The Calculus of Variations and the Euler-Lagrange Equation

[@AaronQuitta]

A little more explanation at a slower pace or visual animation instead if stop-motion-style would make your explanation easier to follow, still really enjoyed it!

[@XanderGouws]

Thanks! Yeah, they could use a bit of work haha - but I'm improving step by step.

[@kevinvigi.mathew6350]

agree,slower and more detailed and with more steps

[@mygoogleaccount4404]

@@kevinvigi.mathew6350 its fine if you pause and work some stuff by hand, it would be bad as a real lecture but as a youtube lecture its as good as anything else.

[@jim2376]

Ditto. Speak slower. (Or I can reduce the video speed to 80%.)

[@mikevaldez7684]

​@@jim2376or you could speed up the calculability of your brain 😁

[@ShredEngineerPhD]

This is by far the best introduction to the Euler-Lagrangre formalism I have seen. Thank you! :) It is nice too see a simple introduction to functionals and a full derivation of the equation!

[@mayabaalbaki6904]

This is the best explanation of variable calculus so far on the internet

[@rafaelaassuncao9729]

I think that you need to continue but with longer and more detailed videos this video was to fast

[@XanderGouws]

Ye, I kinda just put all the clips together without really considering pacing. It's something I'll try to work on in the future :)

[@aerodynamico6427]

Haven't you heard of the "Playback Speed" adjustment feature on KZbin videos? Slow it down!

[@hughjones4049]

@@aerodynamico6427 I can also recognise the pause button (or space bar) and the arrow keys to (re)wind backwards and forwards.

[@unflexian]

Your channel is amazing, very well made & interesting videos. Please more!

[@XanderGouws]

Thank you!

[@Abbas950able]

The most simplest introduction ever!!

[@liubo8147]

The only question i think is the prime of j(e=0)=0 implies that the "principal of least action",which means the path will choose the stationary point of j (tangent line), so the prime of j(e=0) is equal to 0

[@Israel2.3.2]

Nice presentation. I'm currently reading the works of Euler and plan on reading Lagrange's Analytical Mechanics after consuming Euler's major physical works. Hoping to understand Euler-Lagrange before 2021.

[@XanderGouws]

Thank you, and best of luck!

[@Israel2.3.2]

For anyone stumbling across this post who is on a similar journey. Kotz is a good text. Goldstine as well if you are interested in the development of the Calculus of Variations from Euler and Lagrange to Hilbert. For applications to analytical mechanics I found Taylors text on Classical Mechanics quite lucid.

[@GonzaloMiguelGS]

Wonderfully Simple! Clear Enough! The Best Music Ever! Thanks A Lot And... Congratulations!

[@XanderGouws]

Thank you so much! I'm glad you enjoyed it :)

[@charleswayne1641]

Another fact I realised, at the actual path (epsilon = 0), Etha(dy/de) is not defined: either (+C) or(-C). Also, at the end points, Etha(dy/de) undefined, but epsilon(e) = 0. So, I make an important correction: E-L equation, after all is a "Necessary Condition" for the Functional to be at the extremum : namely, (dJ/de) = 0.

[@nanocount09]

Great video! It was super clear and intuitive. It could have been a little slower though. I would love to see a series on the calculus of variations. Or at least a few more videos.

[@XanderGouws]

Thanks! Ye, I'll try to focus on pacing for future videos. I do plan to do a couple of example problems in the future, so stay tuned!

[@aerodynamico6427]

Haven't you heard of the "Playback Speed" adjustment feature on KZbin videos? Slow it down!

[@grantwebster888]

Good video - just one question. At the end of the derivation the final version of the equation references y. In the slide before you use y bar. Can you explain why?

[@XanderGouws]

Great question! We define y̅ = y + ε η(x). At 3:27, we mention that ε = 0 is an extremum of J i.e. at 4:54 the LHS, dJ/dε, becomes 0 when ε=0, and since we're setting epsilon to 0, y̅ = y.

