Derivation of the Euler-Lagrange Equation | Calculus of Variations

[@FacultyofKhan]

After many requests, I've finally made a video on the Principle of Stationary Action. Check it out! kzbin.info/www/bejne/g2GYmquYiLSMbpI

[@admiralhyperspace0015]

Please can you share what you write in these videos as a file in description. I can't possibly take notes as beautiful as you. If you think it would decrease your viewership, you can share them personally because I watch all of your videos and intentionally disable adblocker to support you.

[@Icenri]

The best explanation so far of what a Lagrangian really is.

[@vivekvaghela2274]

The same explanation is also given in book classical mechanics by john taylor

[@shakhwanm]

@@vivekvaghela2274 do you have this book by pdf

[@thecarlostheory]

@@vivekvaghela2274 thx very much for saying that!

[@isaacmelle9303]

I second this statement forsure

[@joycebenbow4836]

I've watched all of these guys videos (I'm a physics PhD, graduated many years ago) and he really is an exceptionally gifted instructor. I challenge anyone to find a better explenation/exposition of any of the topics covered.....hats off to him I really enjoy all of the videos and each one gives me a deeper level of insight and understanding....I'll be making myself a patreon very soon to show my appreciation...

[@annamallett879]

I've watched so many videos and read a number of textbook explanations trying to get my head around Lagrangians and the Euler-Lagrange equation, and this one is by far the best IMO. I still have to pause the video every five seconds and rewatch it every couple of months, but at least I can actually understand it! I even like it better that Feynman's explanation (which is also very nice).

[@ozzyfromspace]

Damn, Faculty of Khan! First time seeing this and you make it look effortless. Your KZbin videos are an absolute treasure, so thank you. Cheers from Michigan!

[@nathanielweidman8296]

Great video. I have been enjoying your channel. There are a lot of k-12 tutorial videos on KZbin but no a lot at the college level, which makes your channel unique and greatly needed. Please do more videos on calculus of variations and keep up the good work.

[@FacultyofKhan]

Thank you!

[@TG-to5nf]

Absolutely the best explanation I’ve seen compared to 4 undergrad text books.

[@ulascilingir]

Thank you for all these videos. 20 years after I take these lessons I use your videos to refresh my memory and every time I wish that you were my professor back in the day 😁

[@umedina98]

You don't stop to amaze me. I was reading this topic in a FEM book and didn't understand NOTHING. But after this video is simply clicked. THANKS.

[@abhinavroy1999]

This is one of the most beautifully explained video in Calculus of Variations I have seen so far. Keep up the great job!

[@dwyerfire]

You saved my ass, finally someone who can just explain something clearly without skipping 'obvious steps'.

[@joelcarvalho5781]

Great video with a detailed explanation on derivation of the Euler-Lagrange equation! I came here after reading the first chapter of Landau's volume 1 and this video helped me a lot :)

[@carlosmagalhaes4953]

Congratulations! This is the best explanation that I've seen so far.

[@erickjian7025]

VERY NICE !!! but one question: 4:41 why do you ADD up the chain rule partials ? Is it because of the way Functionals are defined ?

[@FacultyofKhan]

Thanks! More or less yeah: since we're taking the partial of F(x,ybar,ybar') w.r.t. epsilon, I need to take into account both ybar and ybar' and their dependence on epsilon. That's why I included both and added them. It's basically the formula for the total derivative: en.wikipedia.org/wiki/Total_derivative

[@erickjian7025]

@@FacultyofKhan That makes sense. I have also seen an explanation somewhere: as εη(x) → 0, we could Taylor expand on the integral and you only consider the 1st order variation (to get the expression we got, with using Total Derivative). I wonder is that right? If so, why do you only need to consider the first order?

[@FacultyofKhan]

@@erickjian7025 I believe what you've read is right, but I'm not terribly familiar with the derivation via Taylor expansion beyond reading a few articles online. The reason we only consider the first order is that the E-L equations are the analog of finding the point where y(x) is minimum/maximum/stationary, so where y'(0) (see my previous video for more details on that analogy: kzbin.info/www/bejne/bHnIgpZteLiekNU). Using that analogy, we only require the first-order terms in the Taylor expansion since we're only interested in the 1st derivative; however, if you want to find the nature of the functional (whether it's a maximum, minimum etc.), you'd need the second order terms since now you care about the second derivative as well (this is called the second variation). Hope that helps!

