Talk:Principle of least action/Archive 1

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We really have a MASSIVE duplication problem here! Just look at Lagrangian, Lagrangian mechanics, action (physics), principle of least action, Noether's theorem and who knows what else? Is anyone going to do anything about it?

Phys 01:56, 31 Jul 2004 (UTC)

This has caused some people to think that this principle is a "deep" principle of physics.

Does "some people" mean "some physicists" or "some confused freshmen who've got it all wrong" or what? Michael Hardy 21:43, 13 Sep 2004 (UTC)

Well, it is a deep principle of physics. Anyway, I'm hijacking this article to be history only. linas 22:56, 15 June 2006 (UTC)
Hi, Linas, I added a more accurate translation -- how does that read to you? I'm wondering whether we should start a new article for all this work, though; how about History of variational principles in physics? That would make the title agree better with the content. WillowW 13:20, 16 June 2006 (UTC)


Discovered a copy vio: this edit by User:Jheise on 13 Jan 2004. There a five sentance direct copy from the referenced source "Idle theory". Am removing the copied text now. Shame, its elegant prose. linas 00:14, 16 June 2006 (UTC)


The below is copied here, from the discussion at Talk:Action (physics). Its what lead to the current article

Historical development

(Copied from User talk:WillowW):

To the best of my understanding, Euler discusses the principle in 1744, in "Methodus Inveniendi Lineas Curvas Maximi Minive Proprietate Gaudentes" (online copy given in references section). This is contemporeanous to Maupertius. I have not actually tried to read the thing or to figure out which page it might be discussed on; I am assuming (given the suggestive title "minimi maximi" and historical research by others) that this is the case. linas 18:20, 13 June 2006 (UTC)

Hi Linas I finished the other article and I've printed out the chapter where Euler putatively derives the principle of least action in 1744, which I'll read tonight. I haven't read much Latin lately, so it should be fun! :) In case you didn't check back, I explained on my Talk page how Euler himself did not claim priority over Maupertuis for the principle of least action. I like the sockpuppet hypothesis! ;) Do you know where we can find a PDF of the original 1744 paper by Maupertuis? WillowW 21:01, 13 June 2006 (UTC)
Which chapter did you print out? The thing is 320 pages long!! I made a shallow attempt to find the Maupertius texts online, but was unable to do so. I'm guessing Euler had an ego that did not need to cleaim priority, he'd already done enough (List of topics named after Leonhard Euler) linas 21:53, 13 June 2006 (UTC)

Hi again, it's the first 10 pages of Additamentum 2, the very last chapter. Euler does indeed assert the abbreviated action principle for deriving the trajectory, but not Hamilton's principle for deriving the position as a function of time. It's closely akin to the Jacobi formulation. The opening paragraph is an excellent statement of the whole approach. Still, I think we should give Maupertuis priority, since Euler himself did, don't you agree? PS. When reading yourself, be aware that v stands for the squared magnitude of the velocity vector, contrary to what the article header says and contrary to our modern use of v for speed. Have fun! :) WillowW 12:16, 14 June 2006 (UTC)

Last sentence at bottom of page 309: "Iam dico lineam a corpore descriptam ita fore comparatam, ut, inter omnes alias lineas iisdem terminis contentas, sit , seu, ob M constans, minimum."
I know just enough foreign languages to translate this poorly as "Now say a path basically described in this way, compared to all the other paths having the same endpoints, is a minimum." I took the liberty of replacing "p=velocity times mass" and "dq=ds" in these formulas, and translating "line" as "path". Maybe someone from wikipedia latin can help w/ the translation. I'm tickled also to see the alternating product show up a few pages later. Also how little notation has changed, in that M=mass, and g=gravitational constant. linas 04:19, 15 June 2006 (UTC)
The PDF for this text is here. The description is here. Pfft. There's a better translation there, then mine:
Let the mass of the projected body be M, let v be half the square of the velocity of the projected body, and let the element of arclength along the prescribed path be ds. Among all curves passing through the same end points, the desired one makes the integral a minimum, or, for constant M, a minimum. This principle applies to any number of bodies or particles, but it seems to run into a difficulty when one considers the motion in a resisting medium.
I made the identification just to drive the point home.
Do you have a reference for Euler defending the priority of Maupertuis? linas 04:31, 15 June 2006 (UTC)

Hi, Linas, the factor of two mentioned in the header is incorrect, I think, since Euler clearly identifies as the momentum a few sentences earlier (quantitas motus corporis) and, later on, equates distance travelled . Thus, Euler's equals our modern , where is the velocity vector. I'll find a ref for the "Euler defending", but see this external link for how Euler himself prosecuted König for forgery before the Berlin Academy on 13 April 1752.