[@grantwebster888]

@@XanderGouws Thanks for getting back so quickly. Great answer. I see now where you are coming from. As ε tends to zero y̅ tends to y. I should have spotted that myself. Best video I have seen on the subject.

[@alejandromahillo3536]

Great video despite the fact i had to pause it several times to really get into you were saying. Anyways i really understood this concept and i'm so grateful about that.

[@mathe_ma]

kzbin.info/aero/PL3SiKQGql2rkfgejk1DxmjHXjhWizQWaT

[@XanderGouws]

I'm glad it helped! I'll definitely keep pacing in mind for future videos :p

[@Stelios.Posantzis]

The flow of the exposition is good but showing only one formula at a time on the screen makes this really hard to follow the sequence of steps in the derivation. One needs to keep going back and forth all the time to make sure no tiny nuance in the semantics was missed.

[@stephenhicks826]

Good explanation - pretty fast but helped me a lot thanks.

[@nin10dorox]

Why are both y and y' parameters in F, when y' is determined by y? We can't change y without its derivative also changing, so shouldn't it just be F(x, y)?

[@jasonbroadway8027]

When you say "nowhere", you mean that f does not go up or down? I can work some of the problems in Mary Boas' book, but I found this lecture to be heavy-going.

[@erickjian7025]

since the functional J[ybar] is a function of eta, does that mean the point of E-L equation is minimizing a fucntion(al) J' derivative in terms of eta?

[@mehrdadmohajer3847]

Thx Xander Gouws for the Proof of Langrangian eqution . Very nice done with min. amount of time explaining the procedure how to get there. I may add it seems to me : Although Calculus of Variations is very usefull and efficient Method showing the Proof, represents however : A littel of Eulers application regarding to Subject- Matter. Hope you find it interesting enough to investigate. cheers🍻

[@johnspivack6520]

it would also be good to have more explanation of why there is a total derivative wrt x instead of a partial derivative

[@sydneymakombe3726]

you actually need to do a series of calculus of variation and optimal control theory. Thank you very much

[@charleswayne1641]

One thing about that equation: delta(y) = y - y bar = Epsilon * (dy/de). It establishes the "linearity" of "virtual displacement(delta y)" with respect to "virtual dimension(epsilon)".

[@shuningli7557]

I was confused with what the boundary conditions were, thanks for clarifying this!

[@talllankywhiteboy]

Well made video, but as others have already commented you moved too quickly. Would have liked to see this video at a slower pace with pauses to digest and breakdown what has happened in previous steps.

[@charleswayne1641]

The more I examine it, the more insightful one gets. For example, not only (dy/de) not equal to zero but also (dy/de) = constant (C); Integration by parts, most authors state that Etha(dy/de) = 0 and eliminate/vanish "Etha(2) - Etha(1)".But, Etha is never zero anywhere. So, it's more like "C - C = 0". Here, in the equation : y = y bar + epsilon*(dy/de); when Epsilon = 0, y - y bar is also zero, but, Etha(dy/de) is always "Constant".

[@markgoretsky766]

Excellent job, Xander! Thank you

[@shivangmishra2642]

Beautifully explained...

[@matthewjames7513]

2:55 you mention a "small" change. But eta doesn't actually need to be small. The proof holds for any eta, right?

[@XanderGouws]

Good question - I think you're right. Since we end up taking epsilon to 0, any eta will work as long as eta=zero at the endpoints (and I believe it also has to be differentiable).

[@ritwikchakraborty3319]

You can add that second order derivatives of the functions should be continuous... All time, some students come here to get the information what they have missed...you should remember this... And overall your video is awesome...GOD BLESS YOU...good luck...

[@srallulrich]

On 4:22 du should be dx*(df/dy'), not d/dx, shouldn't it?

