Talk:Brachistochrone curve

History section
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)


 * 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)
 * 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)

Alternate Proof
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) $$\frac{x'}{\sqrt{-2gy}\sqrt{x' ^2 +1}} = c$$

Rearranging gives $$x' =\sqrt{\frac{-2gcy}{1+2gcy}}$$

Letting k=2gc: $$y' ^2 = -\frac{1+ky}{ky} = -\frac{k^{-1}+y}{y} = -\frac{D+y}{y}$$


 * 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) — Preceding unsigned comment added by Alokdube (talk • contribs)


 * 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)


 * 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) — Preceding unsigned comment added by Alokdube (talk • contribs)
 * 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)


 * 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) — Preceding unsigned comment added by Alokdube (talk • contribs)


 * 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)


 * WP-OR talks about articles not about discussion. -Alok 11:36, 20 April 2011 (UTC) — Preceding unsigned comment added by Alokdube (talk • contribs)


 * 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)


 * 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)


 * 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)


 * 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) — Preceding unsigned comment added by Alokdube (talk • contribs)
 * 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)


 * 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? -Alok 06:09, 22 April 2011 (UTC) — Preceding unsigned comment added by Alokdube (talk • contribs)


 * 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)


 * 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) — Preceding unsigned comment added by Alokdube (talk • contribs)


 * 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)


 * 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)


 * 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)

I still think parametric on y is more intuitive and comes straight from calculus http://mathworld.wolfram.com/BrachistochroneProblem.html -Alok 06:52, 10 April 2013 (UTC)

Since this article mentions that the problem can be solved using the calculus of variations, I am just going to add a section on that. Nerd271 (talk) 14:26, 30 July 2016 (UTC)

The Spidey reference
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)

Cusps
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)

Galileo Scholium problem
The part about Galileo trying to solve this problem has been removed. The edit comment refers to a paper saying that one should distinguish between Galileo's scholium problem. Unfortunately I have found only one relevant reference via google to this [http://www.iop.org/EJ/abstract/0143-0807/20/5/301 Johann Bernoulli's brachistochrone solution using Fermat's principle of least time Herman Erlichson 1999 Eur. J. Phys. 20 299-304  doi: 10.1088/0143-0807/20/5/301] and I have no access to this to see what it is about. Can anybody provide an explanation of what the difference is? It sounds like it is notable as there are lots of articles saying he tried to tackle the brachistochrone problem. Dmcq (talk) 11:30, 3 December 2008 (UTC)

For example [] describes something that Galileo did that sounds very like trying to solve the Brachistochrone problem to me, what exactly is wrong with that description? Dmcq (talk) 11:35, 3 December 2008 (UTC)


 * I do have access to the paper. Here are some quotes from it:
 * It is important to distinguish between Galileo’s Scholium problem and the similar, but very different, Bernoulli brachistochrone problem. Figure 1 shows Galileo’s diagram for his Scholium problem. In this problem one is limited to a single plane, or sequence of connected planes which have their end points on a quarter circle, or on an arc no greater than a quadrant of the circle. One seeks the sequence of planes which will yield the minimum time for a particle to slide frictionlessly from the upper point on the circle down to the bottom of the circle. It is assumed that the transition from one inclined plane to another occurs smoothly and with no loss of time.
 * In Bernoulli’s brachistochrone problem one has two points at different elevations and one seeks the minimum-time curve for a particle to slide frictionlessly from the higher point to the lower point. The well known answer to the Bernoulli problem is the unique cycloid extending from the higher point to the lower point. The less well known answer to the Galileo problem is the infinite sequence of planes extending from the starting point to the bottom of the circle, i.e. the circle arc itself extending from the upper point to the bottom of the circle.
 * […] Galileo’s problem included the ‘circular constraint’, i.e. the constraint that the end-points of the inclined planes be on the vertical circle, so that a cycloidal arc was not a possible solution to Galileo’s minimum time to the bottom problem.
 * […] Among prominent writers who have made this misidentification [of Galileo's problem and Bernoulli's problem], we mention Herman Goldstine.
 * -- Jitse Niesen (talk) 17:01, 3 December 2008 (UTC)


 * Actually having read what Galileo wrote and the explanation above I believe now that there are three mistakes!
 * Galileo wasn't originally trying to solve the brachistochrone problem but a related one, what is the fastest path between a point and a vertical line. This was solved later by one of the Bernoulli's and shown to be the half-cycloid starting at the point and ending going horizontally through the vertical line.


