Introduction to Variational Calculus - 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?

[@Freeball99]

Yes, you're correct. A few others have asked about this, so I'll pin this comment to the top of the discussion section in the hope that others see it. The red line should be labeled y_bar. I started out drawing one thing and it evolved into something slightly different. Unfortunately, since KZbin no longer allows annotations, I am unable to correct this on the video.

[@mathjitsuteacher]

@@Freeball99 Thanks for answering so quickly. Your video was fantastic.

[@lioncaptive]

Good catch 💯

[@Mechanical_360]

The red curve represents ybar(x).

[@michaels333]

I probably also would have switched y_bar and y. y is arbitrary and can be written as the sum of the optimal path plus some arbitrary path. Maybe I’m knit picking(?)

[@serchuckseyonski9908]

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

[@brunoazevedo6380]

insightful

[@petermason7799]

Why oh way didn't I know this 50 60 years ago. There is nothing here that anyone with an engineering degree could not understand. Thank you

[@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.

[@Michallote]

Yeah it helped a lot to know we where heading to the generalized form of what Laplace described earlier. Just by adding historic context it unconsciously help you to organise the ideas... Brilliant!

[@mohankrishnan2022]

Yes! The historical introduction at the beginning - succinct but comprehensive - was a great table setter!

[@ako8205]

I too, liked the historical part. Regarding the gallantry of Euler, I read somewhere ("The Music of the Primes"?) that Euler took several weeks to get to Russia where he was invited to work because he was loaded down with creature comforts requested by colleagues already working there.

[@AbhishekSachans]

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

[@motherisape]

most people teach this topic by starting with integral and showing that this integral is stationery. which doesn't makes sense what does it even mean to be stationery. every explanation I see on internet doesn't makes sense this is clearest explanation .

[@NithinGoona]

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

[@akbarahmed3078]

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.

[@jonathanmarshall2518]

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

[@SenzeniNxasane]

Wow you did better than my mechanics lecturer, you made it so simple and understandable, you did what my lecture would never do even if they gave him an entire year to explain, to think our mechanics lecture is for 3 hours but still I did not understand, with you it took 25 min, bravo.

[@RellowMinecraftJourney]

To think our lecturer (let me not speak names) couldn't explain it better 😂😂

[@tshilidzisibara8805]

@@RellowMinecraftJourney Poor Warry. But let's thank him for leading us to this teacher here

[@SenzeniNxasane]

🤣🤣🤣🤣🤣ahah you two

[@ayushtaylorsversion1253]

Im 16 but this is far better than any ecstasy out there

[@stephenhicks826]

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.

[@AnmolSingh-ig3ji]

Wowa💝

[@cheeseinmypocketsvelveeta2195]

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!

[@henryparker3420]

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!

[@GustavoOliveira-gp6nr]

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!

[@vychuck]

Absolutely delightful delivery in less than half an hour, thank you.

[@adamconkle4042]

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.

[@gouravhalder1256]

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

[@evanhagen7084]

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.

[@David-mm6nx]

Words cannot describe the brilliance of this presentation. Best one yet.

[@ylmazcemalunlu3429]

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.

[@jamestucker1126]

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

[@giuseppecerami1764]

This video is a gold nugget for self-learners. Thank you so much!

[@Freeball99]

You're so welcome!

[@gauravkanu2823]

Great video and explanation. Very grateful for the history of classical mechanics and for keeping the concept simple without complicating it.

[@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.

[@ultimatedarktriforce]

Phenomenal explanation I've seen on the internet, no stutters, no delays, no questioning their work, just pure art.

[@theo-zj7dm]

I am a french student and I had trouble finding good mathematical explanations in French, and then I found your video. This is amazing, very well explained and rigorous. You made my day !

[@fawgawtten9515]

The best and cleanest on all internet. Thank you

[@hakankarakurt1100]

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!

[@augustowanderlind7963]

completely agree

[@sonyaraman]

This is the gem, I’ve been struggling to find a good video on derivation of this equation, and there it is. Simply the best 🤝🏻 Additional kudos for bringing in the historical overview of how that used to look like back in time😊

[@squirepegg6157]

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

[@copernicus633]

The best derivation of the Euler Lagrange QE I have seen. Very concise, yet fills in details missing in most other explanations, written or animation.