[@nothing29717]

Thanks this was helpful

[@mattiperkiomaki5009]

Excellent, excellent video. It was rather clear, yet slowing a bit down wouldn't do no harm. Also, slight elaboration on how to change things using Leibniz rule wouldn't do any harm, for someone may not have used it so much as to have immidiate grasp of what just happened when you changed the function into the other form using it. And I'm not at all asking for full derivation, just the general outlook of how you moved things around, just like with the integrationby parts. Perhaps it could be a part of the "slowing down" for the next videos? Anyway, great job mate, you are doing a huge service for people by publishing these

[@XanderGouws]

Thank you so much! Yeah, I intend to try and make my future videos a lot more reasonable with respect to pace :p I'm glad you enjoyed it :D

[@mattiperkiomaki5009]

@@XanderGouws I did, yes, and I hope you'll keep posting videos on this subject. I'm trying to get a hold of theoretical physics atm, and lagrangians are playing a big part of it, and you are explaining all this in a beautiful way. Thanks again, with best regards, Subscriber

[@aerodynamico6427]

Haven't you heard of the "Playback Speed" adjustment feature on KZbin videos? Slow it down!

[@rahulmukherjee37]

I have not seen so simple yet complete explanation of Euler Lagrange equation and that also in 6 mins But the video was a bit fast a little slow paced would be great

[@beoptimistic5853]

kzbin.info/www/bejne/joKsk6FobMmCoKc 💐👍

[@XanderGouws]

Thank you! Yeah, I definitely tried to improve my pacing in my later videos, and will keep it in mind when I get back to it!

[@Kishan_IIT_Indore]

Loved it.. Love from India bro

[@charleswayne1641]

Does Etha(x) have to be a function of "x"? Or, simply a function of Epsillon suffices?

[@XanderGouws]

Yes, η needs to be a function of x so that it changes in the same space that y changes in. Eta can actually be ANY function as long as η(x0) = η(x1) = 0. I hope this page makes it clear: www.desmos.com/calculator/do9njlzp4t Since desmos has limited character options, I used these letters for these things eta ==> g(x) epsilon ==> a y ==> f(x) y bar ==> d(x)

[@nulliusinverba7732]

Succinct & clear, bravo! By the way, what do you do for the case when the functional is with respect to a probability distribution, like an expectation?

[@XanderGouws]

Thank you! I'm not entirely sure! I think you would solve the Euler-Lagrange equation to get y(x), your probability distribution, dependent on two constants. Perhaps you could use one "initial condition" and then also normalize the function. I don't if that's guaranteed to work though - play around with some stuff, let me know what you get!

[@nulliusinverba7732]

@@XanderGouws I tried solving it by expansion, but I'm not quite sure if I did right with the joint probability of the function and the input. Because, in my head, the probability of one should change with the change of the other, but other people seemed to be solving it by treating it as independent of the functional

[@GaudioWind]

I've been trying to better intuitively understand this equation and related problems for quite some time now. I think what I need now is an example with numerical approximations for the terms of the equation. Like, in the brachistocrone problem, for instance, which would be the numerical value for the terms partial F partial y in an arbitrary point of the curve? I mean, if I have the numerical values of x, y and y' in a couple of near points in the the cycloide curve how would I numerically calculate each term of the Lagrange Equation?

[@XanderGouws]

That's a really good point! It can be quite difficult to understand intuitively what some values really mean. I'll see if I can work that into any future videos I make on the topic.

[@nicolasdellano280]

Only 1 thing I dont really get, why does the functional depends on x f(x) and f'(x)?

[@XanderGouws]

Great question! Because the functional takes in f as its 'input', the value of the functional can depend on any operator acting on f(x) i.e. it can depend on d/dx * f(x), etc. Sorry I took so long to respond, but I hope this helps.

[@honeyboiii]

Amazing video I just have a query. Why doesn't the value of the functional depend on second-order derivatives as well?

[@XanderGouws]

Sorry for the confusion - I believe it actually can depend on higher-order derivatives as well, I just didn't show it in the graphic :p

[@duckymomo7935]

So clear, worked examples are nice Nice proof too

[@shivabalaji6668]

How is it useful to mechanical engineering?

