Talk:Brachistochrone curve

message: The "History" section states "Five mathematicians responded with solutions:" but only lists four.  Either the numbers five and four should be four and three, respectively, or the fifth mathematician (fourth published) should be identified.
Perhaps Snell's Law, while it is a clever technique to use in this situation, is not as instructive as an example should be for those looking at the entry.
I am also intrigued and a bit dubious about the use of Snell's Law as proof. Snell's Law is based on Fermat's principle of least time, but that itself could be said to be based on Quantum Electrodynamics, or alternatively in classical mechanics based on Huygens' wave construction. It seems to me we shouldn't get involved in that stuff here - it is really a problem in mathematics, the Physics part just gives a framing to it. Again, Fermat's principle of least time is just a variational principle so it comes back to the calculus of variations at bottom. So I would think the "Alternate Proof" given here is the correct one for this page, although I am no kind of expert on this subject. AdamWGibson (talk) 16:09, 8 May 2008 (UTC)
replies: text: Using Snell's law would have been the obvious way for Newton to solve the problem I believe and makes the solution nice and accessible. Using the calculus of variations would not be true to history, and it certainly wouldn't make the solution to this ancient problem available to a wide audience which should be what wiki aims at where possible. Dmcq (talk) 21:13, 4 August 2008 (UTC)

text: Actually Newton thought light was attracted to and went faster in a dense medium, so he wouldn't have thought light followed Fermat's principle. He knew about Snell's law as the general law about the fastest path when the speed changes though, he thought light followed the path for a different reason which somehow came out with the same law of angles. I'm a bit surprised he came up with his rationale in optics as Fermat's principle was already well accepted as a rationale for the path light followed. Dmcq (talk) 21:32, 4 August 2008 (UTC)
thread_title: History section

message: The aim is to minimise the integral \int{dt} = \int{\frac{ds}{v}}.
Starting with v=0 at y=0 (starting from rest): v=\sqrt{-2gy} (negative sign because y is down)
ds=\sqrt{1+y' ^2} dx =\sqrt{1+x' ^2} dy  (we can use either one),
therefore we minimise \int{\sqrt{\frac{1+y' ^2}{-2gy}}}dx = \int{\sqrt{\frac{x' ^2 +1}{-2gy}}}dy
Substtuting into the Euler-Lagrange equation (this is the most common tool in variational analysis; for this simple problem however, we can derive it here itself)
we get \frac{d}{dy}\frac{\partial}{\partial x'}\sqrt{\frac{x' ^2 +1}{-2gy}} - \frac{\partial}{\partial x}\sqrt{\frac{x' ^2 +1}{-2gy}} = 0 (second term does not contribute as it does not have x explicitly;
\frac{d}{dy} \frac{x'}{\sqrt{-2gy}\sqrt{x' ^2 +1}} = 0
Ck.mitra (talk) 05:33, 29 May 2008 (UTC)
replies: text: Nice proof, check my last link on parametric on y where dt/dy=0 @ x1,y1. fairly easy to compute s(y) -Alok 07:53, 19 April 2011 (UTC)

text: Probably better to only have published solutions as there are quite a few. My favourite graphical method is using Wren's involute which also shows the tautochrone property nicely. Dmcq (talk) 16:43, 19 April 2011 (UTC)
replies: text: You dont get it do you, check out profs in India who would flunk a kid for giving any other solution even if he may be right.Because Wiki says so -Alok 09:20, 20 April 2011 (UTC)
replies: text: I'm not certain what you mean. What I'm saying is that people should not give their own proofs in Wikipedia, they should be based on ones in published sources. Wiki should not be 'saying so' it should be summarizing what other people say. Citations should be given for proofs. Dmcq (talk) 11:06, 20 April 2011 (UTC)

text: Yes but if an individual posts his own proof which is mathematically accurate, is he wrong in doing so? All I am saying is that considering the vast audience wiki enjoys, most people would disregard something as wrong because it is not on the wiki, though the alternative is right. At least if it is accurate let one have it as an alternate in the discussion section-Alok 11:28, 20 April 2011 (UTC)
replies: text: Wikipedia is not a vehicle for new thought, see WP:OR. It is not the place for new proofs. Dmcq (talk) 11:32, 20 April 2011 (UTC)

text:  WP-OR talks about articles not about discussion. -Alok 11:36, 20 April 2011 (UTC)
replies: text: The talk page is for discussing improvements to the article, see WP:TALK, so I assumed you were talking about the article. There's no great harm in a bit of other discussion but it isn't a general forum. Dmcq (talk) 12:38, 20 April 2011 (UTC)
replies: text: The article would benefit from a proof like the one above (properly sourced of course).  The proof which currently appears in the article uses Fermat's principle and Snell's Law, which are descriptions of how light travels through a medium. It's an elegant demonstration that the two are self-consistent, but a  purely mathematical problem should not need to invoke physical laws.  It might even be a case of circular logic to do so, as one might just as well use Fermat's principle to predict Snell's law.  The proof section should use just the calculus of variations. Spiel496 (talk) 18:51, 20 April 2011 (UTC)
replies: text: There's no circularity involved and the proof is perfectly okay. There's no need to obscure the business, the subject is a fairly popular one. One could have a variational proof as well but certainly not instead of one that can be read by someone who hasn't university mathematics. Dmcq (talk) 19:44, 20 April 2011 (UTC)