[@lakshyaagarwal4044]

you guys are awesome, this video helped me a lot, saved a lot of my time. thank you again

[@o.a]

i wish more people knew about this channel. you're making my physics courses a lot easier.

[@randompast]

The step that starts at 4:27 with the chain rule is a bit hazy. Could you elaborate? Why not chain for x and include a dF/dx * dx/de term? Why chain for both y and y' yielding 2 terms instead of 1? I'd really like to see more videos on applications of Functional Analysis and Calculus of Variations especially that which might be applied to physics simulations.

[@FacultyofKhan]

Well, dx/de is just zero. Epsilon is a perturbation applied only to the function y: it has no relation to the independent variable x. However, y and y' both contain epsilon, since by perturbing y, we're automatically perturbing its derivative y'. That's why we need the y & y' terms while dF/dx*dx/de is just unnecessary. And thank you for the suggestion!

[@randompast]

Great elaboration, thank you.

[@ritobrotomohanto6762]

@@FacultyofKhan thanks for the clarification.. I had the same doubt.

[@raptor4916]

could you please do a walkthrough of an action problem? Maybe im just slow but i find the application from this to problems in physics hard to conceptualize.

[@FacultyofKhan]

In my first Calculus of Variations video, I go over two example problems/applications of Calculus of Variations techniques (I've put the relevant timestamp starting at the first example): kzbin.info/www/bejne/bHnIgpZteLiekNUm12s Also, as I said in another reply, I'm going to make a video on finding geodesics (i.e. the path that represents the shortest distance between two points on a curved surface), and then another video on the brachistochrone problem. The brachistochrone problem is more Physics-related, so I think it should cater to what you're requesting.

[@FacultyofKhan]

Hmmm, so I just looked up the term 'action', and it seems as though 'action' refers only to integrals in time, whereas my other reply to you spoke about integrals in space. In that case, I might (but I'm not sure) do an action problem in my Calculus of Variations series (though it would be more appropriate to start a series on Classical Mechanics and then do an action problem from there); thanks for the suggestion! I'll do it eventually though, since your request has a bunch of likes on it.

[@angelmendez-rivera351]

An example of an action problem: suppose you want to find the equations of motion of a particle subject to a sum of forces. To find x(t) (its motion), we solve a functional that minimizes the action (the time integral over the energies). This is the basis for Hamiltonian and Lagrangian mechanics, and the reason we do this is because it generalizes Newtonian mechanics and it allows us to find the equations of motion in general relativity and quantum mechanics, and Newton’s Laws just happen to be special cases of these two above.

[@SeanRhoadesChristopher]

ε = 0, is like a magic carpet, that vanishes from under the feet of an unknown η(x) giving us f(x)! The power of tautologies, something from nothing. Amazing!

[@Zero-ef4sc]

It gives us... f(x) factorial?

[@turboleggy]

beautiful presentation, perfect pace, so clear. The only thing I'm not sure about is why we included the parameter when multiplied by eta.

[@muhammadumar9753]

this video contains all my calculus in a beautiful way from partial derivative, definite integral, extrema etc for the first time it gave me a new way of thinking differently about them. thankyou so much

[@willie5069]

I understand the general form of integrating by parts. Int(v,du)=Int(v,u)-Int(dv,u) but in the part where you use it (about 6 min in) I am having difficulty understanding which parts are t;he u's, the v's and the dv's. I want to be sure I understand this thoroughly and that I am certain of my understanding so having you answer this would help. Also, I find this to be an extremely well done lesson. Was really struggling with the concept and lecture 1 in particular cleared the fog. I am 71 years young and this video helps me to keep learning.

[@FacultyofKhan]

Thank you so much Willie! In terms of your integration by parts formula, v = partial F/partial ybar', du = eta' dx, u = eta, dv = d/dx(partial F/partial ybar') dx. Hopefully that should make sense now. If not, let me know and I can clarify!