[@meeharbin4205]

this is some good stuff. Its a bit fast but I think you explain things well enough for me to pause and understand what you said. But maybe thats just me already watching videos all day on euler lagrange, and realising that i wasnt misunderstanding when people take an entire function or set of points as an input to a function and consider small changes in the input function.

[@XanderGouws]

I'm glad you liked it, I think that my more recent videos have much better pacing!

[@aerodynamico6427]

Haven't you heard of the "Playback Speed" adjustment feature on KZbin videos? Slow it down!

[@meeharbin4205]

@@aerodynamico6427 ight, good point

[@nightknight2032]

Really, nice! But will you continue Calculus of Variations series? It's not a well documented subject in other math channels.

[@XanderGouws]

I may in the future, but at the moment I don't have enough knowledge to! I encourage you to check out 'Faculty of Khan' here on KZbin. There are also some pretty decent free online textbooks from what I remember.

[@likestomeasurestuff3554]

Very nice! I think it would optimize the video quality though if you talked more slowly and took noticable breaks between major steps in arguments and after important points. Animations are hot though :D

[@XanderGouws]

Thanks for the tips! Yeah, it's definitely something I've tried to work on. I'm glad you enjoyed it though :)

[@Harry-ub2fv]

You got a new student/fan today. Awesome videos!

[@XanderGouws]

Awesome! Thank you!

[@fernandogarciacortez4911]

Such a good video! Now I want to explore this topic ha! Keep these videos coming.

[@mathe_ma]

kzbin.info/aero/PL3SiKQGql2rkfgejk1DxmjHXjhWizQWaT

[@drexflea52]

Dude u r a genius.

[@XanderGouws]

I'm glad you enjoyed the video :)

[@drexflea52]

@@XanderGouws woohoo. U replied. So pleased dude. The video is amazing.

[@XanderGouws]

@@drexflea52 Thank you! Being able to reply to most comments is definitely one of the pros of having a fairly small channel haha

[@drexflea52]

@@XanderGouws But I am sure ur channel gonna grow a lot in coming days. Ur channel gonna grow like some other great and popular mathematics related channel like 3blue1brown etc.

[@drexflea52]

@@XanderGouws dude I'm from India. U r from?

[@menot5039]

that's what i was looking for

[@navilistener]

Thank you for this video, it provides a clear derivation of the Euler-Lagrange Equation. However, to be rigorous, at 4:18 du should be equal to d/dx (∂F/∂ȳ') dx = (∂F/∂ȳ')' dx and not equal to d/dx (∂F/∂ȳ'). What I mean is that du is not the derivative of u, but is instead the differential of u, which, by definition, is given by du = u'(x) dx = (du/dx)dx. Similarly, dv should be η' dx and not η'. Despite these innacuracies in the screen at 4:18, we can see that u, v and du are substituded correctly in the next screen at 4:31, so the proof of the Euler-Lagrange Equation is not compromised. en.wikipedia.org/wiki/Differential_of_a_function#Definition I also have a suggestion regarding the presentation. I think you should add some visual queues in the screens where only expressions appear when reading out loud the contents of the expressions, so that the narrator's voice is followed in the image as well while possibily inserting additional information. For example, in the screen at 4:31, it would help a lot to add some colored brackets below each part of the expression stating each element of the integration by parts formula (u, v and du) and make them appear in the screen as you are reading the formula. Apart from these details, great work with this video! I hope you continue doing more videos :)

[@charleswayne1641]

Xander, I just saw your video. I didn't know that you are still young person. You seem very energetic and had lofty aspirations for your life in general. I suppose you were a so called "whiz Kid". Please, go through your favourite college and get a good formal education, not just in mathematics, but such as historical backgrounds why these things happened.... History is very important in understanding issues in Economics, current world affairs, physics, mathematics etc. For example, the equation you mentioned : y - y(bar) = Epsilon*(dy/de) is intended for the English audiences at the time by Lagrainge. Lagrange was very sensitive to the charged atmosphere at the time, especially surrounding Newton-Leibniz dispute. Actually, dy/de suffices without introducing a new symbol Etha. Etha simply means (dy/dx). The functional's derivative with respect to "virtual displacement: epsilon" is simply (dJ/de).