[@eleanorterry-welsh7784]

I'm taking a graduate level classical mechanics course and needed a review of calculus of variations because I had gotten rather lost in a recent lecture. This was an incredibly clear explanation and made the whole lecture I had been totally lost in completely make sense. Definitely going to be watching through more of these as my mechanics class covers more of the types of minimization problems mentioned in the beginning of the video.

[@Ikbeneengeit]

Thanks for the history at the beginning, really helps put the concepts into perspective.

[@dwinsemius]

@22:37. "I know this must be setting your mind spinning". Right. I still remember when Dr. Katz laid this out at the very beginning of the sophomore course that I took in the summer of 1968 at the University of Michigan. It was rather unsettling, but once the fog in my brain distilled and I could see its wide applicability it became such a wonderful elixir.

[@theonionpirate1076]

I've never seen this before but now feel I understand it completely. Thank you!

[@johnhalle6404]

Beautifully done. One of the most lucid and insightful lectures I have heard on any subject. Thank you for investing the time and energy to produce it.

[@johnmosugu]

You simplified this subject. God bless you

[@charleshudson5330]

Excellent presentation. I especially enjoyed the introductory historical perspective.

[@quantusmathema]

you described this very eloquently thank you

[@moussadiaw1682]

Un sujet très rare sur KZbin and well explained. Thank. If possible a video of Euler-Lagrange applied to image processing

[@vinodgopinath7837]

Most complete, thorough and clear explanation of EL equation with its background history on youtube! You are a very inspiring teacher.. Lot of respect from India

[@avatar098]

Thank you for this! My background is in computer science, but recently decided to go back and self study some more mathematics just as hobby. Your explanation truly has put things into perspective for me. Thank you again!

[@MrSlowThought]

You have made clear so many thoughts I've been having on the history of mathematics and physics and the importance of (in hindsight) such simple concepts. You have sketched in some historical connections that I was unaware of, and provided the clues that opened my mind to the Lagrangian and Hamiltonian.

[@pedrocolangelo5844]

I definitively need to watch your other videos. Your way of teaching is by far one of the best on KZbin! I was trying to understand properly calculus of variations for a long time and you are the one who made it possible for me to understand! Thank you so much, professor! The funny part is that I'm not even a physics student, I'm an economics student. Your video is helping several areas of knowledge.

[@jevaughnclarke6174]

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 sooooooo 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.

[@moart87]

You get THIS level math in Economics? Seems more like Econometrics.

[@jevaughnclarke6174]

I had to utilize both principles for Macro and little less so in Micro

[@moart87]

What are the types of problems in economics that you use this on?

[@jevaughnclarke6174]

I had not used hamiltonian nor Lagrange in my econometrics class. Time series models were stressed econometrics along with GLS models. The Lagrangian was used to minimize/maximize utility/ profit functions etc in Micro. The Hamiltonian was used similarly for continuous systems that require optimization with certain constraints on the system variables.

[@xadir]

@@moart87 consumption functions, production functions, growth functions etc. To be fair, proper variational calculus is usually taught at postgraduate level of macro and microeconomics --I had to do it in my MSc course back in the day. Although, I still remember Euler and Lagrange equations from my BSc Econ course as well. It is a common misconception where economics is placed in line with "business studies". Truth is economics is a mathematical science, implementing applied mathematical methodology in both theoretical and empirical research.

[@xhonshameti1749]

This video makes me happy. It’s is obvious you understand the heart of this theory. And it’s obvious that you are genuinely passionate about mechanics. You know know it like an old school watch maker knows it’s watches!

[@miaoshang7732]

I learned this equations from Landao's book and i really appreciate your mathsmatical derivation. They are clear and easy-understand.

[@jeissontoscano1477]

Thank you A LOT, I really mean it! So much useful information is only a few tens of minutes! It's so difficult to find videos of even simple document explaining those concepts in a simple, yet comprehensive and entertaining way... so thank you for you contributions not only for this video but all of them. This channel is truly a gold mine!