 * Galileo was trying to solve a fastest path problem but made a mistake and thought he had a solution using an arc of a circle.
 * Galileo then thought he had a solution to the proper brachistochrone problem as he said Discourses regarding two new sciences (page 239)
 * From the preceding it is possible to infer that the path of quickest descent [lationem omnium velocissimam] from one point to another is not the shortest path, namely, a straight line, but the arc of a circle


 * The quote above refers only to the circle Galileo thought was a solution, that doesn't mean he wasn't trying to solve the problem or didn't think he had a solution to the problem. The circle wasn't an explicit constraint.
 * I think probably this article should have a reference to Galileo's problem and both the difference in his original problem and that he made a couple of mistakes. Dmcq (talk) 21:06, 3 December 2008 (UTC)

Visualization?
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)

Spidey again
JN, if you need a ref that Spiderman 2 references the brachistochrone problem that can be provided (imdb references it...)

If you consider it trivia, I refer to the comment above - the story is accessible to an audience of millions.

Can we talk here...?

Thanks, --M a s (talk) 04:36, 28 February 2009 (UTC)


 * Please see WP:TRIVIA and WP:Handling Trivia. Spidey's reference to the problem is not integrated into the article in any way, it was just stuck into the history section. If a reference was to be put in the best one could do to integrate it is show it as an example of how the problem has affected popular perception or something like that. Currently it says nothing about the brachistochrone and so shouldn't be in the article. It wasn't even an important part of the film and didn't contribute materially to the plot. If no integration or justification is forthcoming very soon then I believe it should just be removed. Dmcq (talk) 11:53, 28 February 2009 (UTC)


 * Thanks, will edit as such. Originally the Spidey reference had the dialogue between Octavius and Peter Parker, which added more of the background.  It didn't contribute materially to the plot, but did contribute materially to the characterization of Parker / Octavius (erudite kid / mad scientist)... The filmmakers didn't put throw it onto the cutting-room floor so they had their reasons for leaving it in. --M a s (talk) 03:53, 1 March 2009 (UTC)


 * The contribution could have been anything, it was random and used nothing specific about the problem. Saying it was an example of persistence doesn't say anything about the brachistochrone. If you could find it used meaningfully in Num3ers or it was a major part of the plot or somebody did something different because of it and not something else - that would be relevant. Dmcq (talk) 12:35, 1 March 2009 (UTC)


 * Quote from WP:Handling Trivia


 * Note that certain kinds of information can be more or less important, depending on the context. For instance, in the South Park episode "Pink Eye," the space station Mir (which really existed) lands on Kenny McCormick (a fictional character), killing him. The overall importance of this piece of information may be hard to define, but it is certainly important to Pink Eye (South Park episode), somewhat important to Kenny McCormick, and not very important to Mir.


 * This reference wasn't even important to Spidey never mind Brachistochrone. Dmcq (talk) 12:39, 1 March 2009 (UTC)

OK, thanks for keeping the conversation here. Will not revert, as if you are different than Jitse Nelson then I'm odd-man-out. Disagree however that it wasn't important to Spiderman as mentioned above. Also, at least one popular expositor of mathematics - Paul Nahin - commented in Dr's Euler's Fabulous Formula - "look for Toby Maguire's casual reference in a Hollywood super-hero adventure flick to Bernoulli's solution to the famous problem of determining the minimum gravitational descent time curve." If MAA or other organizations references Spiderman's reference to the bracistochrone then will revert. If there were an article along the lines of Popular perceptions of mathematics then would agree that it would be an ideal discussion - along the lines of Mir as mentioned above. Best, --M a s (talk) 13:23, 2 March 2009 (UTC)


 * Popular perceptions of Mathematics might be more relevant. However I don't think it would come anywhere near notable enough even for that. I just did a google on 'spiderman Brachistochrone' and it only gave 4 pages, of those only one reference actually linked the two together, and that reference was in a blog which no longer carries the reference. As you pointed out there's some places like IMDB which carry the reference but the connection is very peripheral. It isn't on the Spiderman 2 page and I don't think it should be there either. Dmcq (talk) 14:23, 2 March 2009 (UTC)

here's a link to it http://www.script-o-rama.com/movie_scripts/s/spider-man-2-script-transcript.html

Hermeticism in the Sciences?
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) here's a link http://www.script-o-rama.com/movie_scripts/s/spider-man-2-script-transcript.html
 * see Hermeticism (history of science) --Jbergquist (talk) 23:17, 15 May 2009 (UTC)

On the proof by Jakob Bernoulli
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)


 * 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)

Galileo's scholeium problem
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)
 * 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)


 * 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)


 * 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)


 * 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)

Image
what about this image. -- Raghith 08:29, 4 October 2011 (UTC)

Yes it is a good one, in fact I thought there already was something like this here but obviously there isn't. Dmcq (talk) 08:50, 4 October 2011 (UTC)

That image is incorrect, a brachistochrone does not have an ascent at the end. At least not, when it is also supposed to be a tautochrone. 108.171.129.168 (talk) 11:58, 10 February 2017 (UTC)


 * I disagree. What if the the start and end points are at the same altitude? The path would have to go down and then up, if the particle is ever going to get there, right? Spiel496 (talk) 19:57, 11 February 2017 (UTC)

