Talk:Perturbation theory

Linear combination of atomic orbitals molecular orbital method

I have edited the history section as the LCAO method was introduced by Sir John Lennard-Jones in 1929, not by Fano. However, in what way is this method perturbation theory? What is the unperturbed function? What is the perturbation? I would call it a variational method in contrast to perturbational methods in quantum chemistry. --Bduke 08:06, 10 June 2007 (UTC)

This is explained a bit here. However, variational methods can be used to derive non-perturbative results (i.e. effects that vanish to all orders in perturbation theory). So, in general, variational methods are not equivalent to perturbation theory. Count Iblis 14:23, 10 June 2007 (UTC)

Indeed but LCAO itself is not perturbation theory. It is used in Hartree-Fock theory get the unperturbed reference in methods like MP2. It is the section on LCAO that I think should be removed from this article. LCAO is a variational method, which as you say is not equivalent to perturbation theory. --Bduke 01:12, 11 June 2007 (UTC)

Yes, I agree with removal of that section. I'm not sure if other editors want to give their opinion, so let's wait a few days... Count Iblis 13:05, 12 June 2007 (UTC)

epicycles

The sentence about 17th century epicycles in the history of PT sounds strange to me. In the first place epicycles became less important after Keppler's work of around 1610. In the second place, if epicycles have anything to do with PT, then the origin of PT goes back to Ptolemy (150) and Hipparchos (100 BC). Any expert opinions? In any case a source is indispensable.--P.wormer 09:28, 12 June 2007 (UTC)

In one sense there is nothing either old or new about epicycles, they are merely geometrical analogues of circular functions. As it was expressed by H Godfray in 'An Elementary Treatise on the Lunar Theory' (4th ed 1885) at pp.63-64: our expressions, composed of periodic terms, are nothing more than translations into analytical language of the epicycles of the ancient; (Godfray went on to contrast the modern method of using gravitational theory to derive the periodic functions, as against the narrow repertoire of methods available to the ancients who had to infer their results from laborious observation).Terry0051 (talk) 16:37, 4 March 2009 (UTC)

Epicycles have nothing to do with the PT. Moreover the phrase "need for more and more epicycles that eventually led to the 16th century Copernican revolution" is not true and based on epicycles-upon-epicycles myth. I have removed this and replaced it with the text from Encyclopedia of Mathematics. I believe this is not a problem because WP:COMPLIC. Alexei Kopylov (talk) 05:36, 12 June 2014 (UTC)

History

Perturbation theory has its roots in 17th century celestial mechanics, where the theory of epicycles was used to make small corrections to the predicted paths of planets. Curiously, it was the need for more and more epicycles that eventually led to the 16th century Copernican revolution in the understanding of planetary orbits. [The previous sentence is mistaken: it is a common misunderstanding that Copernicus did away with epicycles. However, a close examination of Copernicus' great treatise, the De Revolutionibus reveals that, not only does Copernicus freely employ epicycles, but that he commits many of the same offenses in his planetary models as both he and the Tradition had accused Ptolemy of doing.]PtolemyGalen 17:36, 28 August 2007 (UTC)

Are the dates above (and in the actual article) correct? The wording seems to imply that people were still doing epicycles a century after the Copernican revolution, and that this continuing study of epicyles led (or fed into) perturbation theory. I would have thought that scientists and mathematicians would have given up trying to work with epicycles by the 17th century. Mcswell (talk) 16:15, 29 June 2008 (UTC)

The language in the article is heavily anachronistic (see prochronism): especially where it says "The earliest use of perturbation theory was to deal with the otherwise unsolveable mathematical problems of celestial mechanics: Newton's solution for the orbit of the Moon,...". Problems with this language include the following:

(a) Newton's consideration of the problem of the deviations from undisturbed Keplerian experienced by three bodies moving under their mutual gravitational attractions, (in Bk.I, Prop.66 and its corollaries, in the 'Principia', 1687) was not a 'use of' perturbation theory as one is said to make 'use of' something actually in existence already: rather, Newton gave the first description ever to define or consider such a problem (as recognised for example by expert commentators such as F-F Tisserand, 1894, Traite de Mecanique Celest, e.g. vol 3, ch 3); it set a new agenda for the recognition and investigation of disturbed motion under plural or multiple gravitational attractions, and contributed the first steps to the creation and recognition of perturbation theory as a subject.