[@shuningli7557]

Does dy/de refer to the same thing as dy/dx?

[@aaronseldes767]

I don't know much about this but maybe we can derive all subjects from first principles of where every function comes from from first principles meaning trigonometry, calculus and what every aspect of the explanation of the functions i.e trigonometry, calculus differential equations, calculus of variations (which I have nothing to know about) and showing all principles and how they are derived and how they all work together to create reality in mathematical sense and how it relates and connects to all subjects math, physics, biology and basically every imaginable subject and how different fields of math relate to different fields of science physics, chemistry , biology earth sciences and how they connect based on there form , reality, and structure and how all reality is in a sense connected based off of all there splendid characteristics and where everything comes from from theoretically religious secular or science sense and connecting and trying to understand the truth of all reality and how everything is derived mathematically and how to determine shape of functions based off symbol +,- x or divide and how it all relates and connects as a whole. BSD (with G-d's help we will find it.

[@XanderGouws]

That definitely sounds like it could be really cool - I'll try to do something like that in the future :)

[@aaronseldes767]

@@XanderGouws I really want everyone to work on it for the betterment of humanity. Have a great day.

[@aaronseldes767]

@@XanderGouws or in the past (quantum physics) or present

[@SoppingWetDog]

Good vid. Have you considered doing some work on statistics,

[@dipakdulal1964]

does this equation determine minimizer or extremizer?

[@XanderGouws]

I don't think it does, unfortunately.

[@absolutelymath3399]

How are these videos made? + I really enjoyed this.

[@XanderGouws]

Thank you! I make these videos with a combination of: Unity, After Effects, Photoshop, and Sony Vegas :) But honestly the tools don't matter that much: you could probably make even better videos with power point and a screen recorder.

[@matheusperes6573]

@@XanderGouws I thought you used manim...

[@XanderGouws]

Ye, I thought about using it when I first started, but it was easier to use tools that I'm already fairly familiar with.

[@charleswayne1641]

I am sorry that I said "mean value theorem" -> there exist some (dy/de) = 0 ; Lagrange has already established the direct proportionality of "y" and "y bar", therefore, (dy/de) should not be zero. In order to apply .... , forget about mean value theorem. I just read some convincing argument that Etha =(dy/de) should not be zero anywhere between two points. Best way to think of this is sort of like "thickness" or the "contour lines". Here, you can think of the usual graph made of "x" and "y"(keep in mind: "x"and "epsilon" are independent variables); the actual path should be thick line, and as it spreads out, it gets thinner and thinner...., in the case of "contour lines", some kind of "origami",the paper is folded and the crease represents the actual "path", here, the height of the "abyss" has nothing to do with either x-axis nor y-axis, sort of like smooth inclined Grand Canyon.

[@ahmedjasim9626]

Great Vid, you need to slow down to allow us catch up with you this Topic is tough.

[@mathe_ma]

kzbin.info/aero/PL3SiKQGql2rkfgejk1DxmjHXjhWizQWaT

[@beoptimistic5853]

kzbin.info/www/bejne/joKsk6FobMmCoKc 💐👍

[@XanderGouws]

You're 100% right! I will keep pacing in mind in future videos :p

[@hasnaingillani2587]

this was very helpful! thanks!

[@mastershooter64]

now optimize multivariable functions on manifolds

[@turkserisi1979]

find the optimum of J=int[x'^2(t)-2tx(t)]dt please

[@jasonbroadway8027]

Going nowhere? If y is a curve, does J then just depend on x? Confusion

[@noism5429]

clear and concise

[@XanderGouws]

Thank you! Glad you enjoyed it :)

[@NovaWarrior77]

Thank you for your work good sir!!!

