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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)
The aim is to minimise the integral .
Starting with v=0 at y=0 (starting from rest): (negative sign because y is down)
(we can use either one), therefore we minimise
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 (second term does not contribute as it does not have x explicitly;
Ck.mitra ( talk) 05:33, 29 May 2008 (UTC)
Rearranging gives
Letting k=2gc:
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)
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)
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)
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 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 [ [1]] 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 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)
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)
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)
here's a link to it http://www.script-o-rama.com/movie_scripts/s/spider-man-2-script-transcript.html
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
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)
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)
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)
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)
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)
Crazy2be ( talk) 05:57, 19 February 2017 (UTC)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.
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.
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[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)
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In Johann's proof -> direct method -> analytic proof... there is the equation . Shouldn't it be instead? The latter should hold because is being held fixed, so that , 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)
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)
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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)
The aim is to minimise the integral .
Starting with v=0 at y=0 (starting from rest): (negative sign because y is down)
(we can use either one), therefore we minimise
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 (second term does not contribute as it does not have x explicitly;
Ck.mitra ( talk) 05:33, 29 May 2008 (UTC)
Rearranging gives
Letting k=2gc:
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)
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)
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)
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 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 [ [1]] 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 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)
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)
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)
here's a link to it http://www.script-o-rama.com/movie_scripts/s/spider-man-2-script-transcript.html
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
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)
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)
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)
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)
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)
Crazy2be ( talk) 05:57, 19 February 2017 (UTC)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.
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.
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[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)
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In Johann's proof -> direct method -> analytic proof... there is the equation . Shouldn't it be instead? The latter should hold because is being held fixed, so that , 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)
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)