[@yuthikasenaratne7250]

the best derivation of the eular larange equation seen so far( espeacialy about that apsolone) others just skip over that

[@bird5119]

This was such a good explanation in a college lecture format that it triggered a Pavlovian reflex: at 22:25 i felt the itch to put everything away in my bag and start to walk out the lecture hall while the professor is still talking

[@FranFerioli]

Thanks a lot. The fact that you pass from y_bar(x) to y(x) when eta is small is key. A good intuition for this is considering that eta parametrises a whole family of y_bar(x) curves all similar (proportional) to each other, but at different "distance" from y(x). When eta ==> 0, Int [y_bar(x)] ==> Int [y(x)] so you can make the substitution.

[@manmis007]

People who have some depth to the interest they have would love this......grt job sirji. .....

[@fisicayquimicahoy]

That's completely and utterly great!! it's the best lecture on Euler-Lagrange equations I ever saw. Thank you very much

[@yamsh638]

I would give thousands thumps up to this

[@jaideepganguly]

Excellent presentation, crisp and succinct! Thank you!

[@tusharmadaan5480]

Reignited my passion for calculus of variations and optimal control. Beautifully explained!❤

[@aryadebchatterjee5028]

u are the best teacher I never had actually well I am an eighth grader and I started learning calculus in grade 7 and none of my teachers supported me and helped me when I faced problems I wish I had a teacher like u to help me out back then I would have way easier and much less frustrating If I had a teacher like u keep up the good work man !! love your videos

[@aniketsengupta9137]

It's great that you are working hard from such a young age. Kudos to you. If you are learning calculus from such a young age you must be brilliant because I couldn't even understand basic trigonometry at that age. Teachers won't support you for such things, you need to take advanced coaching for that advanced stuff.

[@EconJohnTutor]

The best introduction into this concept ever. Thank you so much!

[@euereren]

This is pure art

[@adityabaghel1270]

Thank you so much for this wonderful video! Beautifully explained

[@GuidoNagel-h1q]

I dont really comment much in videos, but you deserve one. Really good explanation, clear, concise and also you speak really smooth and easy to understand (im not a native english speaker). i didnt know anything of calculus of variations like 20 minutes ago but now i know how to start it, Thanks For the video Man!!. Hope you have a great day.

[@nihilisticboi3520]

Beautifully explained! This is elegance at its best. Thank you so much for this lecture!

[@Freeball99]

Glad it was helpful!

[@beauanasson3570]

Damn, this content is great. So concise yet so clear, cheers.

[@garvinmugala7003]

Mathematical and scientific beauty. Wonderful presentation of the lesson Sir. Just what i needed for the morning.

[@moatazabdelrahman5691]

In love with the history part, gets me really interested! and 19 Yo!!.. goodness!!

[@Cherem777]

Excellent video. As someone watching for the first time, I liked how you pointed out some areas where other’s explanations fell short. Thank you!

[@Freeball99]

Glad you enjoyed it!

[@horacioguillermobrizuela4295]

Excellent video. Thank you so much for your effort to keep it clear and simple. The historical briefing at the beginning was quite enlightening for me

[@NeelDhar]

I have honestly watched so many videos before this on this topic, and I swear that in 6 minutes you have explained the concept much better than all those videos. All the other videos spent far too much time on the math before breaking down the concept. Love this video.

[@chenweizhi8609]

Very very easy to follow, nice video!

[@BarunChatterjee-f7e]

Beautifully explained

[@wargreymon2024]

Good editing, Intuitive and comprehensive. Your voice is soothing. This is the best explanation on Larangian mechanics, no one on KZbin even comes close.

[@Freeball99]

🙏 I'm telling my wife what you said about my voice! 😇

[@jesusfuentes7589]

'... and that's it, we're done!' Brutal, absolutely brutal! Many, many thanks - great lesson!

[@brandongammon6978]

Great refresher, perfectly explained !

[@chiragkshatriya9486]

Sir, One of the best video on Euler-Lagrange Equation on KZbin till date. Could you please make a whole series on ‘General Theory of Relativity’ from scratch to the final equation and it’s solutions like this video.

[@devsutong]

history... motivation... derivation. perfect 🔥

[@Fishtory]

Excellent stuff! Love the history tour in the beginning as well!

[@GoutamDAS-ls1wb]

Thank you very much for a presentation of extraordinary clarity! One of the best expositions on the topic on KZbin!

[@Freeball99]

Glad you enjoyed it!

