The Brachistochrone, with Steven Strogatz 

About your challenge... I do not love to be dunned and teased by foreigners.

Do you challenge me because you think you've got an unusually clever solution, or because you think I'm Newton?

i have an excellent proof for this but this comment section is too small to contain it

I rarely ever comment on RU-vid, but I wanted to say that your content is amazing and most importantly thought provoking! Keep up the good work and thank you for the time you put into your videos!

Really neat! I wish I had this video when I took physics to learn the connections between different concepts.

Just wanna say that all your videos are beautiful, well edited and animated, insightful, and a joy to watch. Thanks for the awesome content, 3Blue1Brown.

This solution might interest the skateboarding community! Does any skateboard park exploit the solution to the Brachistochrone problem, I wonder?

The best math in youtube by far

The answer is "no" no, I can't :)

This channel has a superb standard of videos and never fails to present new insights into old problems. Thanks for all the effort!

I heard that you were doing a essence of calculus series. You should consider also doing a essence of multivariable calculus. Multivariable is a class that many people take in all of maths, engineering and science and is surprisingly poorly understood. Thanks! :)

"Newton stayed up all night....solved it..."

The animation at 4:39 ~ish using a space filling curve is magnificent, and for such a one off display is magnificent

Makes me feel special that I was subscribed to you before Vsauce. And I love that more people are tuning in, you need more exposure, your videos are simply the best math videos on the net.

You're insanely intelligent, and an inspiration to me. I'm going to try this. If there's one thing I think I've learned from your videos, it's that nothing consistent is off-limits and there is always a different reference frame from which to view your problem.

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

This is amazing, honestly keep it up I need more insight into certain math problems and the history that surrounds them

Glad you're posting more videos. These are some of the most beautiful on RU-vid.

Yet another beautiful presentation of a very clever proof. It's been a very nice surprise to hear Steven Strogatz participating on one of your videos!

Who's here after the vsauce video?

I was really in doubt about this interview concept when you first presented it, but after just 30 seconds with talking and animation in perfect unison, I was convinced. Great job!

The animation is so good! his must take you such a long time to do, this was a very fun video to watch

This was very fun to watch! Actually a nice recap with some additional information about a calculus of variations lecture some time ago. Thanks for that!

easily one of the coolest math ch. vocal , video quality, use of math language... everything real nice

I love the math and graphics and the cute pi sliding around

Awesome. Your videos are fantastically well-made!

This explantion is so clear and nice, also good and easy to see. Really love it.

Your channel is pure culture to enthusiasts of math like myself. Pure gold. Please do not stop.

I now need to figure out how to work "brachistochrone" into a conversation.

Strogatz, the chaos man. His Nonlinear Dynamics lectures made life much easier for me.

I know it may be late but I was looking for a satisfactory explanation of the warm up challenge in the comments and I could not find any. As so I have tried solving it myself and I finally came to a possible solution. First we must label θ as the angle between the vertical at the center of the circle and the point on the circle's circumference which is tracing the cycloid. We can find the tangential velocity at that infinitesimal point in time if we consider the point of contact C of the circle and the horizontal surface as the instantaneous center of rotation and by using *v = wr* (Eq. 1). If we label the point where the particle is as L, then we can see this becomes *v = wl* (Eq. 2) where l is the length CL. Using the cosine rule we can see that the length l is simply *l = 2Rsin(θ/2)* (Eq. 3). We can also equate the velocity if we consider conservation of mechanical energy since there is no friction. Thus K.E. = P.E. of the particle which is *1/2 mv^2 = mgy* (Eq. 4) where y is the vertical distance from the horizontal surface to the point L. This reduces to *v = sqrt(2gy)* (Eq. 5). Substituting Eq. 3 and Eq. 5 into Eq. 2 yields *sqrt(2gy) = 2wRsin(θ/2)* which can be squared to become *2gy =4 w^2 R^2 sin^2 (θ/2)* (Eq. 6). We can substitute y here with the parametric equations of the cycloid. If we take the y axis downward as positive then it varies slightly that the one in the video as it will be *y = R(1 - cosθ)* (Eq. 7). Substituting into Eq. 6 gives us *2gR(1-cosθ) =4 w^2 R^2 sin^2 (θ/2)* (Eq. 8). Lastly, we can use the trig identity *sin(θ/2) = sqrt((1-cosθ)/2)* (Eq. 9) and transform Eq. 8 into *2gR(1-cosθ) =4 w^2 R^2 ((1-cosθ)/2)* from which we can cancel out the various term to reduce into *g = R w^2* which finally yields *w = sqrt(g/R)* (Eq. 10). This shows that the angular velocity is constant and that the particle's trajectory due to gravity does follow that of the trajectory of a rotating wheel with constant rotation. I hope this helps anyone in the future who might revisit this wonderful video.