[@XanderGouws]

Any time! Glad you enjoyed it :D

[@ericroberts5119]

6 minutes to describe the subject matter is a bit ambitious ( I had to pause to confirm statements - but that necessarily isn't a bad thing- just that there are some gaps that could be filled ). Overall nice effort and explanation! Thanks! P.S. in your Integration by Parts dv = n' dx ( you have dv = n' )

[@XanderGouws]

You're definitely right. If I ever come back to this topic I'll definitely go over it more slowly.

[@leonmozambique533]

nice vid, it’s speed is fine if it’s review but it’s not if you’re trying to learn it for the first time.

[@XanderGouws]

100% agree. I'm glad you liked it though.

[@aerodynamico6427]

Haven't you heard of the "Playback Speed" adjustment feature on KZbin videos? Slow it down!

[@mack_raymond]

OK this was great! I would be interested in worked examples. I think your animations were nice, and your thumbnail pic should show that you are doing animations. Animations make it sooo much easier to visualize and I am more likely to click on a video that uses them....

[@XanderGouws]

Thank you so much! In the future, what specifically do you recommend I change about my thumbnails? I'm still trying to find the balance between being informative about the video topic and the video presentation.

[@mack_raymond]

@@XanderGouws What are your goals for this channel? If you're open to a more general audience, pictures and graphs in thumbnail are nice like in 3b1b. What is your background? I am a undergrad physics major and I've been considering alternative video formats to teach math, but I'm not quite at the level where I can explain these higher level topics. Happy to discuss.

[@XanderGouws]

I mainly wanna aim my channel towards people that are pretty good at math already - like I want to still be able to use actual equations and stuff in my videos. I'm actually still in high school, but have taught myself quite a bit of math at home. I'm going to study applied maths at university next year though! :) I think the best way to start making math videos is to start with _mildly_ easy stuff. Also, your videos will gradually get better over time.

[@chenlecong9938]

Has it gotta be a definite integral?

[@mathe_ma]

Yes

[@younesberdiafe8461]

instructions unclear xdd , i guess you forgot to say that by definition y is an extremal so we let y bar equal y by letting epsilon approach zero in the defferantial equation. And i think we can make etha allways satisfies boundary condition by introducing another function function let say w(x)=(x-x1)(x-x2)*etha(x) by i guess it will make the function w varies over x1 and x2 makes our lifes harder i think..., and yeah keep it up very nice animation clean work and clear explanations .

[@XanderGouws]

Yeah, I mentioned that at 3:00 and 3:20 (but I was talking pretty fast so I wouldn't blame you uf you missed it). In this case it's _easier_ to just let η(x0) = η(x1) = 0. Also, if we multiplied η by some (x - x0)..., but η had (x - x0) in its denominator we would end up with a point of discontinuity :O. And on top of that, there's plenty more ways to get a function equal to 0 where you're not multiplying by a quadratic term.

[@XanderGouws]

But I'm glad you like the channel

[@charleswayne1641]

Sorry, correction : Etha = (dy/de); Another thing I realised, because of the "Mean Value Theorem" There exists at least some Etha = zero; (dy/de) = 0, anywhere between the initial point and the final point. Thus, this tells me that the E-L equation is sufficient condition, Not the necessary condition.

[@jwfundsd]

Excellent!!!!! Thanks!!!!!!

[@XanderGouws]

Glad you liked it!

[@RoboMarchello]

Thanks💖

[@danielschwegler5220]

perfect, thank you very much!

[@XanderGouws]

Glad it helped!

[@hansstephani5869]

Cool Video!

[@charleswayne1641]

Actually, Etha can't be the function of "x". If it is, we cannot factor it out. Then, Etha has to be there in the Euler-Lagraingian.