[@AA-gl1dr]

Wow this is art. I’ve hated math my whole life and you’ve made it digestible and palatable. You’re a skilled teacher

[@erdi749]

...looking for a path that minimizes a function. What is a path? It's a function. So we are looking for a function that minimizes another function.. voooov! wonderful explanation, never thought of variational calculus like that!

[@alvaros9038]

The best explanation I have seen so far! Thank you

[@Arriyad1]

Thank you! At last I understand it - taught to me 44 years ago. Now that I got it, I’d suggest not to call the variable x, as that might induce the viewer to believe it is the first coordinate in the 2 dim plane, where points live called A=(x1,y1).

[@jaafars.mahdawi6911]

Not yet done watching but couldn't resist pausing to throw a word of appreciation and gratitude. Keep it up, sir.

[@briandwi2504]

Absolutely brilliant. So clear, thank you.

[@jonathanaarhus224]

The fact that we can minimize any arbitrary functional integral with a single first order differential equation is mind-blowing.

[@luffis1985]

"Euler case you weren't aware was quite the mathematician of his time" Quite the understatement. I'd say he was quite the mathematician of any time.

[@Freeball99]

Agreed...or quite the mathematician of ALL time.

[@ai_serf]

My calculus teacher made me fear the concept of variational caculus, that it was so advanced and abstract. You make it comprehensible and logical. Maybe it's because I'm older and have a lot more experience, but I absolutely treasure the historical background.

[@williammartin4416]

Excellent and effective explanation

[@paaabl0.]

Very good lecture, thank you. Love the historical intro!

[@rangamurali7667]

Beautiful, word for word, line by line, breaking down the mathematical poem, syntax ..speechless! Brings back memories of college days I wrestled with trying to figure. Can you plz do Maxwell equations? Am sure there are many to catch up, we ask for more and more. Our sincere thanks! Awesome!

[@goitomkfle8805]

Great video with great explanation of the core concepts, and I also appreciate the comments very much!!!

[@Eigenbros]

Excellent video. Really high quality and touched upon many things that typically get glossed over

[@ricardowilsonaguiardacruz4554]

I enjoy correlation of history to physics and math. Very good!

[@mehrdadmohajer3847]

Well done . Thank You ALL 👏

[@AbhishekSachans]

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

[@thescientist7753]

taking a class on lagrangian mechanics next semester, can't wait!! also hearing about how Lagrange discovered this stuff at only 19 makes me feel bad abt myself lmao. same w hearing about Eulers work, but its inspiring. I think part of the problem is that it seems many of the students in my classes like to take formulas at face value and go off using them with no solid understanding of what any of it means but I dont like to move on until I have a complete conceptual understanding of the topics enough to derive them myself, maybe it will serve me well later in life but for now at least I can see the beauty in some of it that makes it all worth it. Seeing things like this make me so excited because I just know that once I really have a thorough understanding of all this ill be able to see the poetry within the math as I apply it. Still trying to figure out why it must be a function F[x,y,y'] with the y' explicitly included. I also think the eta(x) on the graph should be y bar, not sure. Fantastic video though!! it was my first introduction to the topic and it was better explained than anything I've seen in university and I can tell its definitely not the simplest thing I've learned so kudos!! :) thank you

[@Freeball99]

You are correct, the red line in the figure should be labeled y_bar rather than η. F can be extended to higher derivatives of y, i.e. F = F(x, y, y', y'', y''', y''''). F can also be extended to include additional independent variables (this is what we do when we introduce the parameter ε). I didn't extend it too much in this video because it gets very mathematically tedious and I didn't think it would add anything. Still, I wanted to show how the derivatives of y are treated i.e. we integrate them by parts. Higher order derivatives are integrated by parts additional time depending on the order of the derivative. We use these derivatives in calculating the strain energy (as I have shown in some subsequent examples). Good luck next semester!

[@ravikantpatil3398]

Deep understanding of the problems and urge to learning to students interested compelled to increase interest on the subjects ❤❤

[@ducciom.gasparri9727]

Best. Explanation. Ever. Now my plan for preparing for the intermediate mechanics exam is to watch all of your videos... and then go back to the Goldstein for the details :)

[@alonsosainz5214]

Impressive video. I have been looking for a good explanation for a while, yours was the best by far.

[@jwilliams8210]

Wow! That was an absolutely extraordinary presentation! Just awesome!!

[@microcolonel]

I'll watch this a few months from now and know exactly what you're saying.