Talk:Fermat's principle

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WikiProject iconFermat's principle has been listed as a level-5 vital article in Science (Physics). If you can improve it, please do.
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Issues resolved or rendered moot as of 30 September 2019:[edit]


Rejected in June 2003. — Gavin R Putland (talk) 11:58, 29 September 2019 (UTC).Reply[reply]

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The style here was taken from the style used at Poincare conjecture. Nice, isn't it?
it's revolting. A plain BLOCKQUOTE tag will suffice. When we one day switch to XML we'll be able to use tags like THEOREM and have a consistent look across Wikipedia. Until then, please don't impose your sense of aesthetics upon the reader -- Tarquin 10:56 19 Jun 2003 (UTC)
I'm sorry you don't like it. It's not my idea: I copied from another article, and various people seem to have propagated it to various other appropriate articles. I agree, a piece of Wiki syntax <theorem> </theorem>, which generates appropriately styled DIVs would be ideal. For the moment, IMHO, this makes various articles containing statements of theorems and axioms look much nicer -- The Anome 11:01 19 Jun 2003 (UTC)

Better image[edit]

I think this image should be translated and used or perhaps just added and leaving the original image. Any thoughts? Cristan 23:00, 13 November 2006 (UTC).Reply[reply]

☒N Stale. Implicitly rejected? For the time being, and perhaps for the long term, I have cited a standard rare-to-dense refraction diagram in connection with the lifeguard analogy. — Gavin R Putland (talk) 13:03, 29 September 2019 (UTC).Reply[reply]


 Resolved (I hope) by: (i) adding a "Derivation" section; (ii) mentioning, in the "Analogies" subsection, the deduction of the classical path of a particle from its wave function; and (iii) linking to the article on Snell's law, which includes a derivation of that law from Fermat's principle. — Gavin R Putland (talk) 11:58, 29 September 2019 (UTC).Reply[reply]

Not entirely.Extent content to see why it is ambiguous. Nikosbinis (talk) 11:22, 22 August 2023 (UTC)Reply[reply]
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Is there any proof of the principle that's worth putting in here? The discussion of variational principles didn't really talk about how this one was derived. --Joe (11/19/2006)
Here's a book that says it needs no proof because is the definition of a ray: book. What then needs proof is Snell's law as a consequence of this definition of ray. Dicklyon 22:32, 9 December 2006 (UTC)Reply[reply]

If you look for an experimental proof you must put a sensor at the B point.In fact,there is an undefined number of rays progressed from A to B,but sensor detects the strongest=less absorbed:followed the shortest path.Light is not thinking,not choosing route. But what is the fastest route if light cannot reach B?

When it comes to the sciences I've always thought definitions of physical processes are not truly definitions of form but observations of character which therefore merit derivation from more fundamental principles, just as Newton's definitions of simple motions for point masses can and should be derived from limiting considerations of quantum mechanics. (talk)
Does this principle explains underwater refraction if the point B is at the bottom of the sea and sunlight cannot reach it? Be carefull about math:it is not a trick!You cannot put B on a "ray" because P & B define the ray!It is exactly the B point whitch defines the ray.  (talk)10:19, 8 January 2015 (UTC)Reply[reply]

Better statement[edit]

 Resolved (I hope) in the rewrite of August 2019, if not earlier. — Gavin R Putland (talk) 11:53, 29 September 2019 (UTC).Reply[reply]

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Since the very first edition of this article, the principle has been stated as "The actual path between two points taken by a beam of light is the one which is traversed in the least time." I see at least three things wrong with this statement, and would like to fix it or replace it with one from a verifiable source; but if this one has a source, someone please point it out. I would change the word order to avoid garden-path constructions like "two points taken", change beam to ray, change which to that, and maybe make it read easier. I'm looking for a better statement of it. Dicklyon 22:32, 9 December 2006 (UTC)Reply[reply]

Extremum principle of mechanics[edit]

 Resolved (I hope) under the headings "Analogies" and "Modern version". — Gavin R Putland (talk) 13:21, 29 September 2019 (UTC).Reply[reply]

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From the article: "In classic mechanics of waves Fermat principle follows from the extremum principle of mechanics (see variational principle)."