[@willie5069]

Great. As it turns out I lucked out and had it right. Appreciate your reply.

[@dagreds]

Unbelievable that you have that little subs. Keep up your Great work!

[@admiralhyperspace0015]

You are a God send. Only if you were my permanent teacher. I would really appreciate if you would actually share your notes in description to be saved.

[@PhysicsBanglaTutorial]

Your explanation is so good. Thank you so much dear.❤️

[@SaraAlSenani-o1o]

Big Thanks❤❤❤❤ The Best Explanation For E_L Equation Ever!

[@reddykiranrsr]

I am amazed with this mathematics and enlightened with your Explanation. Keep it up your good work..

[@garyrandomvids2098]

Really good explanation, easy to understand and follow. Thank you for this!

[@jameshuang9568]

You are my absolute hero, life saver and true love

[@FacultyofKhan]

💖💖💖

[@turtleyoda7703]

Could you explain better why the derivative can be moved inside the integral?

[@tonyjing7305]

Here's a great explanation: kzbin.info/www/bejne/sJO6mpuBbs6VnsU

[@dontnotsubscribelikeshare2542]

lovely video. and thanks to your robotic voice, can understand even at 2x speed

[@TheTrueJonny]

Thanks! question: for the step at 6:23 you insert ε = 0 into y bar to reset it to y, however as far as I know the right way to do this would be to first differentiate and then set ε = 0, since really the whole expression started off as a derivative with respect to ε, so setting it to 0 beforehand makes no sense. Why are we allowed to do this here?

[@kurotama_doublebread]

I've seen this proof before, but now I understand it fully; thank you

[@mellowjellofello3541]

what an excellently treated derivation of the Euler-Lagrange equation!

[@Deepayan00]

Please explain the procedure of moving a derivative inside an integral more at 4:07. There is a lot going under the hood. :)

[@piero8284]

For those wondering why you can interchange the integral and derivative operators, look up the Leibniz integral rule. I also did not remember of seeing that one.

[@neosrinath]

Great video!! However I have some questions: 1. Can we write a mathematical statement on "Suppose y makes I stationary and satisfies the above boundary condition". Although you said it in words, it dosen't make sense unless there is a mathematical statement? 2. I am not convinced in the equation, ybar(x) = y(x) + epsilon*eeta(x); where y(x) represents extemal function. There is no mathematical relation which tells us that y(x) is extremal. Also, if I choose/assume (for example), y(x) to be non-extremum function which satisfies the boundary conditions, do we still get the same Euler-Langrange's equation? 3. You rightly mentioned that the integrand I, is only a funtion of epsilon. Then maybe in one case, I would expect I = 2 + 3*epsilon. Then what does dI/d(epsilon) = 0 mean in that case. 3=0? 4. In the final steps, epsilon = 0 can only be applied after evaluating the integral. Then how can one apply epsilon = 0 or ybar(x) = y(x) inside the integral and get Euler Langrange's equation? Thanks in advance

[@FacultyofKhan]

Thanks! My responses: 1. Sure, I guess? I'm not really writing out a formal proof as much as teaching the derivation, so I think saying it should be enough? 2. There isn't a mathematical relation that tells us y(x) is an extremal because that's what the video is trying to derive. It's an underlying assumption that y is an extremal. Essentially, the equation ybar(x) = y(x) + epsilon*eta(x) is looking at what the functional would be like if we add a variation (epsilon*eta) to the extremal y. It's like saying what the derivative of f(x) would be at a point slightly far away from a local minimum. Also, you wouldn't get an Euler-Lagrange equation if you chose a non-extremal: the Euler-Lagrange equation, by definition, allows you to determine the extremal function by solving a differential equation. 3. For most practical purposes, you wouldn't end up with I = 2 + 3*epsilon, but supposing you did, then in that case, you can't really apply dI/depsilon = 0 because I has no stationary points (it's essentially a straight line; there are no maxima or minima). It's like trying to find the local maximum/minimum of a straight line: there is no possible answer. 4. We can apply epsilon = 0 inside the integral because we've already moved the derivative with respect to epsilon inside the integral at 4:08. Hope that helps, and if you have more questions, please let me know!

