Lagrangian Mechanics - Lesson 1: Deriving the Euler-Lagrange Equation & Introduction
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- Опубликовано: 3 окт 2024
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Lesson Description:
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In this lesson, the basic tenants and principles of Lagrangian mechanics are explored. In particular, we establish the fundamental backbone for this branch of physics: Calculus of Variations. This is accomplished by first deriving the Euler-Lagrange Equation. Then, it's used to prove that the shortest distance between two co-planar points is a straight line connecting them.
Before proceeding to subsequent lessons, it is ABSOLUTELY ESSENTIAL that you understand the concepts conveyed in this video. Remember, physics is like the philosophy of life translated into the language of mathematics. What do you think will happen if you can't speak the mathematical language essential for explaining it? You'll ultimately struggle to express your ideas and do the physics. Well, Calculus of Variations is the language fueling Lagrangian Mechanics, so it's imperative that you understand it before proceeding. Otherwise, you won't be able to convey your ideas and will struggle with concepts!
If you need any assistance with this information, feel free to check out our services or send me an email:
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ed@thekaizeneffect.com
Discussed Topics:
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Introduction to Concepts 0:17
The Principle of Least Action 7:18
Deriving the Euler-Lagrange Equation 12:52
Example - Proving the Shortest Path Between 2 Points 50:21
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This was so helpful thanks dude!
I stumbled upon this video in trying to solve the brachistochrone problem, I was having so much trouble because I kept setting up this integral whose value depended on which function I was integrating but it was not all clear which function was the correct one to integrate. i was so puzzled upon thinking of a way to solve it. The trick with the epsilon causing a new function to fluctuate around the correct path is ingenius. This truly is a beautiful piece of mathematics, a window into nature's soul, revealing her elegance and simplicity. Thank you for making this so accessible and clear and keep up the good work!!
That's quite an eloquent way to put it; I agree wholeheartedly and really enjoyed reading your comment :)!!! Thanks so much for your support, and I'm glad that I was able to help you grasp the beauty inherent in these concepts. If you're still working on the Brachistochrone problem, we've also put together a video on that topic; maybe it can further help you with that: ruclips.net/video/UI0FE8NhU7U/видео.html .
Wolowitz is this you? Dude, you're awesome.
haha ... well then he's a better teacher and/ or speaker than our nerdy sheldon ...
Thank you very much for this lesson.
The enthousiasm and feeling for a clear explanation is very impressive.
I am a just retired senior and try to study physics a little now.
However known with math for mech eng, the first thing i realized was the necessity to become familliar with
Euler Lagrange. Your explanation gave me a much better understanding of this framework.
Thanks!
Frans
Hey, that's super awesome,and I'm glad that you've chosen to keep learning and growing - even as you enter "retirement". Keep it up, and let me know if there's any way I can help with anything; learning and soaking up knowledge is actually a really great way to stay young and engaged :). Thanks for your comment and support!
Ok how don’t you have billions of views? You’re actually passionate about learning unlike professors who just want to get to the point and get the job done.
Wow, thanks for the compliments; that means a lot to me :)!!! I really enjoy helping students feel the same sense of wonder that I once felt while learning these subjects. If you want to help us get more viewers, please feel free to subscribe and hit that notification bell. Thanks for your support!!!
As a retiree,finally finding time in life to start learning beautiful maths from a young man like you is very gratifying!
You are tearing it up kid!
Bravo!
Well, first and foremost, congratulations on retiring; that's super awesome!!! Secondly, thanks so much for your support, kind words, and excellent question (other comment); it really means the world to me. Your time is super precious, and I'm honored that you find my content worthwhile enough to invest that time into. It's also quite beautiful to find so many retired people choosing to continue learning - especially by choice. That's a true testament to a human's insatiable hunger to understand his/her surroundings and mental capacity. Take care, my friend!
@@TheKaizenEffect Thank you for this video. I am also retired and am studying math beyond the introductory calculus course that was the limit of my undergraduate experience. Your thoroughness is amazingly easy for me to appreciate and follow. This calculus of variations is an eye-opener. I can see why you were thrilled by it when you first saw it. Your enthusiasm is infectious. You did a great job. I will be re-visiting your video until I can derive the Euler-Lagrange and the y=mx+b example without assistance. I will then look at your other videos. I do not believe that you can do better than you did here. But I will be happy if I am wrong.
@@julianbeckford2409 Hey, thanks for the kind words and the opportunity to teach you in your retirement; that means the world to me :) !!! I'm usually very busy, but, if you have any suggestions or need help, please feel free to reach out to me. Also, glad to hear that retirement has given you extra time to look more into your interests; you have my respect for choosing to use it wisely. Take care, and hope to see you on other videos!!!
Definitely the best introduction to Calculus of Variation
Thanks so much for checking out the video and your kind words! I'm glad to hear that it was of some use to you :).
I just watched this video for pure recreation, but it was worth the hour I spent on it. Nice work!
Awesome, I'm glad that you enjoyed the video, Joaquim. I always worry that longer videos will struggle to keep someone's attention, but it sounds like this one made it work. Thanks for your comment and support :)!
Great Video man..U almost made me feel the Lagrange's Equation.
Haha, I try to make my videos filled with as much energy and passion as possible. Thanks for the comment, and hope it helped you learn the material!
Had an 'aha!' moment when you said computing the derivatives is the only hard part, and substituting those derivatives in the Euler-Lagrange equation simply spit out equation of a line, that was truly amazing.
It has been few years I've been away from analytical maths(lost touch after college) so some of the concepts in the video were hard to understand, but like you said I'll have to go over this lecture again(maybe a few times :) to understand it completely.
What I liked best about the video is that you first spent time explaining the intuition behind converting the problem (from, say cartesian space) to a more abstract space (that is how I understand logarithms, and fourier transforms too btw) and finding the solution there. Coming back to our regular space requires just plugging the derivatives into the euler lagrange equation.
Explaining such philosophy greatly helps put things in context as it answers what question really we are trying to answer. Following the content becomes much easier then. If you are not doing that in other videos, please do so :)
TBH I was a bit skeptical about following through the whole video because I'd like to think I understand problems best when there is a geometrical intuition behind it, and have run away from any kind of analytical math all my life (even though I understood geometrical intuition behind complex calculus operations I'd still fail terribly in exams because I have a problem stating the questions analytically). This video helped me change my perspective. I wish I had teachers like you in college.