Can someone explain what exactly is meant by the extrem principle of mechanics? The text doesn't really make it clear and the link to 'variational principle' isn't that helpful either, I think.--BobaFett 12:12, 19 April 2007 (UTC)Reply[reply]

In short, all classical mechanics laws can be derived by requiring physical systems to obey some "minimal value laws". These quantities that are minimized are called "Action", and are usually some line integral on the trajectory of the object. The "law" states that only trajectories that have an extremal value of this "Action" integral are possible. The trick is to find what the action is (This pretty much sums up modern physics). Fermat's principle is a minimum on the line integral , which sums up to the total time travelled. In other words, Fermat's principle says that only trajectories that have an extremal value of the total time travelled are possible (total time travelled is the action). This whole line of thinking in terms of minimizing values, can be is summed up by the Principle of least action.--Dean Mark 17:59, 23 April 2011 (UTC)

"Incompleteness of the principle"[edit]

 Resolved.  The first part (May 2009) was promptly resolved by deleting the offending passage. The second part (January 2011) refers to an earlier version and (I hope) has now been resolved in the "Derivation" section. — Gavin R Putland (talk) 09:37, 30 September 2019 (UTC).Reply[reply]

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The whole half-paragraph that attempts to explain the "incompleteness" of the principle seems to me to be (1) original research and (2) completely wrong. The "optically shortest path" (which I assume means the path that minimizes time) is not around the brick, as suggested. It is through the brick. Indeed, if Fermat's principle were to fail in this case, or in any case where the "dependence" (of what?) is not "contiguous" (whatever this means), then Fermat's principle would have no applications at all. This paragraph sounds like it was written by someone who doesn't understand Fermat's principle -- or maybe I don't?

I think the misunderstanding is like this. Fermat's principle, indeed, does apply to local minimum. A derivative exists only for continuous fuctions. The person who wrote this paragraph took the logical step of assuming that because the derivative does not exist, there is no local minimum. But this is false; consider, for example, the absolute value function, for which 0 is a minimum which is not differentiable. (i.e. it is a critical point.) But this is not a flaw of Fermat's principle, which is not concerned with the type of minimum!

Therefore, unless someone can explain to me why I'm an idiot, I'm going to delete it shortly. (But please do tell me if I've misinterpreted something.) Firegnome (talk) 19:25, 13 May 2009 (UTC)Reply[reply]

Obliterate it. (talk)

There still is a problem with the derivation. It says:

But if there is a shortest route AB0, and the optical path varies smoothly through it, then a considerable number of neighboring routes close to AB0 will have optical paths differing from AB0 by second-order amounts only and will therefore interfere constructively. Waves along and close to this shortest route will thus dominate and AB0 will be the route along which the light is seen to travel.

The problem that needs explaining is why the waves _do not_ travel along the shortest route. The shortest route from P to Q is a straight line between them. It is, however, not the fastest route between P and Q -- it spends a greater proportion of its length travelling in whichever of the two media where light travels more slowly. Should you have two media where the speed of light in that media is the same, no diffraction will happen. The path will not bend. (talk) 14:43, 7 January 2011 (UTC) Laura CreightonReply[reply]

Not really "time", since velocity is phase velocity, not group velocity[edit]

 Resolved.  This section seems to refer to an earlier version of the article. In any case, the article now distinguishes the "ray velocity" from the "phase velocity" and the "group velocity". — Gavin R Putland (talk) 10:36, 30 September 2019 (UTC).Reply[reply]

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The "velocity" used here in deriving Snell's law is the phase velocity, not the group velocity, so calling it a "least time" (or even a stationary time) path is somewhat deceptive.

— Steven G. Johnson (talk) 04:44, 5 August 2009 (UTC)Reply[reply]

If the stationary quantity isn't time, what is it? Collin237 —Preceding unsigned comment added by (talk) 13:00, 6 April 2011 (UTC)Reply[reply]

Not Least, but Extremum[edit]

 Resolved, but not by claiming an extremum. The stationarity in the traversal time is either a local minimum or something more complicated; it is never a local maximum if all modes of variation are taken into account. (Search text for "maximum".)