[@madhamanish8959]

Very clear explanation ... Thanks a lot

[@mujtabaalam5907]

1:27 but isn't the lagrangian often non-zero at the endpoints? 2:19 how do we know that y bar makes I stationary? For crazy eta I don't see why y bar would still need to make I stationary

[@hongzheliu9199]

Thanks a lot for the great video, Faculty of Khan. I have a few questions. at 1:47, we define the function y_bar(x) = y(x) + εη(x). I wonder if it is sufficient to represent any arbitrary function. Why do we define y_bar(x) using algebra addition instead of another way, like multiplication y(x)η(x).Why do we have to introduce a constant ε here instead of just defining y_bar(x) = y(x) + η(x).

[@FacultyofKhan]

The definition y_bar(x) = y(x) + εη(x) is sufficient to represent any arbitrary function that slightly deviates from the extremal y(x). Since eta(x) is arbitrary, you can plug in whatever you want for eta to make y_bar(x) whatever you want. You don't necessarily have to have multiplication. As for your last question, we introduce ε because our goal here is to introduce a small variation on the function y(x) so we can perform our derivative and derive Euler-Lagrange (as I do in the remainder of the video after 1:47).

[@miiortbiiort4610]

Thanks for explaining stuff a bit more detailed :) I was breaking my head over this at first

[@stevenschilizzi4104]

Thank you! This was a very clear and straightforward exposition. Of course, it has left me hungry to know what other methods are needed to figure out the nature of the stationarity. Do you have a video on that too? - Ah yes, perhaps it’s your last inthis series, the 2nd derivation. I’ll check it out. Thanks again!

[@barryhughes9764]

Great video, thanks. Finally coming to terms with this derivation. If I may, could you kindly clear up one small detail in this derivation. When integrating by parts you have said that at x1 and x2 eta of x=0, since we have forced it to be zero at these points, and thus the expression outside the integral sign equals zero. But what about the expression inside the integral which has the same bounds of integration, and yet we assert that eta of x cannot be zero. A little confused by this 'apparent ' ambiguity. Any clarification would be very much appreciated, other than this 'sticking point' I completely understand the derivation. Thanks again and kind regards.

[@FacultyofKhan]

Happy to help! The expression inside the integral involves an integral of eta times d/dx(partial F/partial y'): the integral of this expression is not necessarily zero, and is quite different from the simple eta. The other thing I'll note is that when you integrate by parts using definite integrals (i.e. Integral a to b f(x)g(x) dx = f x Integral a to b g dx - Integral a to b (*integral* *g* *dx*)df/dx dx), then the bolded expression does not have limits applied to it. Instead, the limits are applied to the integral on the outside. Hope that helps (I realize it's late for your question, but maybe someone else will be helped)!

[@knivesoutcatchdamouse2137]

@@FacultyofKhan This explanation did in fact help me, even 8 months after you posted it, so thank you!

[@curious_ben]

Great derivation. Looking forward to more of your videos

[@mungy27]

thanks for doing more than my math professors

[@mirsarfraz7143]

We really enjoyed the derivation of Euler Lagrange equation (Sarfraz and Liaquat)

[@ignatiusjacquesreilly70]

At 5:00, when the derivative of ybar is taken, you arrive at ybar'(x) = y'(x) + εη'(x). Why is there not another term on the right hand side, ε'η(x), due to the chain rule? Is ε a constant or a function?

[@AyushGupta-zc4lh]

It's an arbitrary constant.

[@pierreretief]

at 4:10, is it fine to just move the derivative into the integral?

[@mamalojzka]

Great video. Very understandable.

[@physicsstudent6313]

The one point that I don’t understand (which is critcal to the whole thing) is “what does the product of epsilon and eta mean”? It seems as though epsilon could have been expressed as a function of x with out the inclusion of eta.