Thank you for the wonderful lecture, man!
Wow, thank you so much for the kind words and the excellent feedback. Seriously, that is MUCH appreciated and helps me out a lot. Awhile ago, I was wondering why so many people enjoy this video, and I think you nailed it right on the head; having a philosophical discussion prior to delving into the mathematics really puts things into perspective. It's a shame that, sometimes, this is neglected in a traditional classroom, and some textbooks seem like cookbooks when they don't provide a lot of perspective. So, I will definitely take that into account for future videos; I actually enjoy the philosophical elements of the material, and it's a ton of fun to talk about.Thanks again, and I hope you have an awesome night :D!
Just a small note. They did in fact have the algorithms to solve the differential equations for a long time. Euler developed a method in fact, called Euler's method. Modernly however you can solve those methods with computers and do it very quickly which is a game changer.
That's definitely true! Yeah, most of the mathematicians back in the day were WAY ahead of their times, especially Euler and Lagrange. Unfortunately, it was not until computers came around that we were able to unleash these techniques and utilize them to the fullest. If the greats could have used things such as the Finite Element Method and Finite Volume Method back in the day, I can't even imagine what else they'd be able to come up with; computers and advanced solution algorithms really opened the doors to a lot of possibilities. Thanks for the comment!
Ed, I am an old geezer that enjoys math, and have always struggled with calculus of variations, and your video cleared a lot of things up. thanks!
Hey Brayton, I'm glad that the video helped to clear up any confusion you had :)!!! Also, the beginning of your message made me laugh lol. You're never too old to learn something new, right? Keep going!!!
I can't find more than 4 videos on this topic that you did, and would like to see the entire course. what got me started was the brachistochrone problem, which I wanted to be able to derive and then explore. Do you have more videos?
Hey Brayton, glad to see you around again, and thanks for checking out all of our videos on this topic :)! Actually, I just finished up a full-fledged, extended Udemy course on this topic - which consists of 18.5+ hours of video and 79+ lectures (along with quizzes and using this content to develop physics simulations). If interested, I should be releasing it very soon and will have more information shortly. You'll definitely see a video trailer for the course on this RUclips channel when it's released :). Thanks again for your support, and I hope to see you in the course if interested!
I tried to understand the Lagrangian through several avenues, none made clear to me. But your explanation of Lagrangian made it very clear. I could see how beautiful the Lagrangian is, working of universe is captured in one simple equation. Thank you very much. Keep up the good work. You have become one of my favorite physics teacher on U tube.
Please can we have more videos on using Lagrangian/Hamiltonian on Quantum mechanics (Dirac's work)
Hey Sanjay, thanks sooo much both for your support and kind words; it really means a lot to me :)!!! I'm also glad that our videos helped to steer you in the right direction. If you want more Lagrangian Mechanics videos, I've developed a Udemy course which expands on these ideas (more examples, using it in physics simulations, etc) - which you might find intriguing. If you're not interested in purchasing a course, no problem either :); you're more than welcome to stay on our free RUclips, and I'd be really glad to continue connecting with you here.
Right now, there are currently no more plans for Lagrangian material (aside from finishing the Kepler's Laws series), but there will definitely be more physics/math videos!!! I'll also take your suggestions into consideration. Right now, I'm trying to focus on 2 more Udemy courses - which consume a lot of my time. However, you should see new material almost every month. Thanks again for your support and happy learning :)!!!!
My lecturer and uni hasn't cared or helped at all this year and I've really struggled with my course. I have an exam in 2 days and super stressed. But this video has helped so much and been so much more useful than anything my lecturer has done all semester! Amazing video, thank you so much!
Hey man, I'm super glad to hear that this video has helped you out so much; that's awesome!! Best of luck on your exam, and let me know how it goes!!
This was a really great video. I really had an "aha" moment when I finally understood it
VERY HELPFUL .. KEEP MAKING HARD PHYSICS VIDEOS LIKE THIS. IT HELPS A LOT
You are a superb teacher because you are obviously deeply concerned that the student should understand every step. Thank you!
Thank you for the kind words, Alan; I really do enjoy teaching and guiding students through the same journey in physics I once had to walk. Hope to see you around in the comments section in the future; please subscribe and hit the notification bell, if interested. Take care!!
I have watched 3 videos on intro to Euler-Lagrangian mechanics and yours by far is the best. Of course I am going to rewatch it until it get it perfectly understood.
I am tempted to try some other coordinate systems and geometries now - such as a torus, sphere and saddle.
Oh and most of the other people I will probably go back and watch them with the understanding you gave me, but the different thing is I am subscribing to you. The way you approach understanding is simply outstanding.
EDIT: lol, my temptation is your next video. Brilliant man.
Hey, thanks for the kind words; I'm glad that my videos helped you understand the material so thoroughly :)!! If you like my videos, I'm currently working on finishing up a full-blown Lagrangian Mechanics course through Udemy. and can let you know once it's finished. Thanks so much for your support!
I am looking forward to those videos. As you can see from my avatar, I am a fan of KSP. I wanted to join the modding community that is working on N-Body orbital mechanics, but they were using Hamiltonian Mechanics and thus discovered my old Newtonian Kinematics were not going to cut it. So I am climbing the Euler-Lagrange pyramid to get to Hamilton.
Ohhhh, sweeeeet; that's super awesome, actually! Yeah, I'm definitely not a video game developer at all, but the course has some simulation examples alongside the Lagrangian Mechanics. So, it sounds like that may be of some use to you. I believe that I run through ~5 example problems and show you how to write programs to visually simulate the motion. At one point, I think that some crude physics engines for video games/animation used similar approaches to what my course shows. Hamiltonian Mechanics is definitely the way to go for that stuff though; it's much more efficient and tailored for easy computational solutions.
Yeah its a bit of a large bite to chew through. They are using Runge-Kutta-Nyström interators and so its one of those goals with a distant end point. Even if I don't get there, the journey will be worth it.
Hey Kurt, how's the journey through Lagrangian Mechanics treating you? It's been awhile since we discussed your project, but I hope you're making some awesome progress! If you're still interested, we'll be releasing the long-awaited Udemy course this upcoming Wednesday, May 24th and offering all supporters a limited 20% discount. More details on how to get it will be provided in our course trailer, which is set to release on this channel in the morning. Thanks again for your continued support, and I look forward to working with you inside the course or on future RUclips videos :)!