 — Gavin R Putland (talk) 15:17, 29 September 2019 (UTC).Reply[reply]

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The major summary is incorrect. Fermat's Principle is based in an extremum, not a minimum. For example, reflection inside an ellipse is a maximum. The minimum is to just go directly to the other focus without reflection. (talk) 15:29, 8 September 2009 (UTC)Reply[reply]

You are right that the principle is based on extremum values, but your example is incorrect. The language of minimum or maximum relates to the local extremality of the path compared to infinitisimally close paths. The path you stated above is a local minumum. The existance of another unrelated path that is faster does not change this. You need to look harder to find paths that are a local maxima.

-- Dean Mark 00:02, 11 May 2011 (UTC) —Preceding unsigned comment added by DeanMrk (talkcontribs)


I added a reference to Ptolemys work and his influence in Alhacen´s work.--Knight1993 (talk) 15:07, 10 April 2010 (UTC).Reply[reply]

 OK.  In later amendments, Ptolemy has been dropped for want of specifics; but Alhacen is still mentioned in the Notes, and other characters have been added. — Gavin R Putland (talk) 15:03, 29 September 2019 (UTC).Reply[reply]

The "Modern version" is not clearly stated[edit]

 Resolved (quietly) by defining "ray velocity". While a negative index doesn't mean a negative group velocity, it does mean a negative ray velocity, hence a negative contribution to the notional traversal time. — Gavin R Putland (talk) 10:14, 30 September 2019 (UTC).Reply[reply]

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With the advent of negative index metamaterials, the fermat principle as stated is not clear enough. There is a reference to "stationary optical paths", but this is very obscure. The correct modern formulation of the principle is: , [1][2] . Where is the index of refraction of the media. In words, "The variational Fermat principle states that the optical length of the path followed by light between two fixed points, and , is an extremum. The optical length is defined as the physical length multiplied by the refractive index of the material."[2] This is different than the minimal time formulation when you consider the fact that can be negative.

The modern version is very different than the original version, and this difference should be more clearly pointed out. As an example, suppose we have two half infinite slabs held side by side, one with , and the other with . If we place a point source in the positive index slab at position , its image will be formed in the negative index slab at position [3]. Notice that every ray originating in the source, and arriving at the image, has an optical path of zero. That is, all paths have an extremal value. On the other hand, no ray, except the one traveling in a straight line between source and image, has an extremal path in terms of time. Or, stated differently, according to the original fermat principle, only the ray traveling in a straight line between source and image will exist, but in experiment, other rays connecting source and image exist too. Hence the need for the modern formulation.


  1. Viktor G Veselago 2002 Phys.-Usp. 45 1097 doi: 10.1070/PU2002v045n10ABEH001223
  2. Chapter 1.5 in: R. Marques, F. Martin, and M. Sorolla. Metamaterials with Negative Parameters. Wiley, 2008.
  3. Pendry, John B.; Smith, David R. Physics Today, Volume 57, Issue 6, pp. 37-44 (2004). doi: 10.1063/1.1784272

DeanMrk (talk) 11:12, 23 April 2011 (UTC)Reply[reply]

--edit-- DeanMrk (talk) 16:43, 23 April 2011 (UTC)Reply[reply]

Same old error[edit]

 Resolved  by defining the "ray index" and distinguishing the "ray velocity" from the phase and group velocities. — Gavin R Putland (talk) 10:39, 30 September 2019 (UTC).Reply[reply]

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The velocity in the index of refraction is phase velocity, not group velocity. While the speed of a wave packet is its group velocity, not the phase velocity.