I must say, I love this channel, you are almost souly responsible for kindling my love of math, thank you

It is interesting to note that Huygens in 1672 showed by graphical means, that a cycloid was a tautochrone, that is that starting at any point on the curve an object sliding along it will reach the bottom in the same time, thus a cycloid should be the ideal path for a pendulum on a clock. Hugens partly got to the solution as the result of a good guess, the mathematics of a cycloid is pleasingly simple and given that Mersenne had already showed that a circle didn't quite work, the cycloid was the next obvious and easy curve to try. The two problems are clearly linked and I am sure that Newton knew this and proceeded along similar lines.

"Heeyy, Vsauce, Michael here" brought me here.

My preferred and the most elegant solution I've seen to the brachistochrone is by that of the calculus of variations in which is kind of pops out.

0:42 that joke right there earned an instant sub .... also the face that he timed it so it hit exactly at the 42 seconds mark .... this guy is awesome!

I watch your videos all day, I watch for knowledge and for fun! I can't get enough of this! Your expository is like Nothing I have ever seen before. I'm about to start my PhD in applied Mathematics.... I feel deeply blessed that I came across your channel. You sir, are a breath of fresh air.

Could you have an intuitive way to see that "rate of change" and "area under the curve" are opposites?

one of the best maths/physics videos i have ever seen

Brilliant! These videos of yours are just outstanding!

As a Cornell alumnus, I know Prof. Strogatz is very famous among math and science students for being a great educator, if you could not tell from the video. It's always a pleasure to see/listen to his work.

Are you gonna cover a solution to the challenge question at some point?

Your videos are unbelievably well animated and very interesting!!

I know nothing about higher math yet I love watching 3Blue1Brown videos. Imaging how these concepts occur in the world is why I watch. This concept reminds me of alpine skiing and skateboarding.

Wow, i am amazed by every aspect of this video!!!; what software do you use to make these animations.

This is a fantastic explanation. I'm taking precalc this year, and I can intuitively understand this. That is so cool haha. Came from Vsauce. It was an alright video, but he never really explained the reasoning behind the property, which was what I wanted to know. Glad he introduced me to this channel though.

I saw this problem in Physics olympiad national qualifier 25 years ago. Of course, I couldn't even tough the starting point. It has been forgotten for more than 2 decades, and suddenly pop up in front my eyes. One of the biggest mysteries of my life is solved. Thank you!

Fantastic video! One of my favorites. More videos like this!!!!

7:38 Case Sensitivity: The g in the potential energy term, mgy, is not "the gravitational constant." The gravitational constant, G, is the proportionality constant in Newton's law of universal gravitation that relates the attractive force to the masses involved and their relative position. G = 6.67E-11 m^3/kg•s^2. You are referring to g, which represents the acceleration due to Earth's gravity measured at the surface, 9.8 m/s^2. An innocent slip as I know you know this. Another great video!

Woohoo, new video!!!!!!!

What a brilliant story. As you say, awe inspiring. When we learned Snell's law, it was taught without reference to that path being the fastest, hence the path light will naturally take. The facts were taught but not the fundamental principle or reason. Of course that leads to other questions.....

Amazing content as always. Keep it going.

This video is really great! Why cannot I watch it when I was an undergraduate! The first time I read the details for the solution of Johann Bernoulli was from Ernst Mach's Science of Mechanics.

I'd like to see Newtons solution

the challenge at the end is just tooooo good, like its really interesting and exciting just think and ponder on it

Great video - you keep interesting me in things that I had no idea about

"By the claw the the lion is revealed"

The Most Casual "Oh Interesting" I've ever heard 5:55

I love your videos. The quality is impeccable and the content is inspiring but i just can't watch them yet.

your videos rule dude, that part on light as a solution was like, genuinely mind blowing

Really awesome solution, I paused the vide and tried to solve it myself. My first instingt was to use calculus, but I didn´t do very good. However this is how far I came and if someone could help me to continue this calculation it would be really nice. The variable that we want to minimize is the time, and the variable we are looking for is the function between the two points. So we need a relationship between the time and the function. We know that t=s/v, there is a general way to work out the distance between two points on a function. And it says that the distance from the point x=a to x=b on the curve f(x) is the integral from a to b of the function sqr(1+(f´(x))^2). The distance between the points doesn´t really matter so I´m going to say that it is one just to make things easier. This means that s=0integral1(sqr(1+(f´(x))^2))dx We know that v=k*sqr(f(x)) in any given point, but since the speed changes over time we can not plug it in to the formula. Also I´m going to say k=1 just to make things easier. It won´t affect the final curve any way. Correct me if I am wrong but the average speed between x=0 and x=1 is going to be 0integral1(sqr(f(x)))dx. So if we plug this into our formula t=s/v we get t=(0integral1(sqr(1+(f´(x))^2))dx)/(0integral1(sqr(f(x)))dx). So I have a formula for the relationship between the time and the function. Now I just have to find the derivative and set it equal to zero I guess, it will probably end up with i differential equation. There is just a little problem, I have no idea of how to take the derivative of that function, and I don´t even know if it is right for that matter. Is there even an algebraic way to do it? If not, then I have no idea about how to solve this problem algebraically. It would be easier to find the derivative if the variable was x, but it not is f, we are looking for an entire function here! Does anyone know how to continue this solution?

Is there a way to support you?