This is what i cant help but say after finsishing watching your explanation,"Dude,you are awesome!.You actually got me do some calculus solving ,its hard to make someone who despises a subject into getting interested,curious about it.But you managed to pull it off on me."
Hey man, thanks so much for your kind words and support; that means the
world to me :)!!! Also, I'm glad that my content has inspired you to
both pursue calculus and do it effectively. To be honest, I think that
most people don't necessarily despise calculus/math; they just dislike
the way it's presented in-class and through standard material. In
reality, physics and math are both very intriguing subjects, but the
beauty inherent in the content only becomes apparent if someone teaches
it in an exuberant, passionate manner. That's what I strive to do in all
of my videos; so, I'm glad that I was able to awaken your interests
through my content :). If you ever have any questions, feel free to
reach out to me, and I'd love to further assist with anything. Also,
we've sort of postponed posting videos for the time being, as I'm
currently developing a full-fledged online course. Hopefully that will
be completed and released sometime soon so I can start posting free
content again. Take care and continue exploring!!
Honestly you should consider teaching..You have a gift of explaining things very well..not many people have it...I wish you all the best and thanks for the amazing lecture!(:
Thanks for the compliment, buddy; I try to help you guys as best as I can, and I really appreciate it :)! In the past, I have taught a few university classes and will be releasing my first Udemy course on May 24th. If interested, feel free to check out our social media for the release; however, you are more than welcome to just continue watching our free content :). Thanks again for your support, and I'm glad the lecture seems to have helped you!
Your work is inspiring, man.
Thanks, man! That which is inspiring had to have some kind of inspiration beforehand though, right? So, thanks for that; hope all is well.
I know a teacher when I see one :) You're good.
Thanks for the kind words and support, Andre. I hope the video helped to clear up any confusion you had with this material :)!!!
Well, you just blew my mind. I seriously can´t thank you enough. I'm learning this in Classical Mechanics and I was having some difficult to understand. I am totally recommending this to my classmates! Thanks once more, I will keep watching your next videos!
Amazing explanation!!!
Tanks for your kind words, Sunny; I'm glad that the video was of use to you :)!!
Lagrange lived and worked mostly in the 18th century, the 1700's. For example his attempt to solve the 3 body orbital mechanics problem, led to his discovery of the 6 stable points in any 2 body orbital system. His publication in the 1760's of the paper derived and described what we now call the Lagrange points in orbital mechanics, and which JWST is happily exploiting now.
Spectacular video!
I am 14 years old (turning 15 this February) I try to learn a new field in Mathematics every week, I am really interested in Calculus of variations and when I found about it I started doing a lot of research. I was struggling to understand functionals, but this video made it tremendously clear and now I’m trying to learn tensors. What do I need to strengthen, to get into MIT?
Awesome, thanks for checking out the video and your kind words :)!! If you're studying tensors and calculus of variations at that age, you're definitely way ahead of the curve; keep going!!! However, I'd recommend focusing on the math / physics / whatever intrigues you - as opposed to "how to get into college x, y, or z". If you're talented at what you do and love it, your work will find a way to ensure your success and status. What's more important is what you do while at college, as opposed to the college you attend. It sounds like you'll be intrigued wherever you go though, which is great. You could try to look into uses of these mathematical tools (Lagrangian Mechanics in physics is a direct application of calculus of variations) or perhaps study Linear Algebra / Complex Analysis. There's always something more to learn - regardless of how old you get lol. Hope that helps, and keep moving forward; you've got a bright future ahead of you.
Insanely well done! Of all the explanations of this topic, you, by far, have given the best.
Hey Luke, I'm glad to hear that my explanation has helped you understand the material :). Thanks for your support and kind words!
Not well done at all. A fine, intelligent, explanation ruined by incompetent, amateur presentation.
Given the quality of the material, this was a fine rehearsal that should never have seen the light of day. Take it down and do it over, without the echo and with a usable camera. Puh-leeze!
Thank you so much for uploading the video. I am taking a senior level PDE class which is covering these stuff. It clears many things.
Any time, my friend; I'm glad that the video helped you understand stuff better :)!
The Kaizen Effect You are welcome. I am going to see the remaining videos by tonight on Calculus of Variation.
Awesome, I'm also finalizing a Lagrangian Mechanics course on Udemy at the moment, which has a huge calculus of variations section within it. Maybe that will be of some use to you once it's released :)
Thank you so much
Thank a lot for this video!! What a great satisfaction when one can finally understand these concepts!!
such a great explanation of lagarangian mechanics u did it with a great accuracy..i love this lecture it helped me alot
Awesome, it means the world to me to know that this has helped you through the learning process. Thanks so much for the kind words and support :), and best of luck with your studies :)!
Thank you this was really easy to follow
No problem, I'm glad that the video was of some use to you; thanks for the kind words :)!!!
Cheers from Poland YOU ROCK ! :)
Haha, thanks, my Polish brother!!! I'm glad that you liked the video and appreciate the support :)!!!
You're better at explaining and deriving than the MIT prof on open course with his hands in his pockets! Keep up the good work.....kinda remind me of L Susskind where he let's you see the big picture. Others get so mired that you can't see the forest for the trees. If you know what I mean.
What a great lecture. Thanks a lot, you've made my life so much easier.
Any time, my friend; I'm glad that the video was of use to you :)!!! Thanks for your support!
Thank you very much sir. I learnt lot from this. Keep making these videos.
Thanks for the kind words, Samartha; greatly appreciate it and glad that the video helped you out :)!! If you'd ever be interested in taking a full-fledged Udemy course, I'm launching a classical physics course this Wednesday, May 24th and would love to see you there.
I paused at 9:50 to think about the formula. I guess people would say I am over complicating it, but If θ is the angle of slope with respect to the x axis, tanθ=dy/dx, and you can use the trig identity sec^2=1+tan^2 to get this formula, as secθ=ds/dx
This is excellent stuff. My final exam from Analytical Mechanics is in about a week, so I am so glad I found this channel. Could you recommend any text books that present lagrangian mechanics is such a comprehensive manner? Thanks again for the video.