[@alexismiller2349]

6:34 this might be a dumb question, but why is the integrand equal to zero everywhere? Could it not also just go positive and then negative such that the integral evaluates to zero?

[@yuvallotenberg9586]

Your videos are really good and well made. I really like watching them. Could you please do an example of a problem in calculus of variations using the Euler Lagrange equation?

[@FacultyofKhan]

Thank you, and I will! I'm going to make a video on finding geodesics (i.e. the path that represents the shortest distance between two points on a curved surface), and then another video on the brachistochrone problem. After that, I'll add more lectures depending on my book(s) and on the requests I get.

[@yuvallotenberg9586]

Faculty of Khan Thank you so much for the fast response. You are really doing an amazing job and I hope you will keep posting videos.

[@utkarshpuri3739]

Amazing explanation

[@hiZarki]

His robotic tone spooks me! Ahah nice video though

[@awesometimewithphysics500]

Best 👍❤️❤️❤️❤️❤️❤️❤️❤️❤️❤️❤️❤️❤️❤️❤️❤️❤️❤️❤️❤️❤️❤️❤️❤️❤️❤️❤️ I really like your lecture series sir .

[@xukai4821]

nice talk, looking forward to the next videos.

[@rahulkagdiyal8180]

I understand the proof...but one question: we introduce function eta(x) with the BCs eta(x1)=eta(x2)=0.Why do we equate them to zero...what is the reasoning behind it??What are it's implications??

[@jamesferguson9403]

Thanks again for these. This is really solidifying my understanding of this subject! I have a specific question about applications in classical mechanics. Disregard if you're unfamiliar with this subject. Often, the E-L equation is applied directly to systems with multiple generalized coordinates to obtain so-called equations of motion through time. But in the derivation of the E-L equation, y(x) is varied between definite limits- the imposed boundary conditions. How is it that can we apply these equations to dynamical systems with only initial conditions if the E-L equations are derived using boundary conditions for the dependent variable y?

[@FacultyofKhan]

No problem, glad you like these videos! Even in classical mechanics where we need to find the equations of motion through time, the E-L equations there are *also* derived using the time-equivalent of 'boundary conditions' (e.g. x(t1)=x1, x(t2)=x2, so initial and final states). For example, refer to the following source (Morin, Section 6.2): www.people.fas.harvard.edu/~djmorin/chap6.pdf While deriving the E-L equations, it's assumed that you know the initial and final states of the particle. However, when applying the Euler-Lagrange equations, you don't really use the initial and final states explicitly. You just find the Lagrangian (i.e. L = Kinetic - Potential Energy), take the relevant partial derivatives, and find x(t) by solving the resulting differential equation. In many cases, you just leave the differential equation and don't even solve it, as you can see in some of the examples linked above. The reason I specifically framed my derivation in terms of boundaries and y(x) is that in future videos, I'm going to find geodesics and solve the brachistochrone problem, both of which specifically involve boundary conditions and paths in space, instead of paths in time. If I start a series in Analytical Mechanics (which I might depending on my schedule/videos in queue), then I will do examples where I find x(t). P.S. Sorry for the multiple replies/deletions; I just needed to do a bit of research to clarify my thoughts.

[@jamesferguson9403]

So I guess the "fixed endpoints" are more of an abstraction used in the proof, and when we apply the equations, they work for any path in time- the notion of these endpoints are gone during application. They could really be any variable "endpoint" in your solution that is based on the initial conditions specified for your particular solution. Is this correct, or am I just spewing nonsense?

[@FacultyofKhan]

From what I understand right now, I think you're correct. When you start out with two initial conditions x(t1) and dx/dt(t1), you can, for a second order differential equation (i.e. the E-L equation), use these two initial conditions in conjunction with your solution to find the final state x(t2). So in a sense, specifying two initial conditions for a given differential equation is equivalent to specifying an initial and final condition, since either may be used to find the solution to your 2nd order differential equation (i.e. the path of stationary action). The derivation of Euler-Lagrange requires the specification of the endpoints, whereas the application of E-L to find the equation(s) of motion uses the initial conditions. I'm not an expert in Lagrangian mechanics (though hopefully I can become one quickly), but hopefully that clears things up!