This does not appear to fit the description of being equal to the tautochone curve. the curve should never exceed the floor limit for such a curve since it wouldn't be able to gain energy to escape from the lower point. 74.214.226.120 (talk) 05:46, 13 February 2017 (UTC)
 * "it wouldn't be able to gain energy to escape from the lower point"? This isn't the tautochrone problem; it's the brachistochrone. The mass is released at a point above the end point, so it will certainly have enough energy to get there.  Spiel496 (talk) 01:29, 16 February 2017 (UTC)

I have removed this image from the article, since it is incorrect, as the above contributors noted. However, I think it would be a great addition to the article to have a correct version, so if someone wants to do that it would be much appreciated! Note that the "fastest" curve should match the Tautochrone one. Crazy2be (talk) 00:34, 14 February 2017 (UTC)


 * That was a far cry from a consensus. Two logged-in contributors think the image is okay; two IP addresses say it's wrong, just because. While I concede there is a relationship between the brachistochrone and the tautochrone, I don't see how that eliminates this image. The article says the brachistochrone curve is a cycloid. I see no evidence that this image contradicts that statement, so I'm putting it back in.   Spiel496 (talk) 01:29, 16 February 2017 (UTC)


 * My apologies, I misread one of your statements because I didn't not realize whom it was a reply to. The image messes up the indenting, so it looked like a 3:1. However, assuming the brachistochrone curve can have a lip at the end depending on the ratio x/y of A - B, then the following from the introduction is quite misleading. Going through the history it looks like it's been rephrased quite a few times, but the current incarnation certainly isn't the clearest.
 * "Incidentally, for a given starting point, the brachistochrone curve is the same as the tautochrone curve. More specifically, the solution to the brachistochrone and tautochrone problem are one and the same, the cycloid."Crazy2be (talk) 05:57, 19 February 2017 (UTC)


 * I think the tautochrone is restricted to a portion of the cycloid ending at the horizontal. That doesn't mean the brachistochrone can't include more or less of the cycloid. I think the image here is just a segment of the curve in the Cycloid article. As is the tautochrone.   Spiel496 (talk) 02:08, 20 February 2017 (UTC)


 * Ok, thanks for your thoughts and your catch of my incorrect removal. Please let me know what you think about the new lede, hopefully that should clarify things for others. Thanks! Crazy2be (talk) 02:44, 21 February 2017 (UTC)


 * I think the new lede looks great. Spiel496 (talk) 05:35, 22 February 2017 (UTC)

Johann Bernoulli's solution - Lots of problems.
This section has a lot of problems. To name a few: (in no particular order) 1. "The conservation law" ... there is only one AND the reader should know what it is? Nonsense. 2. v = √(2gy) ? out of nowhere?? why not just add the two preceding steps? y = ½gt² so t = √(2y)/√g and v = gt = g√(2y)/√g = √(2yg) For a particle in vertical free fall, where g is constant, y increases as t increases and do=0, v0=0. 3. The variable "s" pops up here with exactly no explanation nor definition. 4. "the speed of light increases following a constant vertical acceleration" it "follows" a constant acceleration of what?? follows?? Why not just say that light rays have velocity which increases as if under uniform vertical acceleration? 5. Snell's Law gives you sin(theta) = dx/ds (assuming s is path length)? And the reason that the angle which starts out at 0 changes from 0 is ...? (My optics is 40+ years old, but I thought vertical rays did not refract, regardless of the medium?? Explanation (at least) is needed here! ie explain why if dx/ds starts at 0, why will it change?) 6. Half of the section, it seems, is not about Johann's solution, but about Jakob's proof. This is a non sequitur.

External links modified
Hello fellow Wikipedians,

I have just modified 1 one external link on Brachistochrone curve. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
 * Added archive https://web.archive.org/web/20080329153035/http://curvebank.calstatela.edu:80/brach3/brach3.htm to http://curvebank.calstatela.edu/brach3/brach3.htm

When you have finished reviewing my changes, please set the checked parameter below to true or failed to let others know (documentation at ).

Cheers.— InternetArchiveBot  (Report bug) 00:57, 7 November 2016 (UTC)

External links modified
Hello fellow Wikipedians,

I have just modified one external link on Brachistochrone curve. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
 * Added archive https://web.archive.org/web/20110727210743/http://www.math.umt.edu/tmme/vol5no2and3/TMME_vol5nos2and3_a1_pp.169_184.pdf to http://www.math.umt.edu/tmme/vol5no2and3/TMME_vol5nos2and3_a1_pp.169_184.pdf

When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.

Isolated comment on article
''[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)

Cheers.— InternetArchiveBot  (Report bug) 17:46, 6 December 2017 (UTC)

Regarding Johann's direct analytic proof
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)

Hi I've changed it to correct that, and also to more closely follow Johan Bernoulli's demonstration of his method Mikerollem (talk) 16:01, 6 December 2018 (UTC)

Brachistochrone with reference to spaceflight
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)