Hello Kristian, I'm glad that our videos have been of use to you :)! If you want an applicable textbook, I'd highly recommend "Classical Mechanics" by John Taylor. Also, I just recently launched a Udemy course on the subject, which would definitely help with your final (we have quizzes there you could practice with too); feel free to check it out at udemy.thekaizeneffect.com/. However, we do have a few free videos on our channel that would be helpful for studying purposes too. Let me know if there's any way I can help you in preparing for the exam, and thanks for your support :)!
The Kaizen Effect thanks for the reply! I will definitely take a look at that book. I also enrolled in your course, so I am looking forward to going through that as well.
Awesome, I'm glad to have you on-board, Kristian :)! If you have any questions regarding the course/textbook, please reach out to me, and I'll do my best to help as much as possible. In the course, there's a dedicated Q&A section where I'll be able to guide you, and you can also discuss topics with other enrolled students. I can also try to help out with any topics beyond the course which you'll be tested on. Let's make sure you ace this exam while having fun studying it!
This just blew my mind!
Thank you!
Haha, awesome; I'm glad you enjoyed it, and thanks for the support, Jesse :)!!!
You're a great teacher. But a few questions if you don't mind. At 1:02:59 you have said S may be function of y[x], y'[x] and x...but without an explanation of why this is so. The reason I am asking this is because I've seen another derivation, Euler's, whereby it explains these variation explicitly in terms of the variation from an 'equilibrium position' by showing the changes in y , 'y[x] and x on the 'curve'...if you get what I mean. This derivation just seems to be Lagrange's generalization of Euler's original one. Please correct me if I'm in error...just trying to understand it all, and thanks.
Appreciate the work you put in to explain such complex priblems
Hey Debendra, thanks for your kind words and support; I hope that I was able to help you with learning the material :)!!!
Amazing!!! Love your enthusiasm
Awesome, glad that you enjoyed the video :). If you're interested in seeing more content, please make sure you subscribe and hit that notification bell :)!!! Hope to see you in the comments section again
Just a small question, why exactly do we consider S to be a function of three variables, [y(x), y'(x),x]? Can't the formula for the integral be expressed simply in terms of either y'(x) or y(x) since one can be used to find the other?
Keep up the great work man
Thanks, brother; hope the video helped you learn the material. If you want more videos, make sure you subscribe to the channel :)!!
may god bless you with more money more success really hats off
Well geeee, thanks haha :)!!! Hopefully the video was able to help you understand these concepts, and I also wish you the best of success too. If there's anything I can do in helping you reach that, please do not hesitate to reach out :).
I have a question: in y'=sqrt(A/(1-A))=m, Isn't there any problem about that m can be a complex number? I swear A can be a certain type of complex in order to obtain a real m.
I recently had an answer to what elements of A must to be taken in order to have a real m, and that is 0
This class was just amazing. Thank you very much and congratulations.
Thanks for your support Andre, and it makes my day to hear that the class helped you out. In the near future, I'll be releasing an updated, drastically extended version of this course over on Udemy (20+ hours of video content with problems sets, example problems, and simulation applications). Hope to see you there too :)!
It help me a lot, I do not have words to express how you help me. I am from Brazil and I graduated in a very poor quality university and I am unemployed since September. A lot of bad things happened but I was able to join the best university in Brazil to attend the master's degree. The classes will start in February but I am study from now because there are a lot of things that I have no idea, like calculus of variations and stuffs like that. Right now I am struggle with a question about Optimal Design of Euler-Bernoulli Columns, I almost get it, but it is very difficult for me.
You can be sure that I will attend the other classes.
Forgive me for my English and thank you again.
Actually the issue I'm having difficulty is about Optimum Vibrating Euler-Bernoulli Beams, from the book of Haftka. If you or anybody want talk about this I write more details.
I get it. I understood. And again thank you for this.
If still interested, we'll be releasing the long-awaited Udemy course this upcoming Wednesday, May 24th and offering all supporters a limited 20% discount. More details on how to get it will be provided in our course trailer, which is set to release on this channel in the morning. Thanks again for your continued support, and I look forward to working with you inside the course or on future RUclips videos :)!
Lagrange makes the formulation of the equation of motion much simpler and frankly elegant, but be careful it does not simplify the solution itself you still have to solve the differential equation that you would come up with the Newtonian formulation.
The piece of Calc 3 used around 36 min is partial derivative chain rule
Please please please keep uploading videos! You explain it way too good ;)
Thanks for the support and appreciation; that means a lot to me :)! Right now, I'm putting the finishing touches on a full-fledged Udemy course, and I'll start releasing new videos once that's wrapped up. TBH, I'm really excited to get back to a regular RUclips release schedule; hope to see you there when the time comes :)!
wow that's amazing! Reaally happy to hear that! Can I also have a link to the Udemy course? :)
Definitely!!! Once it's released, I can personally send you over a link. Alternatively, I'll be posting more information on our website (www.thekaizeneffect.com/) and social media (ie both our Facebook page: facebook.com/thekaizeneffect/ and Facebook group: facebook.com/groups/SharpenYourMind/) once it's available. Thanks so much for your interest :), and I really look forward to working with you inside the course once it launches :)!!!
Thanks! Can't wait to check it out. If it's not too much to ask, could you send me a link to my personal email once the course gets launched? My email is: alvaronebreda@hotmail.com Thanks a lot!
Absolutely beautiful! The only thing which is kinda still unclear to me is that when you differentiate a function containing y prime wrt y, why should it equal zero? I mean isn't y prime itself a function of y?
Thanks for the comment :)! If you have some equation that is SOLELY a function of y' (and has no y term in the expression), then the PARTIAL derivative with respect to y will always be zero. When you take a partial derivative, you hold every variable constant and vary that one variable which you're differentiating in terms of. So, in that case, the y' is simply a constant, because it is a function of the independent variable only (x in this case...so, you let x = constnat, and y'(constant) = constant). If you find a case in which y' is also a function of y, however, then it may not necessarily be zero though, because that function may change as you vary y (this would be a very rare case though...can't think of an example of this myself). What portion of the lesson triggered this question though? If you are referring to 43:22 , note that the term containing dL/dY' is zero because of eta(x) at the endpoints (eta(x1) = eta(x2) = 0) Hope that helps, and let me know if you have any other questions.
24:03 Aren't the paths that go backwards for a bit illegitimate? These are supposed to be functions of x if I understood this correctly? Apart from that, I found the video incredibly helpful!