(b) It was only much later that the idea 'celestial mechanics' came in, this expression is due to Laplace who greatly extended the reach of the subject, a century after Newton. Newton did not use the word 'dynamics', either: his preferred expression in the Principia was 'rational mechanics': thus, in the words of the Author's Preface in Newton's 'Principia' (1729 English translation): Rational Mechanics will be the science of motions resulting from any forces whatsoever and of the forces required to produce any motions, accurately proposed and demonstrated.Terry0051 (talk) 16:37, 4 March 2009 (UTC)

You are right. I removed it. Alexei Kopylov (talk) 05:37, 12 June 2014 (UTC)

LINK TO THE WRONG SPANISH VERSION ARTICLE

Hi, just a couple of words to tell the contributors to this article that the "Spanish" link takes to Teoría perturbacional, an article about quantum mechanics, not about mathematics. Someone wanting to correct it? Best wishes, --Mechanismic (talk) 09:47, 31 August 2009 (UTC)

Fixed it, thanks for spotting it. :-) The links are listed at the bottom of the article when you press edit, so it's easy to correct. These errors sometimes creep back via automatic bots if they'r present in other languages as well, but in this case no other language seems to have made the mistake, so we should be safe. EverGreg (talk) 16:06, 31 August 2009 (UTC)

Broken link

Hi, I'd just found a broken link:

• Chapter II: Introduction to perturbation methods by Johan Byström, Lars-Erik Persson, and Fredrik Strömberg

Since I do not understand very well the wikipedia's broken link policy, I just announcing it here. Cheers. Felipebm (talk) 19:54, 22 December 2009 (UTC)

It is still dead, so I just removed the link. It looks like a cache is available at the Wayback Machine, but it doesn't seem crucial to keep it in the External Links (and the Wayback Machine's servers often fail). Maghnus (talk) 21:52, 1 January 2010 (UTC)

Reference for use of feynman diagrams in classical mechanics

"Although originally applied only in quantum field theory, such diagrams now find increasing use in any area where perturbative expansions are studied.[citation needed]"

There is a citation that could be added :

http://arxiv.org/abs/hep-th/0605061

Arjun R. Acharya (talk) 09:21, 14 February 2010 (UTC)

Rename the page

For purposes of clarity, I suggest we rename this article "Educated Guessing". — Preceding unsigned comment added by 50.1.100.202 (talk) 22:46, 18 August 2013 (UTC)

I would rather suggest to invert the redirect with perturbation methods: they are a mathematical method, not a theory: you don't talk about asymptotic theory, variational theory nor numerical theory. This also is coherent with the majority of references on perturbation methods and related subjects like asymptotics. This is a definition for the word theory from Merriam-Webster dict: "a body of theorems presenting a concise systematic view of a subject <theory of equations>". All perturbation methods are based on theories of DE, for example regular perturbation is based among others on Taylor's theorem and implicit function theorem, averaging method is also based on Floquet theory. But these are traditionally belonging to more general or more fundamental mathematical theories: they were not born in a "perturbation theory" framework, so it is historically misunderstanding to call these methods a theory. The rename would also allow to employ the term "quantum perturbation theories" and finally solve the redundancy of these pages. 95.238.49.157 (talk) 17:23, 7 October 2014 (UTC)

small divisor problem

Are there other WP articles that actually give the basic classical and quantum perturbative series to second order? If so, can they be linked prominently to this article? I cannot seem to find them, anywhere. The quantum version is quite easy; it's a one-liner, with 2-3 paragraphs explaining the notation. The classical version is a good bit harder, but effectively follows the same form.

The standard textbook example for the quantum perturbative series is the Zeeman effect, it is normally covered in every quantum-mechanics-101 class in college ... but the WP Zeeman effect article does not give the series! Surprising, as it is simpler than most of the rest of that article...) A quasi-standard-ish article on the classical version is here: Wolfram small-divisors but it's stubby and ugly... there's also Springer EOM small denominators which goes from zero to too-technical too quickly. There is also a scholarpedia article on perturbation theory w.r.t chaotic systems but scholarpedia currently has redlinks (literally) for matching articles on classical and quantum perturbation theory.

The German wikipedia has an excellent article on QM perturbation theory that could be translated to English.

The Chinese wikipedia presents a short, simple account of basic second-order series using very traditional, non-QM notation. It's nice mostly becuase it's short.

The Ukranian version is nice; its longer than the Chinese version but otherwise similar, and not nearly as long as the German version. So perhaps there could be two articles: an intro, resembling the Chinese/Ukranian version, and a full-length one, resembling the German version?

I find it strange that the article never mentions Henri Poincaré or the 3-body problem or the "small divisor problem". I was under the impression that this was the number-one most-famous and most-celebrated perturbative series of all: top-o-the-pops chart, it laid the roots not only of resonant interactions, (red-link) whence follows all that jive-talk about feynman diagrams (particles in QFT are still called "resonances" by the accelerator people who actually measure them) but the small-denominator problem also is a foundational stone of chaos theory (as the small-denominator problem is unavoidable, and can be interpreted as leading to chaotic motion. It's the #1 reason why the orbit of Elon Musk's Tesla cannot be estimated beyond 100 million years.. resonant interactions with planetary bodies...). 67.198.37.16 (talk) 15:52, 14 September 2020 (UTC)

vandalism!?