This means that integrating over n does not imply the time of travel, because generally .
-- Dean Mark 23:39, 10 May 2011 (UTC) —Preceding unsigned comment added by DeanMrk (talkcontribs)

I have two notices: First, the principle of Fermat is most of the times used for describing phenomena of light rays and a ray of light travels with the phase velocity (there is no group velocity for light rays because a light ray does not consist of wave packets). Second, I think the expression "time of travel" in this article means the time one point on the electromagnetic wave needs to cover a given distance and the velocity of such a point is the phase velocity.
So in my point of view the main explanation for your given problem is that "travel of time" does not refer in this article to the time a wave packet would need to cover a distance. I have reformulated the mentioned paragraph. (talk) 13:56, 19 March 2012 (UTC)Reply[reply]

Deleted final sentence of introductory section[edit]

I removed statement about Hamiltonian optics that seems better suited to and appears at that page; also who the heck is Winston - either Newton or Lagrange, but even correcting that - it seems out of context to bring this up here. Netrapt (talk) 23:29, 17 December 2011 (UTC).Reply[reply]

 OK, but moot due to later amendments. Hence it is also moot that "Winston" could be William Whiston. — Gavin R Putland (talk) 14:11, 29 September 2019 (UTC).Reply[reply]

External links modified[edit]

☒N No longer relevant; the reference has been replaced by a later version. — Gavin R Putland (talk) 13:46, 29 September 2019 (UTC).Reply[reply]

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Complete rewrite[edit]

(Hidden history)
@Dicklyon, Firegnome, Stevenj, and Netrapt: ("FYI") I'm working on a complete rewrite of this article, with a view to addressing the unresolved issues raised above. The current draft (an early one at the time of writing) can be found at User:Gavin R Putland/Fermat's principle — just in case I go under a bus before it's finished. — Gavin R Putland (talk) 05:36, 7 August 2019 (UTC).Reply[reply]
I don't understand why a complete rewrite is the way to go. You can't update and reorganize incrementally toward a better article? It would be easier to review changes than to compare old and new articles in toto. Dicklyon (talk) 06:03, 7 August 2019 (UTC)Reply[reply]
@Dicklyon: I used the incremental approach for Fresnel equations, and in retrospect I should have done the same for Fresnel rhomb. But my contributions to Augustin-Jean Fresnel and Total internal reflection began with "big bang" rewrites and seem to have been welcomed. I think the present case is more like the latter two; but I suppose I could compromise by starting with those sections that have the closest counterparts in the old version. — Gavin R Putland (talk) 02:08, 8 August 2019 (UTC).Reply[reply]
That sounds like a good approach. Dicklyon (talk) 03:01, 8 August 2019 (UTC)Reply[reply]
Update @Dicklyon, Firegnome, Stevenj, and Netrapt: Except as noted below, the draft at User:Gavin R Putland/Fermat's principle now covers at least the same ground as the live version. If there is no objection within (say) 48 hours, I shall make the draft live (with the empty sections commented out), and fill in the empty sections as time permits.

The issues addressed so far include the following:

  • I hope the "Derivation" makes it clear why traversal time is more important than length.
  • The "Derivation" doesn't merely treat Fermat's principle as the "definition" of a ray, but explains why such a theoretical definition ought to correspond to empirical ones (line of sight; narrow beam).
  • The ray index is distinguished from the usual refractive index.
  • The ray velocity is distinguished from both the phase velocity and the group velocity (see "Notes").
  • The subsection on Hamilton's principle has been rearranged for easier and more frequent comparison with the references. (Disclosure: This is the subsection that I know least about; I tend to take Fermat's principle on its own terms.)
  • "Quantum electrodynamics" is not currently mentioned in the draft. Correct me if I'm wrong, but I understand that for matter waves, Fermat's principle for the phase/ray velocity corresponds to Maupertuis's principle for the group velocity. I'm not sure if it's helpful to include that. (And if I can understand it, it isn't really QED!) (P.S.: Matter waves and the classical path are now mentioned briefly under "Analogies", with two references. — Gavin R Putland (talk) 08:47, 28 August 2019 (UTC).)Reply[reply]
  • Euclid is relegated to the "Notes" due to uncertain authorship. (P.S.: After checking against Sabra, Euclid is now mentioned in the References, citing his Optics, not the pseudo-Euclidean Catoptrics. — Gavin R Putland (talk) 23:15, 27 August 2019 (UTC).)Reply[reply]
  • Ptolemy is no longer mentioned; I think the information would need to be more specific if it were to warrant inclusion. Ibn al-Haytham is not quite a precedent and has been relegated to the "Notes". The same "Note" mentions Boelmans, who did scoop Fermat, albeit not in terms of time.
  • Oh, and Huygens' principle is not "earlier" than Fermat's.

