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 RU-vid no longer allows annotations, I am unable to correct this on the video.
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(?)
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!
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.
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.
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!
This is beautifully explained. I’m an practical engineer - my brain responds very well to understanding the motivation behind the mathematics. Thank you!
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😊
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!
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!
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.
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.
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.
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!
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.
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.
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 !
I definitively need to watch your other videos. Your way of teaching is by far one of the best on RU-vid! 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.
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.
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!
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!
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.
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.
@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.
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
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 .
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.
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.
Good editing, Intuitive and comprehensive. Your voice is soothing. This is the best explanation on Larangian mechanics, no one on RU-vid even comes close.
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
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!
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.
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
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!
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 :)
Great video but maybe there is a mistake at 12:25. In the graph you imply that y(x) + eta n(x) = n(x) which contradicts what you write on the left that y_bar(x) = y(x) + eta n(x). I believe in your graph you meant to write y_bar(x), not n(x) at the top.
@Matthew James I noticed this too, though I think you mean epsilon where you say eta. In other words, _ȳ(x) = y(x) + ε η(x)._ That means that the actual _η(x)_ is an arbitrary shape function (not shown in the diagram) with the constraint that _η(x₁) = η(x₂) = 0,_ resulting in _ȳ(x₁) = y(x₁)_ and _ȳ(x₂) = y(x₂)._ Great video! -- I read a whole book on analytical mechanics and didn't really get it until I watched this.
@@matthewjames7513 , they're all just unicode characters -- y_bar, epsilon, eta, subscript 1 and 2 etc. It's a pain having to look up each one, but the result is worth it for something like this. The final touch is to italicise them which you can do in yt comments by surrounding with underscores. Gotta make sure the underscores are bounded by spaces though or yt screws up, so include any punctuation such as periods within the italics, e.g. ȳ(x) = y(x) + ε η(x). Btw, I noticed that some other treatments do away with the scaling constant _ε_ and replace _η(x)_ with a perturbation function _ε(x)._ For instance, see en.wikipedia.org/wiki/Hamilton%27s_principle#Euler%E2%80%93Lagrange_equations_derived_from_the_action_integral . I'm working through the video to check that would still make sense, as it seems that it would be simpler as long as it works out the same. (EDIT: Elsewhere on Wikipedia it gives the same approach as in the video, e.g. see the "Derivation of the one-dimensional Euler-Lagrange equation" section of en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation . I think perhaps having the _ε η(x)_ formulation allows it to separately specify the constraints that _ε_ is small and _η(x)_ is differentiable. I think I'll stick with that, partly on the basis of "don't mess with stuff you don't understand").
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.