Interestingly, Leibniz's letters were genuine, according to these studies from the Berlin Academy itself

Gerhardt CI. (1898) "Über die vier Briefe von Leibniz, die Samuel König in dem Appel au public, Leide MDCCLIII, veröffentlicht hat", Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften, I, 419-427.

Kabitz W. (1913) "Über eine in Gotha aufgefundene Abschrift des von S. König in seinem Streite mit Maupertuis und der Akademie veröffentlichten, seinerzeit für unecht erklärten Leibnizbriefes", Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften, II, 632-638.

Although the originals have never been found, independent copies of the letters were found with the Bernoullis. Therefore, Leibniz seems to have had priority for "Maupertuis' principle" by ~39 years.

Hope you're having fun with technical Latin; it's a little easier (but less moving) than Horace. I was thinking of posting my translation of Euler's first few pages, but I wasn't sure whether that was allowed on Wikipedia? Does it violate "no original research", do you think? Perhaps there's no need for it, but Euler does express the concepts nicely. WillowW 19:33, 15 June 2006 (UTC)

I made a translation request at la:Disputatio:Pagina prima, and got a positive response (but no translation yet). I think a translation would not count as "original research"; however, where would it go? An article called Euler's Lineas Curvas or something like that? WP does allow articles summarizing books, and a book in Latin, of historical significance, is appropriate. linas 22:26, 15 June 2006 (UTC)
Latin is a dead,dead language,
Dead as it can be;
It killed off all the Romans,
And now its killing me.

Hi Linas I looked into the matter a little further and Maupertuis definitely has priority over Euler, as recounted in this reference

Terrall M. (2002) The Man Who Flattened the Earth: Maupertuis and the Sciences of the Enlightenment, University of Chicago Press.

Maupertuis developed his action principle ( is minimal) in 1744 by extrapolating from his ideas about light and published them in the Royal Society of Paris. The article did not appear in print, however, until much later. Meanwhile, Euler wrote his more general action principle ( is minimal) in the fall of 1744, completely independently of Maupertuis. Upon learning of Euler's book, Maupertuis became anxious about his priority and wrote to Johann Bernoulli II to confirm that Euler sent his proofs to the Swiss published after Maupertuis had publicly presented his work. Euler himself gives Maupertuis the priority, e.g., in the title of his article

Euler L. (1751) "Harmonie entre les principes généraux de repos et de mouvement de M. de Maupertuis", Sitz. Berl. Akad., pp. 169-198.

Their letters suggest a cordial relationship in which Euler aided Maupertuis with the mathematics and gently tried to dissuade him from introducing too much metaphysics. An excellent French account is the article

Brunet P. (1938) "Etude historique sur le principe de la moindre action", Actualités Scientifiques et Industrielles, 698, 3-107.

which I have just begun reading. For completeness, the original article by König that assigned priority to Leibniz was

König S. (1751) "De universali principio aequilibrii et motus", Nova acta eruditorum, 125-135, 162-176.

Hoping that this is helpful, and that you like the idea of making a new "history-specific" page, WillowW 22:03, 16 June 2006 (UTC) PS. I may go on a Wikibreak for a few days. ;) PPS. I found out that you can upload translations to Wikisource!

Excellent work! Do you care to amend the article appropriately, or should I? linas 22:41, 16 June 2006 (UTC)

Thanks muchly, Linas! :) (Glows inside)

You can certainly amend the article, or I can do it next week, once I've had the chance to digest Brunet's article (>100 pages of French -- easier than I feared, but still sometimes rough going).

I've been thinking about the seven-fold "actions" (eight now, if you count Maupertuis' less general definition) and how to present them to the lay audience. What do you think of this approach? We write two good articles that specialize in the main versions, Hamilton's principle (full Lagrangian action) and Maupertuis'/Euler's principle (abbreviated action), describing their deriviation, variational principle, a little history, and applications from mechanics through field theories. We then reduce Action (physics) to a kind of disambiguation article, where we spell out the differences/similarities among the various types of "action". The gory historical details would be put into a separate article, as one of the great scientific controversies of the 18th century. Does this approach seem reasonable to you? Till next week, WillowW 08:52, 17 June 2006 (UTC)

Cut foreign translation part

I don't know what language this is, but I moved it here (below). --Sadi Carnot 21:09, 15 August 2006 (UTC)

"Sit massa corporis projecti ==M, ejusque, dum spatiolum == ds emetitur, celeritas debita altitudini == v; erit quantitas motus corporis in hoc loco ==  ; quae per ipsum spatiolum ds multiplicata, dabit motum corporis collectivum per spatiolum ds. Iam dico lineam a corpore descriptam ita fore comparatam, ut, inter omnes alias lineas iisdem terminis contentas, sit , seu, ob M constans, minimum."