[@jamesferguson9403]

Yes it does! Thanks for struggling through that thought process with me and for making these vids. Not all heroes wear capes!

[@FacultyofKhan]

No problem, and thank you for the kind words!

[@Robert-pq3sb]

Why is I a function of ε and not ε AND x????

[@prostatecancergaming9531]

Because you already intergrate over x.

[@sarasara2818]

Thank you very much for that clear explanation please give us some references you use it

[@omarabu-khalaf6043]

Hello, Thank you for this amazing series. I just want to ask about the integral in the moment( 6:10) in the lower right corner, in pink, the term partial(F)/Partial(y-') * integral(eata'),( the term that was cancelled out), the limits of integration should not have been put for the integral involving eata, but rather this integral must have been computed indefinitely and then the whole term ( which is u*v in the usual notations for integration by parts) is to be evaluated from x1 to x2, am I right? Thanks

[@cornonthecube5830]

One question, if I want to know how to do something similar in other fields of math, how do I know to even create the y-bar equation to get started? Like, what thought process can I use in order to see that if I were to do this on my own, I would need to create the y-bar equation?

[@brandomiranda6703]

To understand the difference with 1D and several D calculus, we usually define the derivative by having epsilon approach zero (with a limit). But instead you took the derivative wrt epsilon and set it to zero which seems slightly different. Do you mind clarifying why you did this instead of doing the usual "epsilon -> 0" but instead you did 1) dI d/eps = 0 AND 2) eps = 0 exactly zero instead of approaches. Thanks for the channel! Love it.

[@bankaikun94]

For the integration by parts, 5:42. Why is eta integrated indefinitely? Shouldn't it also be integrated definitely? If so, it would just be 0, canceling out the entire integration.

[@FacultyofKhan]

The definite integral only applies to the outside integral (not the integral of eta') which combines the integral of eta' and the derivative of dF/dy'. You can revisit the formula for integration by parts just to be sure: integral(u dv) = uv - integral (v du), here I've set u = dF/dy' and dv = eta'. Hope that helps!

[@kristianrigasz8737]

@@FacultyofKhan may I ask why do not we include constant after indefinite integration of eta' in this part? Because it would subtract eventually? Thank you.

[@abdurrahmanlabib916]

To make ybar as general as possible why dont we also multiply y by an arbitary function alongside addition of eta?

[@johnspivack6520]

Good video, quick question. The derivative wrt x that appears in the second term looks like a total derivative, not a partial derivative. Is this so, and if so can you explain why it is not a partial? Thanks.

[@dsazz801]

It was so clear. Thank you :)

[@keshavshrestha1688]

Best explanation!!! But i have question, what is the difference between lagrangian and lagrangian density.?

[@Ivanko71]

Thank you so much.. I have a better grasp at the topic now💫💫💫

[@dk_3562]

Thank you very much, Sir. God bless you.

[@yuxiang3147]

Does the definition of "stationary function/functional" exist? Is there such a thing technically? You can say a stationary point to a function (or a stationary function to a functional) but I haven't heard of the term stationary functional.

[@physicsstudent6313]

As a follow-up to my previous question (with no reply so far) epsilon has been defined as a constant, variable, and a variable constant, and a function. What is t?

[@juliegregory8187]

All good but one question. The notation F(x,Y, Y') seems to imply that there can be an extra non-trivial dependance on x in F that is separate to the dependance specified in Y(x) and Y'(x). For instance in the brachistochrone problem maybe a wind resistance term depending only on x was involved? When you go through the chain rule of differentiation it is dismissed as " x is an independent variable by itself". A bit confusing to me but possibly obvious?

