sourcing. I just wanted to express my skepticism about the existence of accessible first sentences in textbooks about integrable systems. Purgy (talk) 10:51, 13 June 2018 (UTC)
 * Sure. But I think that also imposes limitations on what is achievable within the constraints of Wikipedia.   Sławomir Biały  (talk) 10:56, 13 June 2018 (UTC)

Integration: integrable vs. integral
In the preceding discussion and in some versions of the lead, it seems that there is some confusion between integral and integrable. For this reason, I have added a hatnote to the article, and removed the link to integral.

In fact, here, integration and integrable have nothing to do with integral: in the jargon of differential equations, "integrating" means "solving from given initial conditions". This should be said somewhere in Wikipedia, but I have not found where.

Thinking again to the problem, I'll add a footnote to the article, and an entry in Integration (disambiguation). D.Lazard (talk) 09:46, 13 June 2018 (UTC)
 * I have also edited Integration (disambiguation) for disambiguating between these two notions of integration. D.Lazard (talk) 19:52, 13 June 2018 (UTC)

Incomprehensibility
In my judgement the lead is massively improved in readability, but it is decisive for removing the relevant banner, if the article is now perceived also by others as sufficiently accessible, resp., if there are still wordings suggested for an urgent improvement. I take the freedom to ping. Purgy (talk) 07:22, 14 June 2018 (UTC)
 * I don't oppose removal of the Incomprehensible tag, which I added in the first place. Parts of the article remain incomprehensible to me, but I'm not a mathematician. DonFB (talk) 07:26, 14 June 2018 (UTC)
 * I think the tag is still appropriate, for what it's worth. The lead might be comprehensible (but at the expense of not really saying much), but the article itself is pretty incomprehensible.   Sławomir Biały  (talk) 10:26, 14 June 2018 (UTC)
 * Small strengthening: The first paragraph of the lead might be comprehensible (but at the expense of not really saying much), but the article itself is pretty incomprehensible, even for an experimented mathematician. D.Lazard (talk) 10:51, 14 June 2018 (UTC)
 * Moreover, the linked articles do not help understanding, because they suffer from the same issues. D.Lazard (talk) 10:55, 14 June 2018 (UTC)

Basic questions
This is related to the above comprehensive tag conversation but it would help if the article answers the following type of questions -- Taku (talk) 23:38, 17 June 2018 (UTC)
 * A spinning top an integrable system? (yes?) If so, can we give some details in the manner of say group?
 * Is an integrable system a dynamical system? I thought a dynamical system is modeled by ODE. Perhaps the distinction is not important...
 * What does the phrase like "so so is integrable via a theta function" mean. I think it basically mean some the theta solves the system. It would be nice if the article can clarify this type of the phrase.
 * Relation to "Integrability conditions for differential systems" (in the lede). My understanding is stuff liken vanishing of the Poisson bracket expresses the integrablity conditions on a system of differential equations (cf. above.)


 * Spinning top -- no, because of KAM torus. Unless you describe the top in such a way that KAM can't happen. (gravity is non-linear because it pulls on the center-of-mass of the top with a cosine-of-angle-between-spin-and-downwards term).


 * Dynamical system - yes, sort of. If the integrable system has time, and time is one-dimensional, then its a dynamical system. If there's not "time", then the word "dynamical" is ... confusing.  For example, a topological soliton is very nicely integrable, but there's no time coordinate anywhere. Another example: Lie groups are the solutions of the differential equations of the Lie algebra, they are fully integrable, but there is no "time". Generally: manifolds have a tangent bundle; geodesic flows on the tangent bundle are given by solutions to the Euler-Lagrange equations (or, equivalently, Hamliton's eqns); so, in this sense, tangent bundles look like a special case of symplectic manifolds, and you have a fundamental form etc. defined on them. The point here is that geodesic flows are integrable; heck, many sections of the tangent bundle are integrable in general. Are you ready to call any integrable vector field a "dynamical system"? That's not a rhetorical question; I'm confused about this. Classic reference for all this is Abraham &amp Marsden.


 * Theta function -- this is a special case -- elliptic equations have theta functions as solutions, and lots of things can be reduced to elliptic equations, which is why number theory shows up inside of string theory and inside of fredholm theory, etc. I mean what the heck -- the modular group is a subgroup of the Poincare group. Basically, if you can reduce your system of equations to some collection of things with modular symmetry, then modular forms are a solution, making your system "integrable". Flip side, pretty much everything in sub-Riemannian geometry is non-integrable; yet, if I recall correctly, theta functions do still show up. I cannot remember why. blah blah blah holonomic constraints blah blah blah. BTW theta functions generalize wayyy beyond just the olde-fashioned modular group. I forget how. I think they show up in affine Lie algebras or in discrete subgroups thereof .. or something like that.  Like if you have some group($$\mathbb{C}$$) and can validly define some subgroup($$\mathbb{Z}$$) then you can define theta functions on it. Or something like that.


 * Vanishing Poisson bracket -- I dunno. Not sure. I think that means that there's a constant of motion, and so at least that part is "integrable". For example, some system may be chaotic (and so non-integrable overall), but will still remain on a surface of constant energy (because of conservation of energy; its a constant of motion). Maybe this is a bad example, because although chaotic systems are traditionally non-integrable, they can still sometimes be "solved" if you can specify a subshift of finite type for it, and then fully define all of the solutions to the transfer operator for it. So basically, if its not chaotic, but its ergodic. -- you cannot write down the trajectory of a single point particle (because its ergodic e.g. Bernoulli shift) but you can give an exact solution for the time evolution of pressureless infinitesimal "dust"; i.e the time evolution of a (smooth) function on a manifold, which will split into an invariant part (the invariant measure; sometimes called the Haar measure when you can write it as a group action), and decaying parts (corresponding to eigenvalues less than one of the transfer operator; i.e. the time evolution is not unitary. The Perron-Frobenius theorem says that the eigenvalue one is associated with the invariant measure.). To be "integrable" you have to be able to write down these parts explicitly, exactly.  The point is, I guess, that the subshift is invariant under shifts, thus, the subshift becomes the ergodic analog of the "maximal integral submanifold" its what stays the same as the system flows. (This is "often" the case for manifolds of constant negative curvature; point-particle trajectories diverge, but classic examples are Hopf fibrations which write down horocycles that evolve on the stable and unstable manifolds -- I think you can write Hopf fibrations (err, should I be calling them folations instead?) for all Anosov systems, and I guess Axiom A in general?? something like that. So those are "integrable" I guess, even though they are kind-of-like chaotic. I don't know the details. (Yeah, look at Anosov flow for the tangent bundles of Riemannian surfaces; I think I wrote that article a decade ago, when I actually vaguely understood it...)


 * Ohh, I just looked at the Cambridge PDF you mentioned. Yeah, there's a huge amount of difficult and complicated stuff about integrable systems starting from 1970's onwards .. that is very daunting. There is a reason why this particular article is ... hard to write. 67.198.37.16 (talk) 06:30, 9 January 2019 (UTC)

TODO
Some more models solved via inverse-scattering:


 * Reduced Maxwell-Bloch equation
 * Self-induced-transparency (SIT) equation
 * Coupled non-linear Schrodinger equation

67.198.37.16 (talk) 22:55, 16 September 2020 (UTC)