Comment about teleology

The comment that the apparent teleology in the principle of least action is only resolved in the quantum mechanical version seems a bit odd to me. Surely the apparent teleology can be resolved just by thinking about it in the correct way. I'm putting this in the comments because I'm not familiar with the literature on this controversy, so this is "original research." I agree with pointing out that the apparent teleology has been controversial historically - I'm only objecting to the idea that only quantum physics can resolve the issue.

By specifying some but not all aspects of both the initial and final conditions (the positions but not the velocities, at least in the simple examples I've seen) we are obviously going to be making some inferences about the initial conditions from the final conditions, and it is this "backward" inference that can be mistaken for a teleological causal influence. As a very simple example, consider a classical particle moving freely in a single dimension. Let's say we know that its position at time t1 is x1 and at time t2 we measure its position to be x2. We want to know its velocity at time t1, which we can easily work out (with or without the principle of least action) to be (x2-x1)/(t2-t1), which I'll call v. Now, if we say

"the particle's velocity at time t1 is v because its position at time t2 is x2"

then we're using causal language and it looks teleological, whereas if we say exactly the same thing in these words:

"the information that the particle's position at time t2 was x2 tells us that its velocity at time t1 must have been v"

then it becomes obvious that we're just making an inference: the system's state at later times isn't having a causal effect on earlier times, it's just giving us information about them. This is all the principle of least action is doing: it's helping us to figure out what the previous states of the system must have been in order for certain constraints on the initial and final state to have been met. In short, the apparent teleology arises purely because of the way in which the question is asked.

I agree completely. The laws of physics can usually be expressed either as a differential equation or as the minimization of an integral, e.g., Newton's laws of motion vs. the principle of least action. Specifying the "future" boundary condition in the p. of least a. is not really different in kind from specifying the initial velocity in Newton's equations, as you point out, which I believe most scientists understood correctly from the beginning.
I may have been the one to write that confusing wording; sorry about that! :( I had always meant to come back to this whole topic, but I've gotten distracted as you see. Let me encourage you to improve the wording yourself... :)
...and to become a Wikipedian; it's that link at the upper right corner. I'll be very happy to welcome you on your new user page.  :) Willow 19:09, 21 May 2007 (UTC)

I agree too, so I've changed the page. (I don't have a reference either, but your explanation seems entirely uncontroversial to me, whereas the appeal to quantum mechanics was there before seemed silly.) Nathaniel Virgo (talk) 11:37, 1 February 2011 (UTC)


This should be merged with Hamilton's principle. JRSpriggs 03:56, 30 July 2007 (UTC)


What sort of constraints are allowed?(and why) I was confused about it, and someone said "The principle of least action requires fixing both end points." if true, it is definitely worth mentioning.(I don't have the necessary knowledge, though.) (talk) 14:14, 1 December 2009 (UTC)


"stretched corded ropes between two points to measure distance which minimized the path of separation" doesn't make any sense. Changed to "stretched corded ropes between two points to measure the path which minimized the distance of separation". (talk) 20:21, 11 September 2010 (UTC)

Move the article

to the more accurate title Principle of stationary action instead of the less accurate Principle of least action? What’s the point in using the less accurate title then stating in the article the more accurate one? Surely the readers will get the correct title of the article in their head when they think of "stationary" as "standing still i.e. not changing".-- F = q(E + v × B) 11:13, 10 March 2012 (UTC)

THIS IS ABOUT TO BE MOVED TO THE CORRECT NAME Principle of stationary action, AND MERGED WITH Hamilton's principle. PLEASE DO NOT EDIT YET. F = q(E+v×B) ⇄ ∑ici 15:53, 21 April 2012 (UTC)
Sorry - forget that the edit history will be lost if the move is done. Should have thought of it before...F = q(E+v×B) ⇄ ∑ici 15:57, 21 April 2012 (UTC)
Done. F = q(E+v×B) ⇄ ∑ici 18:25, 21 April 2012 (UTC)
I will revert back to the old version. See also here. Apologies for this... F = q(E+v×B) ⇄ ∑ici 19:17, 22 April 2012 (UTC)
I am not sure that "least action" is wrong. Can you give an example where stationary action is achieved in nature but is not least action? JRSpriggs (talk) 23:49, 22 April 2012 (UTC)
I'm past that mistake now. Maybe I was thinking at the time "least" could necessarily imply "minimum" (which is not always the case?). F = q(E+v×B) ⇄ ∑ici 23:57, 22 April 2012 (UTC)
Yes, of course it need not be a global minimum. A local minimum is sufficient. JRSpriggs (talk) 00:02, 23 April 2012 (UTC)
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