text: Nothing wrong in posting one's views in the discussion forum is there? To a lot of us, when we 1st approach the problem statement, it is not obvious that Snell's law or Fermat's principal is needed at all, so I do like a proof which is purely mathematical and works on maxima minima approach rather than Fermat's principal. On the other had a open book approach is better if one has to dig up possible solutions to the differential equations just obtained.-Alok 04:13, 21 April 2011 (UTC)
replies: text: The proof using Fermat's principle is a notable one which is a much more important reason for inclusion in Wikipedia. There is no general need for proofs in Wikipedia, it isn't a textbook. There is actually a guideline WP:TALK against posting things here which aren't aimed at improving the article but as I said before it's fine if it doesn't go on too long or looks like something in it might lead somewhere in relation to the article. Dmcq (talk) 08:21, 21 April 2011 (UTC)

text: Thanks for your comments, the question if stated as "find the equation of the curve so that ....." would lead to a more accurate mathematical solution is my point, hence I mentioned parametric on y, or as stated above, simply matching the arc length of the cycloid to the form of the equation above. Hope it clarifies my confusion and why I agree with Spiel496's comment. Why would someone trying to find a curve that fits into the given question look intutively at Snell/Fermat's law? </br>-Alok 06:09, 22 April 2011 (UTC)
replies: text: I guess they used Fermat's principle because that's what they were familiar with. The calculus of variations wasn't discovered till quite later. A properly sourced such proof would probably add to the article and I've no objections to that. All I'm trying to point out is the principles of Wikipedia as shown in WP:5P as the talk seems to be about turning the article into a section of a textbook with a single correct proof rather than on having an article that summarizes the topic as described in sources. Dmcq (talk) 08:10, 22 April 2011 (UTC)

text:  I have not seen any reference to Fermat's principal being the sacrosanct reference to a particle in gravity etc during any phase in history.. It was more of a "if i put a block of glass light bends like this" we have done those experiments in school. There is a debate if the velocity of light is a constant or is varying with wavelength / group velocity/phase velocity but again , I think it is a circular reference as mentioned by Spiel496.. -Alok 10:12, 26 April 2011 (UTC)
replies: text: I don't quite understand what you are saying. The main point though is that it is how it was done a long time ago before the calculus of variations was discovered, the proof is famous and can be cited. Any other proofs should work to the basic standards of WP:5P, they should also be reasonably famous and have a citation. Anyway I see the proof there has noi explicit citation so I'll put in one as it was Johann Bernoulli who gave that particular solution. Dmcq (talk) 10:26, 26 April 2011 (UTC)
replies: text: Now that I understand the historical context, I agree that the Fermat-Snell approach fits well into the article.  When I was taught about this problem in school, the professor used the calculus of variations, so that gave me a bias.  I doubt I would have objected if the section had been labeled "derivation" rather than "proof".  That is a minor distinction, but when I see "proof" I expect a series of steps requiring little or no prior knowledge.Spiel496 (talk) 20:24, 26 April 2011 (UTC)
replies: text: I've changed it to solution which I think is about the same but people without much maths might understand better than derivation. Dmcq (talk) 13:20, 27 April 2011 (UTC)
thread_title:  Alternate Proof 

message: Is certainly encyclopedic.
Why?
Because it shows that the story is accessible to an audience of millions!
--M a s (talk) 13:16, 19 June 2008 (UTC)
thread_title:  The Spidey reference 

message: When the word "cusp" is substituted for "point", as User:Anwar_saadat has done several times, I can no longer understand what the article is saying.  Even Fermat's Principle was reworded towards this end:  The actual path between two cusps taken by a beam of light is the one which is traversed in the least time.  The phrase "path between two points" is about a million times more common.  I'm not convinced the cusp jargon is even correct, let alone accessible.  I propose we use the word "point" to explain Brachistochrone problem and then maybe state afterwards that the important points on the curve are really cusps.  Spiel496 (talk) 14:06, 11 July 2008 (UTC)
thread_title:  Cusps 

message: I suggest adding an animated gif to show the meaning of "fastest decent". The motion of the mass along the cycloid can be compared to motion along an inclined plane from position A to B (or any other arbitrary curve), so that readers can see that the mass along the cycloid reaches B earlier than the mass along the inclined plane.
Antony css (talk) 07:49, 14 February 2009 (UTC)
thread_title: Visualization?

message: One wonders about the use of a single constant in Johann Bernoulli's original proof that the curve of least time is a cycloid. Was he deliberately being cryptic and is the proof an example of hermeticism in Science or was it the sort of mistake that someone accustomed to working with line segments would make? One must take into consideration that first reports are often contain errors but then again one has to consider the source. Bernoulli seems to have been asking for help in the solution of this problem. But he could also have been directing the attention of a select few or it could have been the action of someone with divided loyalties. But as published the proof seems to have been corrupted.  --Jbergquist (talk) 21:29, 15 May 2009 (UTC)
replies: text: see Hermeticism (history of science)  --Jbergquist (talk) 23:17, 15 May 2009 (UTC)
thread_title:  Hermeticism in the Sciences? 

