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The Brachistochrone, with Steven Strogatz

10243 ratings | 445020 views
Steven Strogatz and I talk about a famous historical math problem, a clever solution, and a modern twist. ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that). If you are new to this channel and want to see more, a good place to start is this playlist: https://www.youtube.com/playlist?list=PLZHQObOWTQDPHP40bzkb0TKLRPwQGAoC- Various social media stuffs: Patreon: https://www.patreon.com/3blue1brown Twitter: https://twitter.com/3Blue1Brown Facebook: https://www.facebook.com/3blue1brown/ Reddit: https://www.reddit.com/r/3Blue1Brown
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Text Comments (830)
SquidofCubes (13 hours ago)
Is it because the laplacian has to be zero, and hence there can't be any second derivatives -> t(theta) must be linear?
Christopher Smith (5 days ago)
But what are the curves' properties with respect to time when its positioned inverse to gravity? If you turn the curve upside down how do objects act when progressing downward from the center to the edges? Is it also the fastest path? Slowest? Does it have increased structural loading properties due to its unique connection to gravitational forces?
Paul Ebert (6 days ago)
I might be wrong, but I think you missed something at 13:06. The first term in x(t) shouldnt be R*t, should it? You want to make x(t) an length, not length*time 🤔 Maybe R*omega*t would be correct. Nevertheless I absolutly enjoy your Videos. Your great! :)
Saf Ingram (8 days ago)
"Note the displeasure." 😂😂😂
Peter Peterson (8 days ago)
But can you turn a shpere inside out?
James Torres (7 days ago)
I like this curve and this video. Very good job. I am no mathematician but the tautochrone is most interesting because it alludes to a temporal dynamics. It is not easy to see Time only as the parameter here and space or rather location in space the variable only. Could we not rotate both the space time plane with the time theta into an imaginary or other time dimensions? 2 or more times with variation between? Dimensions of curved time. I see it would imply temporal recurrence . Please forgive my ignorance.
imshippyupup (9 days ago)
Lawrence Calablaster (9 days ago)
Penn State represent! #WeAre
John Quill Taylor (10 days ago)
How might this relate to the way the water in every stream, while it always takes the quickest path, yet no two rivers ever seem to follow a similarly-shaped path, and none are straight?
Mitch Roper (10 days ago)
I am NOT a mathematician, so this is probably going to sound stupid, but isn't the relationship you spoke of at the end represented as a straight line because of the constant called you explained earlier?
George Steele (11 days ago)
It has been a while since someone commented. So here's a challenge: How would you prove that blue-pi reaches the midpoint of the curve in the same time t regardless of where he starts? How could you compute t if the diameter of the circle tracing the curve was 1.
ed for ed (20 days ago)
But it is not working with all objects
ilia gildin (21 days ago)
I think because our gravity constant is constant to G=10 plus potential energy and that is the slope Else if we would live on planet with two suns we would get a parabula
But what is the optimal ratio of the radius of the circle the for the cycloid to the distance of AB? 2pi?
7:11 amazing eyes
Sayantan Pramanik (1 month ago)
That angle analogy is what I had used to solve the problem 😅
Omg I remember this! The curve of fastest descent! I remember how we tried to find its equation I college 8) There was so much pain...
Edward Lau (1 month ago)
You got me thinking about another case: So far we're talking about brachistochrone with a homogeneous gravitational vector field. What would the brachistochrones look like under any arbitrary gravitational vector fields? And can we ever associate the vectors with the brachistochrones, like the vectors put weight literally on a tight string to make a brachistochrone?
Wai Shing Tseung (1 month ago)
w00td00t (1 month ago)
Rewatching this, it occured to me that the waves of an earthquake move along curves as well. They dip into the earth before they surface. I have a strong suspicion that these curves might be brachistochrones as well, since the waves move faster the deeper they go.
markdelej (1 month ago)
Does this only work when point B is not somewhere nearly under point A? If point B is kinda below A and only a tine bit to the side then a brachistochrone wont work. How do you know how far to the side B has to be before a brachistochrone will work? Edit: or maybe a brachistochrone with a huge radius would work? Also how do you know what radius the circle used should be to give the correct brachistochrone? Do you use the full semi circle of a smaller radius circle i do you get a bigger circle and not use all of the semi circle?