Gavin R Putland (talk) 15:04, 23 August 2019 (UTC).Reply[reply]

(Hidden history)
I don't like this approach, of replacing an OK article with one that's all new, very long and complicated, and full of section that look they will expand a lot more. If instead you add material to the existing article, and make corrections as needed, we could probably say more easily when certain sections are getting too big and detailed, or too hard for a lay person to appreciate. Dicklyon (talk) 00:47, 24 August 2019 (UTC)Reply[reply]
@Dicklyon, Firegnome, Stevenj, and Netrapt: OK, I shall back-pedal as far as possible from here: I shall curtail the draft, and not expand it beyond what is necessary to address the apparent complaints (except the "History" section, which, I trust, will at least not be considered "too hard for a lay person to appreciate"). And this will be the last time I offer a big-bang rewrite of an article. — Gavin R Putland (talk) 13:15, 24 August 2019 (UTC).Reply[reply]
Update @Dicklyon, Firegnome, Stevenj, and Netrapt: The complete draft for the new version is at User:Gavin R Putland/Fermat's principle. Any objection?? (BTW, the distinction between the ray velocity and the group velocity (Note 6) resolves any apparent contradiction concerning negative-index media. While a negative index doesn't mean a negative group velocity, it does mean a negative ray velocity, hence a negative contribution to the notional traversal time.) — Gavin R Putland (talk) 13:15, 26 August 2019 (UTC).Reply[reply]
Yes I still object to the idea of a "new version". Dicklyon (talk) 14:31, 26 August 2019 (UTC)Reply[reply]
@Dicklyon, Firegnome, Stevenj, Netrapt, Knight1993, Cristan, The Anome, and Tarquin: And I object to breaking up the now-completed draft into a series of incremental edits, because: (1) my "compromise" procedure initially seemed to be encouraged; (2) in spite of ample opportunity, there has not yet been any criticism of what I have actually written; (3) it would be disingenuous to make one "incremental" change when there is wholesale change in the pipeline; (4) the technical part of the new version is not unduly long after relegating parenthetical matters to the "Notes"; (5) examining multiple edits would waste other watchers' time; (6) turning one rewrite into multiple edits would waste my time; and (7) in view of my medical history and my consequently dubious life expectancy, I don't have time to waste. Accordingly, there is nothing for it but to put up the new version and see how it flies. Whatever happens, having seen what could go wrong, I shall not be doing any more "big-bang" rewrites. — Gavin R Putland (talk) 11:39, 28 August 2019 (UTC).Reply[reply]

A specific objection is that the lead paragraph is rather formidable. Why not start with the principle of least time, which is what Fermat said, and then in later paragraphs introduce waves and extrema? And what the heck is stasimochrome? Did you make that up, or can you source that terminology? Is your derivation and approach following sources more generally? There's a ton to check here. Dicklyon (talk) 14:49, 28 August 2019 (UTC)Reply[reply]

@Dicklyon: Thank you. I hope my latest edit addresses most of these concerns. In particular, it belatedly occurs to me that the place to define a ray path as a "path of stationary traversal time" is immediately before we justify that definition in terms of lines of sight and narrow beams. That gets rid of the need to coin a separate term to be used only eight times. Facepalm   P.S.: That edit was on 29 August. Belated signature: Gavin R Putland (talk) 14:55, 29 September 2019 (UTC).Reply[reply]

unnecessarily complicated[edit]

I could not understand this article, I really wish math and science wikis had summaries that curious lay people could parse. not sure why an encyclopedia would require so much prior knowledge like a academic article 2600:1702:1DA0:1700:911C:95B2:7E98:B039 (talk) 20:27, 1 February 2023 (UTC)Reply[reply]

IMO this drive-by comment is not substantiated in any way. Maybe the lede could be a bit cleaner but compared with other pages this one has nice long ramp up. Nice work! Johnjbarton (talk) 16:25, 27 May 2023 (UTC)Reply[reply]