This will be a hand wavy solution to the question but the way I see it, the straight line in the theta-time space indicates a constant rate of change in theta which is the smoothest possible path to the solution as a straight line is always the shortest path: i.e. add any more or less curvature than that exact curve and you'll distort the theta-time curve creating a lower than optimal area under the curve of that graph. Background is only a B.S. in math though so take my intuition with a few grains of salt. Cheers to the work you do, and the tools you've developed - it's all much appreciated!

I’m not sure if I’m looking at the problem you gave us correctly but I love puzzles. It looks like a straight line because t-0 graph is measuring how long it takes to get into a certain position each starting at zero. If you look at the graph side ways where t=y and 0=x. The end points makes a curve.

6:48 "but for now, all you need to know" come ON, i wanna know it all!!!

5:28 I loved your animations of light rays that included the oscillations of the wave itself. Very clever

Wow - I love this (and your other content)!!!

Your videos are so rare, that I auto like them as soon, as they posted

where do you get ideas for your videos? I imagine it takes awhile to organize your thoughts and results to make something you're content with

3b1b: challenge the audience Audiences: meme about newton

Wow, very mind-bogglingly elegant solution! I'll have to work on the challenge now!

Snell's Law is an example that Bartosz Milewski gives of the kind of global thinking that is taken to the extreme in category theory, with its universal constructions. A math instructor of mine also mentions that these kinds of global optimizations (optimizing with the telescope, rather than the microscope, you might say) are done in calculus of variations. I find this duality of local vs global approaches to optimization to be quite interesting.

I watched this way before vsauce, man i feel special!!

Your videos are inspirational. They have allowed me to see mathematics, or as Keith Devlin would say, "mathematics has made the invisible visible." And your videos do that for me in a unique way that has not been replicated by any other learning source. I am studying math education, and I would actually like to learn how to make videos like this so that I can make learning more enjoyable for my students. I'm not much of a programmer, but I'd be willing to learn. Is it a trade secret or can you point me in the right direction? Thanks, Grant. -Ben

Wow. Just wow. This video was the highlight of my day.

thanks for what you're doing, your videos are awesome

I am so happy for you that vsauce discovered your video xD

Wait, so what was Newton's solution? Was it the same as Johann's?

Holy crap, Strogatz! You mean the author of the book I loved in undergrad, Nonlinear Dynamics & Chaos! Loved my Dynamical Systems class in undergrad, but damn...it was a beast! Much much much respect for these mathematicians who pioneer the way in these fields, but even more so for those who can beautifully, eloquently, and effectively communicate these advanced concepts in math to plebeians like myself; such is the man you interviewed here: Steven H. Strogatz.

I love you. This is brilliant. Going to devour Levi's book asap.

4:38 what is with the hilbert curve at the top left?

Physicist (upon seeing the solid normal line at 7:05): REEEEEEEEE

Great explanation, thank you!

What I absolutely loved about this video is, light is shown as a particle and as a wave! xD

isn't it in some way a shortest path problem in 4 dimensional spacetime? also, what software are you using to make those great animations?

First time I'm seeing a video with no dislike.

super cool that you brought strogatz, im a fan of his "non linear dynamics and chaos"

Just stumbled onto this video. Content is fantastic and so well organized! The visuals are SO well made. I’d tie willing to share, what did you use to animate?

Cool challenge. Now I'm going to be procrastinating my homework, and it's all your fault! D:

It's because time and space are an illusion... Great visuals/graphs in the video... You are really great at teaching!

Never knew about the history of the problem .. great job digging into the origin of the problem

one of my favourite problems. never thought I would one day watch such a good video about it, and absolutely free of charge. your videos are really really good. I am repeating the question somebody asked below - any way we can support you?

My educated guess for your challenge is that it minimizes the change in energy. I remember in engineering school we had some examples that were similar where the integral of the work done was minimized when it was a straight line. This also seems to make physical sense in that light will want to minimize its energy state.

I've seen several videos on this today (started at Vsauce). An infinite number of cycloids, with varying diameters of original circle, can be fit to two points. Are of of them solutions to the Brachistochrone problem? That cannot be the case because the largest of them is a straight line. So what parameter determines the size of the curve?

2:49 brachistochrone IS the arc of a circle

Fantastic video!! I'm surprised you don't have more subscribers by now.

You say "challenge", I say "homework"

This is not rigorous and I'll probably revisit this later, but I do have a general idea: The lateral force imparted on the object due to gravity is gsin(theta). As the slope is constantly changing, mapping the axis to sin(theta) renormalizes the curve in a way that reduces it to a straight line, as it undoes the encoding of the function. In other words, the function does not look linear on a normal cartesian set of coordinates is because the coordinate system does not factor in the constantly changing lateral force of gravity. By mapping the function into a set of coordinates that cleverly cancels out the constantly changing force of gravity, the optimization problem becomes linear. I also want to thank you for making these videos. I rarely find videos or textbooks that explains things intuitively, and the animations help so much in this regard.

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*.

I have only the warmest of feelings for Strogatz after listening to his Great Courses course on chaos.