Talk:Stationary-action principle/Archive 1

Archive 1

Why Dirac and Not Feynman

This article gives Dirac credit for adding the principle of stationary action to the quantum mechanical mix, but I read that it was Richard Feynman. I believe it was in the "The Theoretical Minimum", where I read this.

Also, in Feynman's Wikipedia article, it says this:

Feynman's thesis applied the principle of stationary action to problems of quantum mechanics, inspired by a desire to quantize the Wheeler–Feynman absorber theory of electrodynamics, laying the groundwork for the "path integral" approach and Feynman diagrams, and was titled "The Principle of Least Action in Quantum Mechanics".

Should he not at least get honorable mention in this article if not total credit?

User:Lowell Boggs — Preceding undated comment added 20:49, 9 October 2014 (UTC)

Incorrect caption in new image

As the system evolves, q traces a path through configuration space (only some are shown). The path taken by the system (red) has a stationary action (δS = 0) under small changes in the configuration of the system (δq).

Please say why it's wrong here. I didn't add the citation before but this is in Penrose's Road to Reality (only difference is that he draws a configuration manifold depicted as a curled up sheet, I didn't want to copy it completely). As shown the caption has been fully reduced and cited. If the drawing is also wrong for any subtle reason - say so and it can be redrawn.. Thanks, Maschen (talk) 07:35, 12 September 2012 (UTC)

About the diagram in itself - it's intended to be a dual purpose figure, for classical and quantum mechanics. The opacity (in %) of the paths

• in classical mechanics corresponds to variation in action (lower opacity for larger variation, so more intense colours show less variation).
• in QM (path integral formulation) corresponds to the probability of the amplitude occurring (each path is a probability amplitude). Since action (in units of hbar) is the wavefunction phase, the larger variations cancel (nearly "out of phase") while lower variations reinforce (nearly "in phase").

Maschen (talk) 09:24, 12 September 2012 (UTC)

The caption to which I was objecting said "Illustration of the principle. The configuration of the system is described by generalized coordinates q (green), a "path" (blue) is the set of q vectors the system takes from times t₁ to t₂. Each path corresponds to a different time evolution of the system. The path with the least variation in action (corresponding to smallest δq) is the path taken by the system (red).". In particular, the last sentence is wrong, if regarded as a general principle.

The physical path is the path along which the action is minimal. This implies that infinitesimal variations from that particular path will cause no change in the action. "Least variation in action" has nothing to do with it. And what does "smallest δq" even mean? JRSpriggs (talk) 15:27, 12 September 2012 (UTC)

Nothing. Guess I should have just said what the caption is now. There were a number of things running through my head at the time so I mixed terminology up, but that's not an excuse of course... Thanks, Maschen (talk) 16:08, 12 September 2012 (UTC)

Your revised version is OK. Feel free to add it to the article. JRSpriggs (talk) 05:42, 13 September 2012 (UTC)

Will do, thanks again. Maschen (talk) 07:11, 13 September 2012 (UTC)

References

least is not the same as stationary!

Least (together with some smoothness) implies stationary, but the converse is not true. For example, to get the equations of motion of a free particle you need *least* to get that the mass m is positive (OK, let us consider classical physics only ;). Moreover minimized or maximized over any time period, either, is not equivalent to stationary: e.g., use the principle to find a geodesic in a Pseudo-Riemannian manifold (well, this is not necessarily a path of some physical object, but the method goes by using stationary action, anyway). The introduction should be more clear.

Large parts of the sections "Apparent teleology" and "More Fundamental Than Newton's 2nd Law" are - to the best - just opinions of a single scholar; they should be presented as such (I am going to change the sections a bit).78.15.173.82 (talk) 22:07, 3 August 2015 (UTC)

I agree. The sentence in the lead, ' It was historically called "least" because its solution requires finding the path that has the least change from nearby paths.[1]' is incorrect. Martin Hogbin (talk) 14:23, 8 February 2016 (UTC)

Agreed. Also, a valid point re geodesics. StrokeOfMidnight (talk) 14:43, 4 April 2021 (UTC)

The statement appears to rely too heavily on Feynman's Ch. 19. Jbergquist (talk) 17:49, 5 July 2021 (UTC)

Perhaps more emphasis should be placed on the historical perspective.

https://books.google.com/books?id=_iTnBwAAQBAJ&lpg=PR2&pg=PA101#v=onepage&q&f=false

Jbergquist (talk) 17:52, 5 July 2021 (UTC)

Add Examples and Clarify

• I would like to see an example of least action being applied, and have a well-known answer come out to demonstrate how a classical physics equation can come about from this important principle
• When would it be appropriate to use the principle of least action vs. Maupertuis' principle vs. Euler's principle?