Guilherme Gondin (1 month ago)
Did someone looked for a solution for his challenge using the tautochrone curve somehow? It seems pretty logical for me, but I lack the skills needed to try it myself.
henrique bonafe (2 months ago)
Title in portuguese, why?
Aapeli Syrén (2 months ago)
the formula for a brachistochrone is e^(i*theta)+theta it's because e^(i*theta) in the complex plane corresponds to rotation and the additional theta moves the axis forward as the angle changes.
Cran Cowan (2 months ago)
Another way to show the linearity: Imagine an "old time" clock with hands.... Put a pen on the second hand so that it would draw a circle in 60 seconds. Now place the clock under a 'strip chart recorder' whose width is the diameter of the clock. The strip chart moves at the same speed as the tangential speed of the second hand at 12 o-clock (i.e. the length of the second hand times the angular rate of change of the second hand). So in 30 seconds, the clock's hand/pen moves through an angle of pi and goes from one point on top of the strip chart to another point on the bottom. The angle changes linearly with time even though (because of the centripetal acceleration of the pen) the pen draws a curved line (brachistochrone) on the strip chart. The brachistochrone "maps" an accelerating reference frame (the pen) onto an inertial reference frame (the strip chart). But if you put yourself in the frame of reference of the pen, the brachistochrone maps the fastest way through time onto a (relatively) accelerating reference frame through space (the strip chart).
Eric Sageser (2 months ago)
Skiing is the fastest way to slide a brachistochrone
Spencer Taylor (2 months ago)
Newton proves that he is the greatest once again. He is untouchable
TheGiantcube (2 months ago)
The proof is trivial! Just view the problem as a semi-decidable hypergraph whose elements are connected 4-forms
Donald Wallace (2 months ago)
I am by no means a mathematician and my understanding of physics is rudimentary, but when you started talking about expressing the problem in term of time over theta, space time diagrams kept popping in my head. every curve or motion is actually a straight line in space time. I don't know maybe i am, reaching but i feel like there is a connection here.
Jax Blonk (2 months ago)
I'm troubled, because the challenge seems actually really straightforward in solving just on insight, but I'm a depressive, so at the same time, I don't really think anybody would care if I did just write down what was in my head, at this point. It's probably been done by now, I'm a nobody, and that's a good bit of work for something unlikely to be actually desired anymore, but, uh, I mean, I'm all ears if I'm just being mopey; I could give it a shot, if it matters to anyone.
Chaînons Manquants (2 months ago)
Thank you !
Kenji Gunawan (2 months ago)
Brachistochrome or Brachistochrone?
Fake Story (3 months ago)
lion pic reference how is this even...wtf
Razvan Tepeneu (3 months ago)
Yunmin Hwang (3 months ago)
can anyone please explain to me why we get a pair of similar triangles?
Shrimples 362002 (3 months ago)
An answer to your question: the angle of the baristochrone represents where you are on it, and thus it shows ur distance on the curve. The graph can thus be changed to distance in function with time, where every point on the line created represents a point on the baristochrone relative to the starting point A where time is 0 and distance is a int x, and the ending point b. When observing the graph, we see that as distance and time both increase linearly (and thus speed is constant), the line created (which is the distance traveled on the baristochrone) increases linearly as well. This means that this graph of distance in funtion of time effectively turns the problem into “what is the shortest distance between A and B, when speed is constant” the answer is obviously a linear line.