Umm, 2/3rd's of this article was removed in 2017 in this series of edits, and no one reverted them or challenged them. At the risk of being excommunicated, I would like to call these edits "vandalism", because the result is an article that never actually says anything: it is currently a fluff-ball of cotton candy and meaningless words that is devoid of any factual information. Can we get back to an encyclopedia-like article that actually states facts, instead of being chock-full of randomized opinions that speculate about what perturbation theory might be, without ever once saying what it actually is? Sheesh! Come on, people, we can do better than this! This is an important topic, its foundational for hundreds of other WP articles that link to it! 67.198.37.16 (talk) 16:29, 14 September 2020 (UTC)

Oh, and history too: Major 20th century developments include the invention of Feynman diagrams to represent perturbative series, and the KAM torus as a description of weakly chaotic systems. The history section was also blanked by the above edits. 67.198.37.16 (talk) 16:52, 14 September 2020 (UTC)

Some of that material blanked from the page was more textbook-like than is really suitable for a Wikipedia article. Working step-by-step through a detailed example isn't what our articles are for, generally, and such exercises are seldom suited for articles on broad topics like perturbation theory. That said, much of the blanked material could be restored, I think. (Comment copied from WT:PHYS.) XOR'easter (talk) 18:08, 14 September 2020 (UTC)

Looking more carefully, it appears that the Chinese version is based on the blanked English version. It's OK, but the notation is kind-of old-fashioned. I'm not sure if that makes it easier to understand or not. Maybe it would be best if there were multiple articles: this one, which could remain formula-free (but which should include a review of what happened in the 20th-century) and then a quantum perturbation theory which could be translated from the German Wikipedia, a dynamical system perturbation theory which could be based on the scholarpedia article, and a classical perturbation theory article for which I have no promising crib sheets. I'm interested in this, because I am trying to write an article on another topic, and I hoped to refer to this article as background material for the terminology, but since basic definitions are missing, its hard to talk about more modern developments... 67.198.37.16 (talk) 18:18, 14 September 2020 (UTC)

In case it helps, I retrieved the old text and cleaned it up a bit. See User:XOR'easter/sandbox/perturbation theory. XOR'easter (talk) 18:43, 14 September 2020 (UTC)

That looks reasonable. I like how it builds a bridge to greens functions and name-drops borel resummation. and the "places where it fails" is interesting, too. I find the notation in the main batch of formulas to be clunky, but am not aware of anything more "elegant" or more readable (without resorting to notation of the quantum version, which presents it's own issues). Any ideas on how to declunkify? Also, to restore content: should it just be copied from your sand-box into here, or should it be carefully reverted from before? (I have no plans on doing any of this, but my plans often change.) 67.198.37.16 (talk) 19:19, 14 September 2020 (UTC)

Yes, that old version looks familiar, I contributed to that a long time ago (e.g. the part about resummation). I think it is important for Wikipedia to cover this subject from all perspectives and that should then include the way it is used in theoretical physics where does not stick to the rigorous mathematical theory and one uses resummation methods in way that's not rigorous (such methods can be rigorous, but as used in practice the conditions for the mathematical theorems that have been proven often do not apply). See here for lectures given by Carl Bender in this topic. Count Iblis (talk) 19:42, 14 September 2020 (UTC)

I believe that copying the text from the sandbox page and giving the revision number of the version of this article it was extracted from would preserve the chain of attribution satisfactorily. I haven't yet put much thought into how the notation might be improved, though I did snip several intermediate steps that didn't seem to provide much clarity. XOR'easter (talk) 20:55, 14 September 2020 (UTC)

I like the derivation of the perturbation solution of a DE in User:XOR'easter/sandbox/perturbation theory, I think it could be in the article, but at the bottom, certainly not the only example. I agree with 67.198.37.16 at top, the current article does not get across what perturbation approximation is. This is a mathematical technique, it cannot be described adequately without equations, and some examples. On the other hand, I don't think a complicated celestial mechanics example or the above derivation would help general readers understand. We need to start with an example on the level of high school math. First-order approximations are used everywhere in mathematics. How about starting with the approximation of a trig function? $\cos x\approx 1-x^{2}/2\quad (x<<1)$ Anyone who has an acquaintance with algebra could get it. --Chetvorno^TALK 21:09, 14 September 2020 (UTC)

I completely re-wrote the early and middle parts of the article so as to be understandable. There is a section called "examples" left over from before; its ugly and needs a complete overhaul. I expanded the "history" section to include the 20th century. I'm not sure what to write about the 21st century, its arcane and maybe too much in flux; just saying "oh its just strings" or "oh, its just modern algebraic geometry" or "supersymmetry" or whatever is a cop-out. Anyway, I think there is now enough of a skeleton there, that more precise statements can be hung off of it. 67.198.37.16 (talk) 22:16, 17 September 2020 (UTC)