Hey Leonard, thanks for your kind words and support. If the curves are, indeed, solely functions of x, then you're correct; they technically would not be legitimate (that wouldn't make it a function anymore!). I suppose that the figure used is sort of extrapolated away from the mathematics, and I'm using it to drive the point that we're considering as many lines connecting those two points as possible. If it were a function of time (and we were integrating over a different variable), then it would be possible / legitimate.
absolutely awesome,, if i apply what you tought in your video ,,,then what i saw is the shortest path to understanding eulers equation....
sir hope you make more such videos
with regards
Awesome, I'm glad that I can convey the concepts and ideas in a way that's easy for you to understand :)! I will most definitely make more videos like this. In fact, I've been developing a full-blown course which I hope to launch very soon.
thanks a lot sir!! sir will be waiting for that !!
Hey umdbest001, if you're still interested, we'll be releasing the long-awaited Udemy course this upcoming Wednesday, May 24th and offering all supporters a limited 20% discount. More details on how to get it will be provided in our course trailer, which is set to release on this channel in the morning. Thanks again for your continued support, and I look forward to working with you inside the course or on future RUclips videos :)!
Excellent teacher
Thanks, buddy; I'm glad that the lecture was of some use to you :)!!!
What is this energy efficiency you are referring to? I don't understand. Thank you.
please make a video on lagrang equation for non-holonomic systems.
you are a genius
Great video. Intuitively explains Calculus of variations. Thank you very much.
awesome, I'm glad the video helped you understand Calculus of Variations, buddy :)! Thanks so much for your kind words and support too!
hy man, you will be a good profesor ,pero tienes que explicar algo a los que no sabemos nada debes explicar que es trajectoria en el contexto de newton mecanics y y que significa minimizar esa trayectory y otra que es el principio de how the systems evolve in time what is the principle of minimun action que hace que los systemas de cualquier clase al final siempre ocupan los estados estables de menos energia
31:52 It is not clear what you are differentiating with respect to here. Shouldn't you consider Y as afunction of two variables, x and epsilon and take the partial derivative of Y wrt x?
Setting dy/dx = 0 finds the points where y(x) = 0. This may be either a local minima OR maxima. It does not always find a minimum point. dy/dx is slope at a point on f(x)= y
Agreed, it finds the extrema of the function, and this is still the case with the functional; thanks for highlighting that and checking out the video :)
Thanks a lot for such a lesson .. I really appreciate your efforts.
Nice video! You made it easier to understand it, thanks a lot!
I have only one question: you found the *function* with minimum length which conects two points. But is there any analogous formulation that finds the *curve* with minimum length which conects two points? Since the graph of every continuous function is a curve, but not all curves are obtained by the graph of some function, that would be an extension of that construction and would provide a more general approach.
Awesome, I'm glad that the video ended up helping you out, and thanks for the kind words :). To be honest, I don't fully understand what you're asking lol, but, if you're asking how to find the minimum length between points on any curve, I actually made a video about that already (it's the next lesson...they're called geodesics: ruclips.net/video/r16GPW6g-SI/видео.html ). If that doesn't answer your question, try to rephrase or explain what you're looking for a bit more clearly, and I'll try to help you out as best as I can. Cheers :)!!!
excellent approach. Been looking for this for ages. This is not taught in a physics degree, at least not to me. It is assumed.
Thanks for the kind words, and I'm glad it was of some use to you :)!!! If you want more content, make sure you subscribe and hit that notification bell!!!
awesome job, a real gem and pleasure to watch, one question
at 31:50 can not follow how you redefine the derivative of y over e
Thanks so much for the support and kind words; it means the world to me :)! I hope the video helped you learn the material!
Awesome video, thanks
Thanks for the kind words and support; hope the video helped!!! Also, keep playing the guitar, chief :) (I'm a guitarist too)
Great video!
Thanks, man! I'm so happy that you enjoyed the video and seem to have gotten something out of it; that means the world to me :)!
Great video ... But what happens when A is greater than 1 ?
If you're referring to the portion at the end of this video, it's important to note that A is already defined but unknown. In other words, it's defined such that the curve passes through both points, and our values at those points (ie y(x1) and y(x2)) allow us to determine that unknown. So, in that context, A can NOT by greater than 1, because that would give us an imaginary number and, hence, an imaginary slope. In that scenario, how would we connect the two points, and, further, how would we guarantee that the connection is the shortest path (which is our purpose in the first place, right?)? We wouldn't! So, in that case (when A > 1), there is actually no solution, and we wouldn't consider it at all.
When you approach situations like this, however, it's important to let the math guide you - as opposed to testing out each scenario. That's why I took that expression of constants and redefined everything as a new constant C - which allows the math and our boundary values to dictate what the constants are. Whenever you run into expressions composed of unknown constants like this, I'd highly recommend condensing them into a new constant; that's the most efficient safe approach possible.
If you want, you can also plug the expression with the A's in and solve for A using y(x1) and y(x2). You'll find that what I'm asserting above is true; A will be defined such that it is less than 1 (and, hence, allows the slope to be a real number - not an imaginary number). It'd be grossly inefficient haha, but it'll prove what you're looking to reason through.
Hope that helped clear up your confusion, and great question!
The Kaizen Effect Thanks ! I figured it out after watching the second video where Z'^2/(R^2 + Z'^2) = C^2 = A. A has to be less than 1.
I was faced with the first problem in my general relativity book: geodesics on the surface of a cylinder. I needed a refresher on the Euler-Lagrange equations, especially as applied to geodesics. This video was very helpful, although for my level it could have been condensed somewhat.
Awesome, glad that it helped you refresh your memory :). Yeah, this video was definitely made for beginners to Lagrangian Mechanics.
@ 37:00 You have the partial of Y' with respect to epsilon, in order to do that you have to know what Y' is, You said Y' is the derivative of Y with respect to x, how do you know it is with to respect to x ?
Great question; thanks for the comment! If you go to around 32:00 in the video, I lay out the framework both for Y and Y' - which might help you with your question. However, I'll try to explain what I think you're having an issue with here. When I develop the framework, I note that Y is an altered version of y - which is treated as the EXACT solution. So, in a sense, both Y and y are functions of x; they yield the y-values to the curve for a given x-value. However, we must recall what Y was conceived for. It was developed such that we have an independent variable (epsilon) directly tied to the functional (or the curve's length). That's why I treat it as Y(epsilon) and differentiate the functional in terms of epsilon; we want to fluctuate epsilon to see how it affects the functional. In actuality, I probably should have defined it as Y(x, epsilon) - as it is technically dependent on two variables (we need to define x and epsilon first to obtain Y, right?). Now, going back to the framework at 32:00 of the video, we chose to differentiate in terms of x, because we wanted Y' to replace y'(x) during our derivation. Differentiating in terms of epsilon would not achieve that goal. So, the moral of the story is that Y and Y' are emulating the framework for y and y', and they must be treated in the same fashion; failing to do so would invalidate the proof and our assumptions. Hope that helps :)!
