Introduction to Variational Calculus - Deriving the Euler-Lagrange Equation

An introduction to the Calculus of Variations and the derivation of the Euler-Lagrange Equation.

0:00 An Historical Background
3:47 Path Minimization Problems
10:25 Deriving the Euler-Lagrange Equation

[mathjitsuteacher]

In your drawing you tagged the red curve as eta, but I think this is not correct because in this case you would have eta(x1)=y1 and eta(x2)=y2 and you wanted these nunbers to be zero.

From what I understood what you call eta in the picture is actually y+epsilon eta, where eta is a curve which vanishes in the boundary.

Am I right?

[serchuckseyonski]

That is, without doubt, the best explained and cleanest derivation of the Euler-Lagrange equations on the Internet.

[kvasios]

Exceptional! Absolutely exceptional! Only someone with deep understanding could deliver as such. Extra credits for the historical intro... these couple of minutes for providing a timeline of people, events and facts are helping tremendously in putting things into perspective.

[AbhishekSachans]

Most in-depth and elaborate illustration I've seen on the topic. A lot of aha moments. Thank you!

[akbarahmed]

Almost everything I learn, I learn from the internet. It's been like this for the last 5 years and I can confidently say that this is the finest and the most well explained video on this topic I have watched so far.

[jonathanmarshall]

This is beautifully explained. I’m an practical engineer - my brain responds very well to understanding the motivation behind the mathematics. Thank you!

[NithinGoona]

More than 10 years of confusion in my head cleared in 10 mins. Thanks a lot.

[GustavoOliveira-gpnr]

Man, this is the best explanation EVER of euler lagrange equation!

You were very meticulous in explaining the important details (that was holding me back from fully understanding it) that most videos skip through, and you even explained the history behind it! It was perfect! Congratulations!

[stephenhicks]

Thanks so much for this. You've shone a bright light on the Euler-Lagrange equation for me. Thanks. I'm 67 years old but still learning.

[cheeseinmypocketsvelveeta]

Thank you for doing what others couldn't do for me in helping me understand this beautiful principle. As someone who has the calculus tools and has been interested in classical mechanics for longer, discovering the lagrangian is like finding buried treasure in your backyard. Who has been keeping this from me!

[hakankarakurt]

You are on fire! One of the best educational YT channels I’ve encountered so far. Way underrated but I guess when you go deep into detail you somehow sacrifice being mainstream.

Nevertheless, even though the view counts are low, the appreciation of the viewers are high. Thanks for the content. Stay safe!

[jevaughnclarke]

I am a PHD student in Economics. While I passed the classes utilizing Lagrange and Hamiltonian optimization I always struggled with the 'why'. Thank you much as I now got an intuitive idea as to the why.

Please do a full course on Variational Calculus. I will pay to be a part of such a class with you if that is what it takes. Please consider doing a course on VC. Thanks.

[adamconkle]

As someone who has taken Intermediate Mechanics and has gone through this material, this has been the most thorough explanation of the derivation that I have seen. This is just phenomenal.

[dwinsemius]

Great stuff. It's the first time I have heard the word "brachistochrone" actually pronounced. The perspective that the goal is to calculate a function rather than a scalar leads into the need for operators rather than definite integrals very nicely. I wish that I had been prepared for quantum mechanics with this framework.

[evanhagen]

I knew from the instant I heard his voice that this was going to be an absolute banger of an explanation. This video is incredible. Very hard to find content this high quality even from the biggest names on the internet.

[henryparker]

I was reading Landau Mechanics and I couldn't follow the logic. I finally understand it from this perspective, and I was able to work backwards to figure out what Landau was saying too. Thank you very much!

[squirepegg]

You have my vote for clarity; it's a great presentation.

[jamestucker]

Only one of the best explanations of the Calculus of Variations that I have ever seen or heard.

[ylmazcemalunlu]

Maybe I watched more than 15 videos and read various papers on this subject, but mate, this one is far better than the rest you can find on the internet. Why does it always take this much to find quality content? Not sure but this might be my first comment on the platform as well.

[gouravhalder]

I find myself lucky to have found these lecture series on YouTube...😊