[@XanderGouws]

Apologies if I wasn't clear. TL;DR: When I said "factor it out", I meant "factor it out *of the sum*", not "factor it out *of the integral*" ∂F/∂y̅ * η + d/dx(∂F/∂y̅') * η = [∂F/∂y̅' + d/dx(∂F/∂y̅')] * η This is true by the simple rule of distribution. You _are_ correct in saying that it can't be factored out of the _integral_ though - but since we're keeping it in the integral, we're fine.

[@charleswayne1641]

I realised that Euler-Lagrangian = 0 is Not a necessary condition, but a sufficient condition. Just in case, can we also assume dq/de can be zero anywhere between point 1 and 2 ?@@XanderGouws

[@XanderGouws]

@@charleswayne1641 I think it should be fine, as long as its zero at x0, and x1.

[@SimchaWaldman]

Why are we not funding this??? ❤️💚💙🥰

[@williamlee4215]

So clear! Thank you for making this video.

[@srallulrich]

5:00 why?

[@ravisaripalli6735]

A tad slower pace of presentation would be great. Also, there is no explanation given in the final step, as to how you can assert part of the integrand should be zero for the integral to be zero ... for one to arrive at Euler-Lagrange eqn.

[@XanderGouws]

Yeah, for sure - I tried to go slower in my latest vid. If you're curious, that term becomes 0 because of the boundary restrictions we set on η: η(x0) = η(x1) = 0.

[@ravisaripalli6735]

Perhaps I haven't conveyed my earlier comment well. I am referring to the final equality mention at exactly 5m mark in your video. What I was referring to was the integral with Lagrange expression and eta product. I think the explanation that was missing was the fact that since the integral has to be zero for any arbitray eta the lagrange expression must be identical to zero.

[@XanderGouws]

@@ravisaripalli6735 ahh. Yeah you're right.

[@aerodynamico6427]

Haven't you heard of the "Playback Speed" adjustment feature on KZbin videos? Slow it down!

[@ericzeigler8669]

Is y'=dy/dt or is y'=dy/dx ?

[@XanderGouws]

Good question! It depends on the context of the problem. At 1:57, y' = dy/dx, because x is the independant variable. At 2:12, q' = dq/dt, because q is the dependent variable, and t is the independent - although typically we would use dots to indicate time derivatives. For the derivation of the E-L Equation, we're using y and x as a dependent and independent variables, respectively.

[@ericzeigler8669]

@@XanderGouws Crystal clear, thanks.

[@charleswayne1641]

I took a "pot shot" at English :), then I backtracked. And, I discovered an important meaning: That Etha(dy/de) is Constant. Keep in mind /etha(dy/de) is never zero, but, epsilon(E) is zero at both end.

[@samuelj5890]

sick vid my man

[@XanderGouws]

Thank you!

[@va9if]

u run so fast that I can't keep it up but awesome!

[@dodo-js5gw]

Thanks !!

[@XanderGouws]

Thank you !!

[@zigadiyacine3550]

Hi i have a memor for this these can you help mee plzz

[@beoptimistic5853]

kzbin.info/www/bejne/joKsk6FobMmCoKc 💐👍

[@abubardewa939]

Could have been so much better if you were speaking at a slower pace. But thanks for the animation.

[@XanderGouws]

I totally agree. I'm glad you enjoyed the animations :)

[@aerodynamico6427]

Haven't you heard of the "Playback Speed" adjustment feature on KZbin videos? Slow it down!

[@charleswayne1641]

I am sorry, but I have to say that this virtual displacement should only be about the "single variable epsilon" If you include "x" or "t" as a variable in addition to "epsilon", then you should also have an expression "partial of y with respect to "x" .... Therefore, I claim that Etha is a single valued function only of epsilon.

[@XanderGouws]

Eta can be any function in the x-y plane that satisfies η(x0) = η(x1) = 0. You draw it in the x-y plane. It is a function of x. When we define y bar, we MULTIPLY IT BY epsilon. y bar is a function of x and epsilon. η is a function of x, just like y.