My first question TKE : Which is the most "energy efficient configuration" for the cosmos to be in, the "Black Death" configuration of permanent expansion, or the "Big Crunch" configuration" of the "eventual collapse"?
You seem to be implying that the former is "preferred" by the cosmos rather than the latter.
Perhaps I am not correct in this though!
;
(1 of 2) Wow, that's quite a mind-boggling topic to think about; EXCELLENT question!!! Unfortunately, I don't think I can give you the "correct" answer (I don't know!), but I can certainly ponder about it with you :). My contention is that BOTH configurations (ie expansion and contraction) can potentially be the most "energy efficient configuration". In the end, it really depends on the circumstances and what's going on, but, for whatever reason, the current configuration we reside in right now is expansion. I think that the evolution of stars/suns is a great example of this idea. For example, certain stars tend to fall into a contraction configuration (ie they continue to compact matter as densely as possible until they form a black hole), but there are others which become supernovas and move towards an expansion configuration (ie they literally blow up!). These two scenarios would insinuate that each configuration can POTENTIALLY be the most energy efficient configuration, but it all comes down to the current circumstances and what's going on. This obviously then begs a ton of other questions: why do they fall into two separate scenarios; does the universe as a whole operate in the same manner (probably not!); ow can we predict what's going to happen?
(2 of 2) From a scientific perspective, we do know that the universe evolves towards disorder though. In other words, if we use the Second Law of Thermodynamics, this means that systems will evolve such that their total entropy ALWAYS increases. We're not sure EXACTLY why this is the case (or at least I believe we haven't figured this out yet), but all observed behavior has always upheld this behavior. That means that whatever configuration the universe / a star will adopt is dictated by how it will affect the system's entropy. This applies to everything in the universe. So, in that sense, I'd assert that BOTH configurations are the most "energy efficient configuration", but which one prevails is tied to how it'll affect entropy.
I had a ton of fun thinking about that (Thanks, and hope you did too!), but, from a practical standpoint, here's all you'd need to really know. Smaller, compact systems (such as anything we'd analyze on Earth) always move towards disorder, and this is the most "energy efficient configuration" for that system (ie mostly expansive behavior). What Lagrangian Mechanics strives to do is take advantage of this reality and incorporate it into the mathematics. That's why Hamilton's Principle (or the Principle of Least Action) is so POWERFUL and forms the backbone of a ton of ideas/branches of physics. In a sense, it's almost like the backbone of the universe.
Hope that helps, and thanks so much for your support; it means the world to me :)!!!
You said that when you drop an object, it wants to go into the "most energy-efficient configuration." Is that the same as the minimum potential energy configuration? I never heard "energy-efficient" before in physics.
As far as I understand, assuming there is some friction to eventually slow it down, it will end up motionless in a local minimum of the total potential energy, probably down on the floor.
Excellent question! Yeah, strictly speaking, the term "energy-efficient configuration" is definitely not 100% kosher in physics haha, but I think the notion helps to understand how systems evolve and why particles in motion conform to certain paths (which was my goal in using it...I wanted to provide some physical understanding of what's going on when you use Lagrangian Mechanics and employ the Euler-Lagrange equation). As you mentioned though, the object will never reach the "most energy-efficient configuration" possible, since the floor halts motion. Rather, it will meet a local minimum of the total potential energy at the bottom of the floor (definitely agree with that; good clarification!). At the same time, the fact that the object still has potential energy (perhaps relative to another reference point/axis) implies that it still "wants" to be in motion and reach a more energy-efficient configuration. For example, if you removed the floor, it would continue its descent until another local minimum is found (perhaps at the Earth's surface now). This same logic could be applied ad infinitum - which would imply that objects tend towards constant motion (or, perhaps, towards "disorder"/chaos...I'm interpreting that as energy efficient). In fact, this is also why the universe supposedly continues to expand and entropy always increases. Everything seems to be in constant motion, and all objects will move such that the most energy-efficient configuration is obtained (if an electron is excited, it will emit a photon and perhaps jump to a lower orbital; if there is a chemical reaction, "new" compounds may be formed, and heat may be emitted; etc). This would obviously beg a lot of thermodynamic-related questions: why does the universe tend towards disorder (second law of thermodynamics); can we even reach the most energy efficient state in the universe, and what is that; etc? (questions which I don't even know how to approach XD lol)
In the context of what we're discussing though, I think this observation of the universe helps to understand what's being accomplished in Lagrangian Mechanics (think of it like we're scaling the question down to focus just on the analyzed system). According to Hamilton's principle, one may find these paths of motion by minimizing the Lagrangian functional, but why is that true? Where did the Lagrangian even come from? I wanted to provide some physical intuition to explain what's going on and leave my viewers with a fuller understanding of the universe - as opposed to just throwing out the mathematical framework and asserting that "this is how it is" (which, unfortunately, sometimes happens in classes on this subject!). This perspective shows that the path is "chosen" (and the Lagrangian is defined) such that the laws of the universe are met (Newton's Second Law, conservation of energy, conservation of momentum, etc). From a more technical standpoint though, the derivation of the Lagrangian arises directly from the Principle of Virtual Work (red flag for energy, right :) ?) - which asserts similar ideas to what I'm talking about here (check it out; it's some really beautiful stuff!). So, my explanation is more philosophical, as this is more of an introduction of the concepts. If interested, I can try to put out a few videos on the Principle of Virtual Work and all of the technical derivations for the Lagrangian at some point (LOTS of math and boardspace lol), but hopefully that helped you understand the material a bit more while giving you some food for thought :)!
Thanks for bringing this up; I actually enjoyed thinking about it, and it was a really great question to ask (never stop questioning what you hear/read)!
I don't think you can only consider the potential energy. You're right, the object will stop probably on the floor but at that point, its velocity is then zero...so that its kinetic energic is also zero. The way the object has moved led to a minimum of TOTAL energy, the path it took was the one one which minimized the Action. The final state led to the lowest state of energy possible for the system.
