The Brachistochrone Problem and Solution | Calculus of Variations

In this video, I set up and solve the brachistochrone problem, which involves determining the path of shortest travel in the presence of a downward gravitational field. This is done using the techniques of Calculus of Variations, and it will turn out that the brachistochrone can be represented by the parametric equations of a cycloid.

The Brachistochrone is a rather popular topic on Youtube, with pop-science channels like VSauce making videos about it. However, not many people actually derive the equations, so I'm hopeful that this tutorial will be a more rigorous change of pace.

Questions/requests? Let me know in the comments!

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[FacultyofKhan]

I've addressed your requests for a classical mechanics series! Here is my playlist, which starts with the Principle of Stationary Action!

[physicsmathsworld]

😂😂 last words are so hilarious 🔥👌 "I am the kind of person who would like to fall down from sky to ground rather than ground to hell " btw, your channel is one of the most beautiful channel for Physics & maths ❤ for me . Thank you

[schatter]

awesome final comment on sky and hell! great video btw

[siom]

That was actually helpful. An absolutely brilliant explanation, better than the professor at my university!

[michasz]

You're the great teacher! Keep making those wonderful videos!

[ja]

So this is what the textbook meant when they said, "the solution is trivial and we leave it to the reader to solve"

[kacperkazaniecki]

Amazing! So happy I found your channel (at 3am lol). This is the first video I've watched, but I can tell that the quality is outstanding so really well done. One question though, at 7:50 why would we use trig substitutions? Of course, it works and we get a nice result however if I were to be working this out myself, I'd be comfortable up to that point. Then I wouldn't really think of using trig substitutions. So what's the 'insight' in this that I could learn from?

[RaunySilva]

I hope the best for you. I am really grateful, thanks

[healinghub]

I like your its precise and explainatory.
Plz make a series on classical would get loads of subscribers that's something people denand

[saitaro]

Thank you very much, sir, for this series, it was a pure joy to watch it.

[rodrigogarcia]

Thank you. You, Euler, and Lagrange saved my life. This helped a lot on my EE.

[smrd]

Your site fills a crucial educational need. Can you do videos on pursuit problems?

[nickallbritton]

I like your videos, but I'm having trouble putting it all together.

I came to this channel having hit a wall in classical mechanics: I want to understand the principle of stationary action and the Euler-Legrange. What I did before was learned differential calculus when I couldn't move ahead in physics without it, but I'm finding it very difficult to pick up variational calculus?

Any advice to help me put all the pieces together? Thank you a lot for your videos!

[dx-dt]

Well, I think I am going to be here for a while. You've hooked me in, sir! Subscribed :D

[diegoortegadod]

Hello Faculty of Khan!
You have no idea how helpful this video has been for me... I am currently an IB student doing my Internal Assessment for my Mathematics class exactly on this problem.
I have one question, at 9:02, how did you end up with that equation on the left? I understand you replace -C1 for K, but how did (1-cos(x)) turn into (x-sin(x)), and why did you get a value of K2 which is added at the end? This is my only question regarding the mathematics of your video, which is very well explained. Thank you!

[prithvitimalsina]

Please explain why you have taken (dx) out?? what is the meaning of doing so?? What if i take out (dy) outside? ? and what about the brachiostrochrone problem in curved space...

[AkshayPatil-ofii]

hey your way of explanation is awesome.Please make some video on hamiltonian.

[manassivakumar]

at 7:35 while taking the square root why are we not considering the negative value

[CE]

At 5:00, what do you mean by "stationary"? In what sense is T made to be stationary by finding the right function y(x)? I don't know what "stationary" means in this context. Stationary to me means "not moving." But T is a time interval. Time intervals don't move. Mass moves. Electromagnetic waves move (i.e. the energy of EM waves propagate). In what sense does T move when you don't have the right function y(x)? This is the second video on YouTube. In both videos, the point is made that we need to find the function y(x) that makes T stationary. But I don't understand what "stationary" means.

[م.رضوانحسن]

Great explanation, thank you very much.