freakazoid115 (10 days ago)
What's the ratio of the length of the distance of a ball falling straight down and hitting the ball rolling, at the middle of an isochronous curve? And then, what's the ratio of acceleration in regards to "gravity" 😂
Younes Layachi (3 months ago)
I figured all the points in the rolling circle move during the same period of time, so the one that moves the longest distance describes the most efficient path 😋
Nasim Uddin (3 months ago)
float video
Thomas Thurman (4 months ago)
Conservation of angular momentum would be a correct answer.... whether that’s the answer you’re looking for is a different point....
evan pham (4 months ago)
Does it have something to do with the fact that function minimums occur when derivatives of the function are zero? The second derivative of the linear t-theta space functions would be zero, so maybe these functions are some direct manipulation of the time- minimization curves. This manipulation would have to account for the additional derivative somehow. Just some random and probably incorrect thoughts.
James Wilson (4 months ago)
Wonderful, awesome video! Although a little more math would've improved it even more.
s6th (4 months ago)
The engineers have a solution to your brachistochrome problem. They say "jet turbines".
peshu toshuv (4 months ago)
Uhmm I am a bit lost here :D Can someone please explain to me if the velocity of the point moving along the brachistochrone is a constant, or is it changing along the way (assuming only gravity affects the object)?
Yun Wang (4 months ago)
Solution to the challenge: Consider an infinitesimally small time interval dt, when the velocity of the pi-creature is v at angle theta with the vertical line. In this time interval, the pi-creature descends a height of dy = v cos(theta) dt (1). By conservation of kinetic energy, v^2 = 2gy (y being the total height the pi-creature has descended). By Snell's law, sin(theta) = Cv (C is a constant). Take the derivative of both equations above: 2v dv = 2g dy (2) cos(theta) d(theta) = Cdv (3) Putting (1) (2) (3) together, we have d(theta) = Cg dt, so the theta-t function is a straight line :D
Michael Burt (4 months ago)
My attempt at an intuitive answer to why theta(t) = k: The gravitational field is essentially constant (and downward) close to the surface of the earth. Imagine, in the void of free space, this vertical field and then take a large sphere and give it a nudge along an axis perpendicular to the direction of the field. On this axis, in the absence of any other fields or forces, a straight line path will be the shortest distance between any two points (measured on this axis) as well as the shortest time. Since this observed phenomenon is true for the *large* sphere we must deduce as an a posteriori assumption that it is true for all matter *inside* the sphere as well. If any atom or chunk inside failed to meet this requirement (that it was on the path of least *time*), then it would fail for the object as a whole as well. In the absence of any of the information about forces in the perpendicular direction then the large sphere must move in a straight line from A to B, at a constant speed (this ignores the degenerate case where it is given no initial energy and does not move). Given the geometry of a sphere, the angle made by any surface point as a function of time will be constant.
StrangaDanga (5 months ago)
I’ve heard that another property of this curve is that when you use it to connect points a and b, and point on the curve between a and b, when rolled towards b, with arrive at point b at the same time. Source: Vsause
federico saviano (5 months ago)
7:11 Girl I,loose myself up,in those e-e-e-eyes
forestsoceansmusic (5 months ago)
Nature is imbued with this property of doing the most efficient thing, as Bernoulli's solution shows. There was some other new technology I saw recently in which they realised the naturally-occurring way was the most efficient....I remember, it was the best way to merge (and therefore greatly amplify) several laser beams into one, was to focus them all on some jelly-fish protein. It was a Dnews YouTube clip hosted by Trace (who hates jellyfish).
forestsoceansmusic (5 months ago)
Vsauce said the problem was first addressed by Galileo, and that Bernoulli later improved on Galileo's answer.
M O (6 months ago)
Lol, actually, yes this "is" a shortest path problem. It's just not a shortest path in space but shortest in time. If any sort of shortest path is asked, you could start by drawing a straight line. The axis of time is already given by the question itself, so the next question is: what makes sense as the other axis? Well we need theta there to be able to construct a curve. Then there are plenty of ways from there to find the equation of the curve in the x,y-plane. This was my first intuition because I've learnt to often swap time and space. They're both just number-lines. As humans we tend to treat things (or each other) differently once we categorize them differently. What I find beautiful about math is that some of the most elegant solutions just treat numbers for what they are treating them equally.