Thank you for that MyLord
Thanks for your support, buddy :); hope you enjoyed the video!!!
Good video and explanation, however I don't understand the part after you write Y(e) = y(x) + en(x), surely there's some dependence on x from Y(e)?
how did the eraser drop sliently?
At 32:20, why do you take the derivative of big Y with respect to x instead of with respect to epsilon? Thanks.
Great video btw.
Hey Harris, thanks for the kind words :). In the beginning of the lecture, recall that we were focused on reformulating the underlying mathematical framework. Since we could not differentiate the functional using Calc 101 techniques, we had to redefine y(x) and, hence, y'(x) to incorporate a new independent variable - one that allows us to employ our familiar Calc 101 techniques. So, technically, Y(epsilon) is just a reformulation of y(x), and we should treat it in the same manner. In other words, if the original framework defined y' in terms of x, we must stick with that (the y is just defined more thoroughly here!). Perhaps my notation here was a bit inappropriate and came a little too early. As you'll see after this portion, those Y functions will be differentiated in terms of the epsilon terms (that's why we introduced them, right :)?), and that's what I wanted to emphasize. To make it formal and 100% correct, that function should technically be a function of multiple variables, and that obviously affects whether it's a partial/ordinary derivative. Does what I explained above help to clear up the confusion you had though?
Let me tell u your presentation was really interesting. And though I only know one single variable calculus, I was able to follow the whole of your explanation. But being so calculus limited, I Wonder if Euler's geometric view could allow me a better understanding.
Great video
I'm glad that you enjoyed the video; thanks for supporting our work :) !!!
Best teacher ever. Damn. Soooo good...
Haha, thanks so much for the kind words and support, buddy :)!! Hope the video helped you learn the material
Great video m8.
Thanks for watching and your kind words. Hope to see you in the comments section again soon :)!!!
Sadly, I'm still confused about how to derive the equation. I don't think it's a rigorous proof of the equation, or at least, you didn't clarify some details, e.g. why S is a function of y(x), y'(x) and x and later on why Y doesn't include x as an independent variable (since you are using straight d rather than curved d) and why it's acceptable to change Y to y in the end? Maybe it's because I'm a bit slow in mathematics, but I think it better if you cover such details catering to those like me.
I have the same doubts as you have
Thank you so much , it was very helpful !! Ur really good , thanks a lot.
Hey Evellyn, I'm super glad that the videos helped you out :); Thanks for your support!
whoaaa you are awesome dude
No, you're awesome for seeking out this content and working to understand the universe :). Thanks for the support brother, and I hope you enjoyed the video!!
Hi, man!
Could you explain Euler's development - not Lagrange's - of the Euler-Lagrange Equation?
If you can, I ask you to go step by step.
Thanks a lot!
Hello Sergio, I actually wasn't aware that Euler and Lagrange had two separate developments. From what I understand, they're both the same but were conceived at different times. Actually, I've read that both mathematicians were actually in correspondence regarding this development at some point.
@@TheKaizenEffect . Lagrange just generalized Euler's derivation. Euler developed the calculations from first principles...and is easier to understand...at least as far as I am concerned. Have you seen Euler's derivation ?
What a beautiful equation!
I agree, buddy; it's such a beautiful, clever "trick" :).
Google the following: the original euler's calculus of variations.
How exactly do you take the partial derivatives? I am quite new to multivariable calculus and when I looked up partial derivatives, the examples that were shown didn't resemble the one that of the video so I was unclear on that step.
Well, it depends on the application and what you're trying to differentiate. In this case, it's important to realize both what you're differentiating and what a partial derivative is actually doing. We're differentiating in terms of epsilon, and the function we differentiate is L(y(epsilon), y'(epsilon), x). So, we're trying to figure out how the function L changes when variable epsilon changes (and everything else is held constant). Well, when epsilon changes, y or y' changes, and both of these functions affect L directly. So, we'd have to account for both of those changes, which leads to two separate terms. In this case, those terms end up using the chain rule, which leads to the forms shown in the video.
If you read up on partial derivatives, you'll definitely see many different variations, but they all follow the same principle; you differentiate in terms of a single variable (while holding the others constant). For example, you might have a function z(r, theta). If you want to find dz/dr (these are partials), you would just take dz/dr while treating theta as a constant. Another situation similar to what we have here may be the partial derivative of S(y(t),v(t),t) with respect to t. In that case, the full derivative becomes: dS/dt = dS/dy dy/dt + dS/dv dv/dt + dS/dt; we need to account for the contributions of each dependent/independent variable! Here, the t variable leads to a change in S, but we ignored the x (it is treated as a constant) in our L function - as changes in epsilon do not affect x and, hence, L. Hopefully that helps you understand that a bit better. If you need additional help, please let me know; we offer 1-on-1 skype tutoring sessions, and I can always explain it real-time for you better. Thanks for your question and support!
fantastic
Hi, thanks for your work.
Could you explain with more accuracy the meaning of epsilon.
When you introduce epsilon to build the fonctionnelle, you say that epsilon is a constant...variable.
That makes no sense. If it's a constant, it cannot vary by definition. If it varies, in respect of what does it vary ?
What i think :
a fonctionnelle is a function of functions (Y1, Y2, Y3,....).
So there are Epslion1, Epsilon2, Epsilon3...! which define the fonctionnelle by
Y1 = y(x) + epsilon1 . eta(x) ;
Y2 = y(x) + epsilon2 . eta(x);
....
Each Yi represents a different path from a to b
So it appears that Y is a fonctionnelle which depends of y(x), eta(x) and epslion which is independant of x.
What do you think about it ?
Right, you are; epsilon is NOT a constant! In the context of the video, it is referred to as a "constant" to convey how it differs from the eta(x) term. In other words, eta(x) is a function that allows Y(x) to represent any path between the two points, but epsilon is simply a NUMBER (like a constant). A better way to express this, which I believe is used in the video, is to assert that epsilon is an INDEPENDENT VARIABLE - a quantity that exists in and of itself (as opposed to a function, which is dependent on an independent variable...ie, you must define x first to define the function). This particular term is needed and introduced such that we have the same framework developed in calculus 101. When epsilon changes, that change leads to a direct change in the functional we're trying to minimize, and, hence, we can then express the functional S as a function of epsilon. In essence, it allows us to bypass the fact that functional S can not be defined/varied without first knowing the path variable y(x) - allowing us to utilize the derivative concept as introduced in calc 101. Hopefully that helps to clear up any confusion you had, because understanding this concept is ABSOLUTELY essential (make sure you understand what's going on here)! Thanks for your support and question :)!