[@XanderGouws]

η isn't our 'virtual displacement' ε*η is.

[@charleswayne1641]

Your right, epsilon*(dy/de) = delta(y): "virtual displacement" of any function(generalised coordinate) with respect to epsilon.@@XanderGouws

[@charleswayne1641]

Xander, here "x" is indépendant of epsilon.

[@charleswayne1641]

To clarify, I meant to say that the "virtual displacement" = delta(y). The equation you keep mentioning, unnecessary. In fact, you don't need to replace Etha = (dy/de) in the derivation of E-L equation. The equation doesn't say much except: delta(y) = y - y bar = Epsilon(dy/de), which is (ratio)*dimension = The Quantity. Here, the dimension, of course, is Epsilon(virtual dimension).

[@charleswayne1641]

I disagree that Etha should also be a function of x. If it is then you should include a third term according to the rule of Partial derivative/Chain rule. This third term should have a factor "dx/de", Lagrainge cancelled it out, because here "x" and epsilon is independent of each other. Here, Etha = dy/de should only be "single-valued" function of epsilon.

[@johnspivack6520]

a few mistakes in the integration by parts spoil this video unfortunately.

[@bruzote]

What is the intended audience for this? Is it people trying to watch a mathematical equivalence between a starting equation and final equation? Certainly, the pace combined with the formulas suggests it does not matter WHY this is done. Just as Einstein earned a Nobel Prize with a three-page paper but my 80-page thesis was not worth printing except to get me an advanced degree, I perceive this video as testament that someone likes to talk a lot. WHAT is the goal of the video? What IDEAS guide you to the goal? I don't see the latter question being answered. Rather, I see someone showing how mathematical equivalence works during manipulation of equations. Call me disappointed - by almost every Lagrange video that I am finding.

[@isobar5857]

Much too fast for us lesser mortals. Not enough of a mathematical explanation of the derivation........what is being held constant and which are the variables, and more to the point, why? Example, why do we have to consider the change in the gradient when surely just a change in the 'variation' of the height would appear to suffice? Always found this subject to be ill explained and too much assumed of the student. Never found an explanation of this concept yet that has a derivation which is fully consistent [ in my opinion]. But thanks for the try anyway.

[@FATEHmaistro]

Reciting the whole function make it hard to follow you

[@beoptimistic5853]

kzbin.info/www/bejne/joKsk6FobMmCoKc 💐💐💐

[@XanderGouws]

Hmm ok, I'll try doing it differently when I get back to making videos. Are there any parts that were particularly hard to follow?

[@FATEHmaistro]

@@XanderGouws i was lost at the multivariable chain rule ,i hope that you post more in the future.

[@neelchattoraj]

Slow it down to 0.75. Thank me later

[@XanderGouws]

Haha, my bad :p Definitely will keep pacing in mind in my future uploads

[@neelchattoraj]

@@XanderGouws I am incredibly grateful to you for putting together this wonderful explanation. It might just be me who is a bit slow on the uptake, but thankfully youtube allows time scaling so... what a time to be alive! It is not your your bad at all, but 0.75 is like ASMR

[@serdomal8796]

sounds like a rap, but i get it

[@beoptimistic5853]

kzbin.info/www/bejne/joKsk6FobMmCoKc 💐👍

[@Septumsempra8818]

The outro tho

[@shannonsteinmetz7711]

Why make this video? This is you talking to yourself.... Who is this for?

[@XanderGouws]

That's a good question. The goal was to help other people learn the derivation for the Euler-Lagrange equation, but I think it definitely functions better as a review-tool given the poor pacing

[@lowersaxon]

By far too quick.

[@XanderGouws]

You're definitely right! I've tried to slow down more in my later videos :)

[@donegal79]

lecture, not teaching.

[@beoptimistic5853]

kzbin.info/www/bejne/joKsk6FobMmCoKc 💐👍