Matthew Coble (6 months ago)
I took differential equations with Prof. Strogatz so that I could not take more statistics. One of the better classes I took and one of the more memorial!
Rodrigo Resendes (6 months ago)
great video
happily confused dog (6 months ago)
Its a straight line because the fastest way to get to point a to b in this case is actually a straight line. Since you want to maximize the angle theta with respect to time as this translates to the circle going across the surface the fastest.
frankie (6 months ago)
12:00 Illuminati confirmed!
Noopi Grewal (6 months ago)
Saw your video yesterday, really great :) *Here is the answer to your challenge*: particle moves such that speed v is proportional to sin(theta). Differentiate it: rate of change of change of speed is proportional to cos(theta)* rate of change of theta. But rate of change of speed is component of gravity along the curve = g cos(theta). Hence *rate of change of theta = constant*.
Elie Simsch (6 months ago)
I’m not sure about this, but about your challenge I feel like there’s some comparison to energy being both 1/2mv^2 and mgh, where the former compares to the acceleration from gravity on the particle, and the latter compares to the constant rolling speed.
Reidar Wasenius (6 months ago)
GREAT. Thank you!!
Michael Kasprzak (6 months ago)
To bad I already watched the Vsauce video
Patrick T. (7 months ago)
Hints: 1. The complex numbers. 2. Pi.
Rio Lane (7 months ago)
"Time" is constant
Rio Lane (7 months ago)
"Time" is constant
Andrea Bracco (7 months ago)
Because cycloidal path in a gravitational scope makes an harmonic
Mudit Sinha (7 months ago)
Just like you can have multiple circles passing between two points, I'm curious if the same argument stands for Brachistochrones. Can I have multiple Brachistochrone curves between two fixed points? If yes, then which among those will be the fastest?
Luke Dope (7 months ago)
How to graph it?
Sean Prudhoe (7 months ago)
Mark Levi was one of my professors at Penn State, he is full of nice physical intuitions that help solve math problems.
IJ Films (7 months ago)
uploaded in april 1st...
YouTube Administration (3 months ago)
April fools this entire video is fake (not really)
Zes (7 months ago)
not lovex tox, nonerx, nst as storix or up/down or not, say anyx. up can b inferiox
Nyamaa Bayarma (7 months ago)
the next 15 minutes of your life... as if you've got hours before you die
Chan Myae Naing (7 months ago)
Vsauce brought me here!
Daewoon Kim (7 months ago)
the rolling pi creature made me laugh
Bruno Signori (8 months ago)
It means the rate of change of the accelaration is constant
Oscar Patterson (8 months ago)
I wonder if spherical surface mapping could shed any light?
Alan Armando (8 months ago)
My favorite curves: cicloids! 😍
Ricky Pazzi (8 months ago)
Nature is crazy: light minimizes the time it takes to get from point A to point B even though time isn't flowing for photons :)
KazzArie (8 months ago)
VSauce brought me here but I recognized the name from a DE class I had 😁 awesome
Ben Thurston (8 months ago)
I think its interesting this is such a simple curve parametrically but theres not even a closed form for writing it as y=f(x) to get the points on it
Arturas Karbocius (8 months ago)
What is shortest time between one meter height y=1 and one meter long x=1 distance curve quarter of cycle or 45 degree straight slope exponent of 1? or maybe other exponent k ? it took 5 months to calculate and Newton solved in one day brachistochrone curve, very doubtful :/
jesse mckeown (8 months ago)
Snell's Law in Quantum Mechanics comes from conservation of horizontal momentum (which holds because a collection of parallel transparent media is translation-symmetric); what this *really* means for the brachistochrone, I don't know, but there's something about it echoing in that constant-speed wheel.