Thanks ! I do agree with you.
Any time, chief! Thanks for the support :)
**backhands palm for emphasis**
Great video! The only thing I would suggest is reduce the clapping for emphasis; it’s a bit distracting.
Also when he hits board with marker. Tik tik daz dazz. For the rest , he is good in expressing concepts.
Great job! ... just subscribed ...
Thanks, glad to have you onboard and look forward to interacting with you on other videos!!!
Why Lagrangian is defined to be T-V?, is it just by chance ?. What is the physical intuition behind that integrating over T-V at each segment through t increments?
Hey Zeki Zetbek, great question! At this point in your education, it is important to realize that the quantity L = T - V is selected such that stationary points of the functional yield the correct laws of physics. However, this selection is, indeed, NOT selected by chance. Once you understand the basics of Lagrangian Mechanics, I'd recommend checking out a formulation of physics known as the Method of Virtual Work. According to this formulation, particles/systems will evolve towards the path for which the "virtual work" is zero. In other words, the system will choose that path leading to the most energy efficient configuration, and, hence, these other "virtual displacements" or possible paths do not require additional work by the system. So, by setting these work terms to zero, the formulation is able to yield the correct result. It just so happens that these paths also minimize a system's action (this is a principle always found throughout the universe). Using this formulation, you can then work through the mathematics and physics principles to eventually obtain the Lagrangian. That's how the Lagrangian is truly defined as T - V, but it requires a TON of math and lots of summations. That's why I think it's more appropriate to revisit the idea once a student understands the underlying Lagrangian Mechanics. Perhaps I'll post up a video or two on this subject in the future. In the mean time, I hope that helps, and thanks again for the support :)!
Thank you for your answer. I totally get the Lagrangian Mechanics (euler lagrange equation, minimization of action, proof for the identity, calculus of variations etc.) I have watched several other videos and worked through some books as well. What I wonder is for a particle that is under the effect of uniform gravitational potential, let us say free fall or a regular projectile motion, through mathematical derivations can we end up having the Lagrangian as T-V (not using anything about newtonian formulation, that would be quite trivial)?. I will look for the things you mentioned. Thank you again for your effort.
The Kaizen Effect,dude at 36:50, when Y(E) is differentiated-Epsolon is a constant?But how?
And also,I hated Calculus,despised every minute of being forced to do it.But man,you are awesome.I love the way you teach,thank you a lot.
I believe Engineering Mechanics,Discrete Maths wont feel like pain in the neck now
Thanks for the question, and sorry for the confusion. Someone pointed this out previously, and I posted a clarification which I'll post here. It was really a poor choice of wording on my part. Hopefully that helps you out; let me know if you have any additional questions:
"Right, you are; epsilon is NOT a constant! In the context of the video,
it is referred to as a "constant" to convey how it differs from the
eta(x) term. In other words, eta(x) is a function that allows Y(x) to
represent any path between the two points, but epsilon is simply a
NUMBER (like a constant). A better way to express this, which I believe
is used in the video, is to assert that epsilon is an INDEPENDENT
VARIABLE - a quantity that exists in and of itself (as opposed to a
function, which is dependent on an independent variable...ie, you must
define x first to define the function). This particular term is needed
and introduced such that we have the same framework developed in
calculus 101. When epsilon changes, that change leads to a direct change
in the functional we're trying to minimize, and, hence, we can then
express the functional S as a function of epsilon. In essence, it allows
us to bypass the fact that functional S can not be defined/varied
without first knowing the path variable y(x) - allowing us to utilize
the derivative concept as introduced in calc 101. Hopefully that helps
to clear up any confusion you had, because understanding this concept is
ABSOLUTELY essential (make sure you understand what's going on here)!
Thanks for your support and question :)!"
u r the best
No, you're the best for the kind words :)!! Thanks for checking out our content.
At 29:32,n(x1)=n(x2)=0 is being proposed.
But if in our function Y(x)=y(x)+n(x),if n(x)=n(x1) or n(x)=n(x2) (which is the same thing anyway since both are equal to zero),
if that happens,then wouldnt our line's coordinates be this-
(0,some number) to (0,some other number)-if my proposition is right,then our line would be lying exactly on the Y-axis since its both the x-coordinates are zeroes.
Am I making sense?Are my assumptions right?If not,please explain
Hey, great question, but it sounds like you may be mixing up the intent of eta(x) in the redefinition Y(x) = y(x) +epsilon*eta(x). When the function is redefined as Y(x), eta(x) is used to incorporate a variance from the CORRECT solution y(x) (think of this as the line tracing the shortest path between two points). This variance then allows us to use the same tools introduced in calculus 101 (ie vary an independent variable epsilon to generate a change in the function of interest....or the functional for length in this case). At the same time, we NEED to ensure that eta(x) = 0 at the endpoints x1 and x2. Otherwise, the varied line won't go through the two points it has to. So, when x = x1 or x = x2, eta(x1)=eta(x2) = 0, and either Y(x1) = y(x1) or Y(x2) = y(x2) [if that confuses you, just plug x1 and x2 into Y(x) = y(x) + epsilon*eta(x) and substitute eta(x1)=eta(x2)=0]. In other words, the line will pass through the two original points we're trying to connect. So, the coordinates actually become (x1, y(x1)) and (x2, y(x2)), and the framework guarantees that the line will always connect the two points - even when we vary the proposed solution. Hope that helps to clear up any confusion you had :)!
Thx. enjoyed🍻
Hey Mehrdad, glad to hear that you enjoyed it, and I hope to see you around on other videos. Cheers :)!!!
How does the L you have defined here relate to the Lagrangian defined to be the kinetic energy minus the potential energy?
Check out my other video on this subject: ruclips.net/video/yvzYoD_56bc/видео.html. In short, if you're trying to find a system's equation of motion, then the L term becomes the Lagrangian - as dictated by the principle of least action.
Cheers
thanks how can solve Fy-Fy'x-Fy'yY'
What does energy efficient configuration mean?
Hey, man, thanks a lot for answering.