Tstring42 (talk) 00:08, 8 April 2017 (UTC)

Quantum principle of least action

In classical mechanics, the principle of least action determines a unique path for a particle when the momentum of the particle is given by the relation p=mv. However, as in quantum mechanics, if the momentum of the particle is given as p=h/λ then the principle of least action will lead to Feynman's assumption which states that the particle can take any path to go from one position to another in space. Hence, in quantum mechanics, the principle of least action does not need to produce a unique trajectory as in classical mechanics. Please refer to an article posted on ResearchGate entitled ON THE PRINCIPLE OF LEAST ACTION by Vu B Ho for more details. 101.182.50.172 (talk) 11:15, 23 May 2017 (UTC)

Translated Euler quote does not match cited Wikisource translation

In 2 Origins, statements, and controversy -> 2.3 Euler, there is a quote involving the following sentence:

"Now I assert that the curve thus described by the body to be the curve (from among all other curves connecting the same endpoints) that minimizes …"

Noticing that it is grammatically incorrect ("assert that the curve … to be"), I traced the link in the citation given to a Wikisource translation and found the same sentence to be quite different there:

"Now I assert that the true trajectory of the moving particle is the trajectory to be described (from among all possible trajectories connecting the same endpoints) that minimizes …"

What should I do about this? FishyFisch (talk) 21:47, 18 October 2018 (UTC)

Euler should probably have said more about constraints on the variation but this was before Lagrange and his multipliers which allowed completely arbitrary variations. It seems for the deviation of the integral, ∫mvds, to be a minimum the variation in the path has to be transverse to the actual motion. Jbergquist (talk) 17:02, 5 July 2021 (UTC)

The translation does not appear to be well done although one might cute Euler's Latin as being rather convoluted. Confusing v with velocity in Methodus inveniendi may be a mistake since he just refers to a relation between "speed" and height. The translation is a rewrite, a paraphrase, of Euler's work. Jbergquist (talk) 17:12, 5 July 2021 (UTC)

incomplete

Under General Statement, there is no definition for R. It is not attached to anything in the preceding word salad. either link it like you did for stationary point, or explain what it is in the same place. What is "to first order"? If that means the first derivative equals zero, say so. What is the definition of "minimal principle" and where is a link to a page on that? 100.15.117.34 (talk) 18:51, 29 September 2023 (UTC)

As usual in mathematics, "R" represents the real numbers — the ordered field over the rationals which is algebraically complete. There is a reference with a link giving an answer to your last question. JRSpriggs (talk) 19:34, 30 September 2023 (UTC)

Least vs. stationary

According the Feynman in The Feynman Lectures on Physics Vol. II Ch. 19: The Principle of Least Action (which is cited in the article), he uses the phrase "least action" throughout, He also remarks that his principle is a redefinition of the original classical mechanical term, applying it to electromagnetics and quantum mechanics as well. Nowhere does he mention stationary action by name. I do not know how today's mainstream physics uses these terms or how that usage developed historically, but since this article leans so heavily on Feynman for sources I think it needs to make these various meanings and usages a lot clearer; it is no enough to flout a major source's terminology without proper explanation - and equally rock-solid sourcing. — Cheers, Steelpillow (Talk) 08:41, 17 June 2023 (UTC)

The problem you point to is caused in part by the article's name. Physicist generally use "principle of least action", not "stationary action principle". So your observation that Feynman does not mention "stationary action" is normal and expected in physics.

Separately, multiple different applications of variational techniques to physics exist, depending upon what is held constant (eg end points, path length, time). I have not seen any physics reference which sorts this out but its worth looking for.

And in addition, different particular action choices seem to affect the name applied, esp. in the early days.

We're working on the History of variational principles in physics; hopefully that will ferret out some references to help. Johnjbarton (talk) 22:44, 13 November 2023 (UTC)

Maupertuis's_principle#Comparison_with_Hamilton's_principle claims to connect two action principles, but it is completely unsourced. Feynman, in the article link in the first line of this topic, says his action in principle of least action is "Hamilton’s first principal function". Johnjbarton (talk) 23:21, 13 November 2023 (UTC)

Goldstein compares Hamilton's principle with (his version) of principle of least action. The three differences in the unsourced section "Comparison_with_Hamilton's_principle" can be referenced with Goldstein. He gives a detailed comparison. (Page 359 in 3rd edition) Johnjbarton (talk) 04:16, 15 November 2023 (UTC)