message: The derivation of the condition for least time by Jakob Bernoulli was somewhat gnarly and there seem to be some omissions which may have been intentional. The first few proportions stating a law of motion relate the change of distance with time and there is the unstated assumption that speed along all paths is the same including the speed associated with the 2nd differentials. By comparing the neighboring path with the path of least time he is able to equate the 2nd differentials in time. He doesn't explicitely state the condition for least time but directly exploits a law of motion relating the velocity to the square root of the distance fallen. One can find a complete version of Jakob Bernoulli's solution of the Brachistochrone problem in Die Streitschriften von Jacob und Johann Bernoulli : Variationsrechnung, edited by Herman H.Goldstine, Birkhäuser, 1991. One gets the impression that neither Bernoulli is being completely open. It's conceivable that changes were made during the conversions to Latin as a matter of style. A possible connection to Hermeticism is that Johann makes a reference to Apollo in the presentation of the problem. Hermes was the Greek god governing boundaries and it would not have been inconsistent to have things appear to have been done in haste. It should also be remembered that this was done during the Enlightenment Period and there still may have been some resentment over the treatment of Galileo by the Inquisition.  --Jbergquist (talk) 00:07, 20 May 2009 (UTC)
replies: text: The reference to Apollo is, "...thus we shall crown, honor, and extol, publicly and privately, in letter and by word of mouth the perspicacity of our great Apollo." See A Source Book in Mathematics, David Eugene Smith, Dover, 1959, p. 647.  --Jbergquist (talk) 16:44, 20 May 2009 (UTC)
thread_title:  On the proof by Jakob Bernoulli 

message: It is pretty evident from the quote from Galileo he was trying to solve the problem of the quickest descent from a point to a wall - and thought the answer was a quarter circle. He proved another thing altogether but the commentary in the article seems to say now that he was only trying to prove that bit of his result that was correct. He also said something about generalizing to smaller arcs that is also wrong which is in the second bit of the quote but now it just sits there looking silly. Is there a reason for this change or is it just a mistake? Dmcq (talk) 14:54, 29 May 2009 (UTC)
replies: text: The last quote is I suppose a get out of jail free card though I'm not sure where the bit about sophistry came from Dmcq (talk) 22:59, 29 May 2009 (UTC)

text: I removed the bit about sophistry. Galileo compares the time of travel for chords with their arc. He does not consider a complete set of admissible curves so it is a bit of a stretch to say that the arc is the actual "path of least time." In a way it is misleading but I think that is ok in a dialogue. Re Latin "lationem": it seems to be related to the noun  lati·o -onis and latus the past participle of the verb fero (which indicates motion).  --Jbergquist (talk) 03:14, 30 May 2009 (UTC)
replies: text: And I put in a bit about the actual solution plus that Galileo had worked on the cycloid. Must have a look at that article its history section is very short and there's much more to it than that. Dmcq (talk) 09:30, 30 May 2009 (UTC)
replies: text: Galileo's argument is similar to Zeno's paradox about the race between Achilles and the tortoise. Achilles can't win. It is difficult to say how much Galileo knew about the path of least time. There is no doubt that he is acting under constraints. But we can say that he is willing to admit error and acknowleges that constraints may be best.  --Jbergquist (talk) 19:15, 30 May 2009 (UTC)
thread_title:  Galileo's scholeium problem 

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thread_title:  External links modified 

message: [NOTE:the sphere in the perpendiculerly cornered bend appears to be accelerating due to the chamfered end. Perhaps we should edit the .GIF to show a totally _|_ (perpendicular) end's motion.]
This comment was included on top of the article by some unknown users: Moving it to talk page --JPF (talk) 07:24, 11 July 2018 (UTC)
thread_title:  Isolated comment on article 

message: In Johann's proof -> direct method -> analytic proof... there is the equation \text{speed} \propto \sqrt{CG} = \sqrt{x}. Shouldn't it be \mathrm{speed} \propto \sqrt{CG} \propto \sqrt{x} instead? The latter should hold because \angle NKn is being held fixed, so that \frac{CG}{x} = \sin\theta, \theta=\angle ANC being another angle that is being held fixed. — Preceding unsigned comment added by 186.215.54.146 (talk) 03:36, 20 August 2018 (UTC)
thread_title:  Regarding Johann's direct analytic proof 

message: The term brachistochrone is also used (misused?) to refer to spacecraft trajectories in which a constant acceleration is maintained. For example, if a fictional spaceship flies to a distant planet by accelerating at 1-G to the halfway point of the journey, then flipping around and decelerating at 1-G to arrive at rest at the destination, it is said to follow a brachistocrhone trajectory. See http://www.projectrho.com/public_html/rocket/torchships.php#brachistochrone for more detail.
60sRefugee (talk) 00:16, 12 February 2019 (UTC)
thread_title:  Brachistochrone with reference to spaceflight