Mira R (8 months ago)
Maybe this isn’t a very mathematic way of putting it but my answer tot he challenge is: since the curve is the fastest way to go from A to B, it must have a constant speed. If it didn’t, there would be another way to reshape the curve that would even out the speed to make a faster way from A to B.
Tycho Neve (8 months ago)
Thank you so much for these videos, I lost a night of sleep when I discovered them. Can I use one of the diagrams in this videos in an educative Master's thesis about (general) plane curves? If that's no problem, I will 'force' my audience to subscribe to your channel of course!
Finn Ameer (9 months ago)
Question: if there are 30 students and the teacher gives the 1.student back 30 test results, Then student n.1 distributes them until he finds his own test result.then he gives the remaining tests to the next student who doesn't has his result jet. This continues until everybody has there result. How often are the remaining results passed on to the next student, on average?
Joshua Saffy (9 months ago)
Nothing on the Euler-Lagrange?
Alex Frasca (9 months ago)
which path is slowest?
Vanessa Kitty (9 months ago)
A nice paper assisting in answering the question presented may be found here, https://www.google.com/url?sa=t&source=web&rct=j&url=https://arxiv.org/pdf/1401.2660&ved=0ahUKEwjro6GGk4nYAhWo44MKHRZGD4UQFgggMAA&usg=AOvVaw3F3Vn7Czn8whSGNgsCykyf . This is a paper titled, "The straight line, the catenary, the brachistochrone, the circle, and Fermat", by Raul Rojas Freie Universit¨at Berlin January 2014. The conclusion is awesome.
ALPHA_sh (9 months ago)
april first much?
Femi Oyekan (9 months ago)
"Newton stayed up all night....solved it..."
khalid Mukhtar (9 months ago)
Congratulations for the great video.. but I didn't get the question at the end; and since my reply is to a 19 months old question.. mm, let's see: - Since "Y (to the power-2) is constant to T", I don't find it difficult to imagine that it won't take a straight line shape; given T an axis on that diagram.. we are basically converting an exponential graph to logarithmic.
Austin Mackey (9 months ago)
Why do people thumbs down videos like this? Does it hurt their brain and make them angry, I wonder?
Sam (9 months ago)
The idea that nature looks at all possible paths and chooses the best one is surprisingly close to Feynman's model of quantum field theory, which was a crucial missing piece in what is now the most well proved theory in history. On that theory, light (and indeed all particles) do in fact take every possible path. But the paths that are close to the optimal path resonate with each other so much that it makes it nearly guaranteed that if measured, light will be observed along that optimal path while the other paths destructively interfere with each other making it nearly zero chance that you will ever see light moving in some odd curve or other non-straight path.
welcomeblack (9 months ago)
Nobody ever shows the brachistochrone with an endpoint that goes up (or down). This makes me sad, since it creates a lot of false intuition
Yan Yeomen (9 months ago)
Are you up to make a video about solving brachistochrone problem using Euler-Lagrange equation?
Scott Smith (9 months ago)
I thoroughly enjoyed this regardless of the fact that i had no clue what was going on 90% of the time.
Gunslinger 256 (9 months ago)
Let me get this right. You want a real world analogy of how to follow the path of a brachistochrone without changing direction? Isn't that lightspeed?
thebullybuffalo (10 months ago)
centripedal acceleration?
Ayush Aryan (10 months ago)
The relation at 7:30 ,sir how can u treat as a particle
Iustin Raznic (10 months ago)
My favourite piece of math, realy realy neat
Robin Falk (11 months ago)
Miles Watson (11 months ago)
Make. A. Podcast.
3Blue1Brown (11 months ago)
+Miles Watson http://www.benbenandblue.com
J.J. Shank (11 months ago)
Is it just me, or is your voice really muffled?
C. Weis (11 months ago)
I just wanted to tell 3Blue1Brown I hated math in school and just did not get it. Your awesome visuals with explanations has given me an entirely new outlook on math!

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