Introduction to Variational Calculus - Deriving the Euler-Lagrange Equation
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- Опубликовано: 1 июл 2024
- An introduction to the Calculus of Variations and the derivation of the Euler-Lagrange Equation.
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INTEGRATION BY PARTS: www.dropbox.com/s/f5wlzbqhzqr...
0:00 An Historical Background
3:47 Path Minimization Problems
10:25 Deriving the Euler-Lagrange Equation
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?
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 RUclips no longer allows annotations, I am unable to correct this on the video.
@@Freeball99 Thanks for answering so quickly. Your video was fantastic.
Good catch 💯
The red curve represents ybar(x).
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(?)
That is, without doubt, the best explained and cleanest derivation of the Euler-Lagrange equations on the Internet.
insightful
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
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.
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!
Yes! The historical introduction at the beginning - succinct but comprehensive - was a great table setter!
Most in-depth and elaborate illustration I've seen on the topic. A lot of aha moments. Thank you!
You have my vote for clarity; it's a great presentation.
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.
@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.
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.
This is beautifully explained. I’m an practical engineer - my brain responds very well to understanding the motivation behind the mathematics. Thank you!
More than 10 years of confusion in my head cleared in 10 mins. Thanks a lot.
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!
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.
Wowa💝
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😊
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.
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!
completely agree
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 !
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!
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.
I definitively need to watch your other videos. Your way of teaching is by far one of the best on RUclips! 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.
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.
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.
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.
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
Words cannot describe the brilliance of this presentation. Best one yet.
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!
Only one of the best explanations of the Calculus of Variations that I have ever seen or heard.
Absolutely delightful delivery in less than half an hour, thank you.
Thanks for the history at the beginning, really helps put the concepts into perspective.
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!
Thank you very much for a presentation of extraordinary clarity! One of the best expositions on the topic on RUclips!
Glad you enjoyed it!
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.
Phenomenal explanation I've seen on the internet, no stutters, no delays, no questioning their work, just pure art.
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!
This is the only good explanation I found in the entire youtube after searching through videos for hours. I finally understand this concept. Thank you so much for this wonderful video. I would recommend this video out of all the videos on this topic out there, trust me this is the one you need to watch.
Glad it helped!
Most in-depth and elaborate illustration I've seen on the topic. A lot of aha moments!
I learned this equations from Landao's book and i really appreciate your mathsmatical derivation. They are clear and easy-understand.
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!
Damn, this content is great. So concise yet so clear, cheers.
Excellent presentation, crisp and succinct! Thank you!
Great video and explanation. Very grateful for the history of classical mechanics and for keeping the concept simple without complicating it.
That's completely and utterly great!! it's the best lecture on Euler-Lagrange equations I ever saw. Thank you very much
The best introduction into this concept ever. Thank you so much!
People who have some depth to the interest they have would love this......grt job sirji. .....
Excellent presentation. I especially enjoyed the introductory historical perspective.
Impressive video. I have been looking for a good explanation for a while, yours was the best by far.
I find myself lucky to have found these lecture series on RUclips...😊
Wow! That was an absolutely extraordinary presentation! Just awesome!!
I've never seen this before but now feel I understand it completely. Thank you!
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.
You get THIS level math in Economics? Seems more like Econometrics.
I had to utilize both principles for Macro and little less so in Micro
What are the types of problems in economics that you use this on?
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.
@@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.
Awesome videos, very clearly laid out and well organized. Keep it up!
The best and cleanest on all internet. Thank you
This video is a gold nugget for self-learners. Thank you so much!
You're so welcome!
Beautifully explained! This is elegance at its best. Thank you so much for this lecture!
Glad it was helpful!
Great content.Explained with great care and clarity. Thank you!!
You simplified this subject. God bless you
Excellent video. Really high quality and touched upon many things that typically get glossed over
Good editing, Intuitive and comprehensive. Your voice is soothing.
This is the best explanation on Larangian mechanics, no one on RUclips even comes close.
🙏 I'm telling my wife what you said about my voice! 😇
Excellent stuff! Love the history tour in the beginning as well!
Not yet done watching but couldn't resist pausing to throw a word of appreciation and gratitude. Keep it up, sir.
wow-by far the best explanation of calculus of variations i have seen in undergrad or now in graduate school. This is the first time the concept really made sense. Beautiful idea and great explanation. Also, you have an excellent voice for these types of narrations. Could be a professional narrator haha
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
Reignited my passion for calculus of variations and optimal control. Beautifully explained!❤
Sir,
One of the best video on Euler-Lagrange Equation on RUclips 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.
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!
Glad you enjoyed it!
Wonderful stuff. So clear and concise. Really appreciated 🙂
...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!
The best explanation I have seen so far! Thank you
My gosh this is so great! Wonderfully explained and made so many things very clear!
I never understood anything about this subject until now, best video i've watched all week you have my sub
Mathematical and scientific beauty. Wonderful presentation of the lesson Sir. Just what i needed for the morning.
Great refresher, perfectly explained !
extremely well explained, please keep making great videos like this!
Absolutely brilliant. So clear, thank you.
Best! It fits my brain perfectly! I love the historical background too!
Thank you for being such a great instructor.
This is excellent. Completely new to me and very interesting. Very well presented.
Thank you so much for this wonderful video! Beautifully explained
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 :)
Thank you for this amazing video. So clear, thorough and easy to follow.
Great video with great explanation of the core concepts, and I also appreciate the comments very much!!!
Great explanations! Your video lectures are awesome!
In love with the history part, gets me really interested! and 19 Yo!!.. goodness!!
Very good lecture, thank you. Love the historical intro!
This video rescued me today. Thank you for such an elaborate explanation.
Im 16 but this is far better than any ecstasy out there
This was brilliant and very clear! Thanks so much!
Many thanks for such a clear exposition of Euler-Lagrange equation. I use it a grea deal in my research on particle physics.
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.
Un sujet très rare sur RUclips and well explained. Thank. If possible a video of Euler-Lagrange applied to image processing
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
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.
This is pure art
the best derivation of the eular larange equation seen so far( espeacialy about that apsolone) others just skip over that
that was sure lotta fun, thanks for making it accessible!
This is incredible work- thank you for this!
Thank you for this excellent explanation! I finally understand!
'... and that's it, we're done!'
Brutal, absolutely brutal! Many, many thanks - great lesson!
que buen video, estoy empezando a estudiar este tema y estaba muy perdido. Una maravilla tu explicacion, saludos desde Argentina!
I enjoy correlation of history to physics and math. Very good!
I like the brief history about how it's developed, thanks so much
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!