Derivation of the Euler-Lagrange Equation | Calculus of Variations

In this video, I derive/prove the Euler-Lagrange Equation used to find the function y(x) which makes a functional stationary (i.e. the extremal). Euler-Lagrange comes up in a lot of places, including Mechanics and Relativity. The derivation is performed by introducing a variation in the extremal via a parameter epsilon, and setting the derivative of the functional with respect to epsilon to be zero.

My previous Variational Calculus video was very positively received, so I thought it would be appropriate to continue the series and upload the second video sooner rather than later. Also, you'll notice that the writing here is smaller, but that's because the screen I'm using now is bigger because of my new desktop.

Questions/requests? Let me know in the comments!

[Icenri]

The best explanation so far of what a Lagrangian really is.

[nathanielweidman]

Great video. I have been enjoying your channel. There are a lot of k-12 tutorial videos on YouTube but no a lot at the college level, which makes your channel unique and greatly needed. Please do more videos on calculus of variations and keep up the good work.

[annamallett]

I've watched so many videos and read a number of textbook explanations trying to get my head around Lagrangians and the Euler-Lagrange equation, and this one is by far the best IMO. I still have to pause the video every five seconds and rewatch it every couple of months, but at least I can actually understand it! I even like it better that Feynman's explanation (which is also very nice).

[ozzyfromspace]

Damn, Faculty of Khan! First time seeing this and you make it look effortless. Your YouTube videos are an absolute treasure, so thank you. Cheers from Michigan!

[joycebenbow]

I've watched all of these guys videos (I'm a physics PhD, graduated many years ago) and he really is an exceptionally gifted instructor. I challenge anyone to find a better explenation/exposition of any of the topics off to him I really enjoy all of the videos and each one gives me a deeper level of insight and understanding....I'll be making myself a patreon very soon to show my appreciation...

[ulascilingir]

Thank you for all these videos. 20 years after I take these lessons I use your videos to refresh my memory and every time I wish that you were my professor back in the day 😁

[TG-tonf]

Absolutely the best explanation I’ve seen compared to 4 undergrad text books.

[umedina]

You don't stop to amaze me. I was reading this topic in a FEM book and didn't understand NOTHING. But after this video is simply clicked. THANKS.

[turboleggy]

beautiful presentation, perfect pace, so clear. The only thing I'm not sure about is why we included the parameter when multiplied by eta.

[abhinavroy]

This is one of the most beautifully explained video in Calculus of Variations I have seen so far. Keep up the great job!

[willie]

I understand the general form of integrating by parts. Int(v, du)=Int(v, u)-Int(dv, u) but in the part where you use it (about 6 min in) I am having difficulty understanding which parts are t;he u's, the v's and the dv's. I want to be sure I understand this thoroughly and that I am certain of my understanding so having you answer this would help. Also, I find this to be an extremely well done lesson. Was really struggling with the concept and lecture 1 in particular cleared the fog.

I am 71 years young and this video helps me to keep learning.

[Peter_]

I like when math videos are straight to the point like this, instead of babbling about irrelevant formal theorems and lemmas all over the place for a century before getting to the stuff that actually matters.

[dwyerfire]

You saved my ass, finally someone who can just explain something clearly without skipping 'obvious steps'.

[lakshyaagarwal]

you guys are awesome, this video helped me a lot, saved a lot of my time. thank you again

[raptor]

could you please do a walkthrough of an action problem? Maybe im just slow but i find the application from this to problems in physics hard to conceptualize.

[joelcarvalho]

Great video with a detailed explanation on derivation of the Euler-Lagrange equation! I came here after reading the first chapter of Landau's volume 1 and this video helped me a lot :)

[PhysicsBanglaTutorial]

Your explanation is so good. Thank you so much dear.❤️

[stevenschilizzi]

Thank you! This was a very clear and straightforward exposition. Of course, it has left me hungry to know what other methods are needed to figure out the nature of the stationarity. Do you have a video on that too? - Ah yes, perhaps it’s your last inthis series, the 2nd derivation. I’ll check it out. Thanks again!

[o.a]

i wish more people knew about this channel. you're making my physics courses a lot easier.

[yuvallotenberg]

Your videos are really good and well made. I really like watching them. Could you please do an example of a problem in calculus of variations using the Euler Lagrange equation?