[@mcdudelydoo3116]

One thing that confuses me is what exactly is meant by a parameter? I get that epsilon is a parameter but what doesnit mean? Is it just like a constant or what

[@eamon_concannon]

6:28 The explanation for why the derivatives wrt ybar become derivatives wrt y when e = 0 is not complete here I think. Here is what I think is the complete explanation, By chain rule, dF(ybar, ybar', x)/dy = dF(ybar, ybar', x)/dybar * dybar/dy = dF(ybar, ybar', x)/dybar since dybar/dy=1 and dF(ybar, ybar', x)/dy' = dF(ybar, ybar', x)/dybar' * dybar'/dy' = dF(ybar, ybar', x)/dybar' since dybar'/dy'=1 When e = 0 , F(ybar, ybar', x) = F(y, y', x)

[@Life_outside_is_manipulated]

I have a doubt, why do we need epsilon =0, given that eta is zero for the boundary condition ?

[@elumixor]

Amazing video! Thank you very much!

[@matthewjames7513]

Is there a discrete version of the Euler Lagrange equation? If so, couldn't that be used to find the 'best fit' to n data points?

[@aarifhussain3700]

No words great sir

[@umedina98]

Math is simply beautiful...

[@jagadeeshwaran1512]

What is a Sufficient condition and Necessary condition?

[@omerakcan5536]

I couldn't see any possibilities except he must be the prophet of our century.,

[@wolspanewolspane9663]

can someone please help! What is a stationary function?

[@mathswitha.c6403]

Muri Chibaba Prof.Respect!!!!!

[@FranFerioli]

Thanks for the nice video. I would explicitly note the fact that the function F inside the integral I depends on x, y and y' (i.e., F=F(x,y,y') as you wrote - and not, say, on y''). Otherwise you end up with a different expression than the E-L equation (I think, I'm no maths professor just an internet weirdo : ).

[@FacultyofKhan]

Thank you for the kind words and advice!

[@ethan_martin]

Great video. This was super easy to follow. The only thing I don't understand is why y' is a parameter of the functional in the first place. What is wrong with the concept of minimizing the integral of a functional F(x, f(x)) versus the integral of F(x, f(x), f'(x)). Obviously, you wouldn't end up with the Euler-Lagrange equation at the end of the derivation, but I don't see what is wrong with it?

[@FacultyofKhan]

Thank you! A lot of well-known functionals contain y' (e.g. the distance functional which involves the integral of sqrt(1+(y')^2)), so it's beneficial to use y' as part of F. In fact, some functionals even contain higher order derivatives like y'', y''', etc. Also, simply using F(x,y) restricts us in the types of functionals we can manage, since y' isn't necessarily something that can be easily determined by modifying y using simple algebraic operations in a fixed pattern. Like you can't always get y' from y by multiplying y by a constant, or squaring y, etc. That's why y' needs to be in F(x,y,y') on its own. On the other hand, if I had something like F(x,y,y^2) or F(x,y,2y), then yeah, you could argue that the 3rd argument is useless and that we're better off with F(x,y). However, for something like F(x,y,y'), you can't necessarily omit the 3rd argument. Hope that helps!

[@ethan_martin]

Yeah that makes total sense. Thanks!

[@dindamaulina7610]

that was a great video, but i have a question, how we can get natural boundary condition from euler-lagrange equation? i need an example for that. thank you :)

[@itsincredible5966]

is this the derivative of euler lagrange equation using variational principle?

[@essadababneh5871]

Great video! I guess my question is pretty late since this video was made five years ago. Since y changes as x changes and eta changes as x changes, shouldn't epsilon be adjusted so that we can land on any perturbed function point we want? Wouldn't that make eta depend on the values of y and eta in turn which depend on the value of x. So wouldn't make eta depend on x? This is confusing me a lot. I hope I can get some clarification.

[@nabeelsherazi8860]

No, epsilon will be a single value that "scales" the perturbation function eta. We can make it land on any function by changing the perturbation function eta, but epsilon will just let us scale that deformation to be further or closer to the original y. For example, let's say we want the function y = 2x to look curvier, and land on the point y(2) = 6 (which it doesn't). We might make eta the function cos(pi*x), and then choose epsilon to be 2, so that y + epsilon * eta at x = 2 equals 6. This is just an example, and using cos as your eta might not match other boundary conditions, but if you graph that function and change epsilon, you can see how it deforms y, and how setting epsilon = 0 recovers the original y.

[@hansameena1237]

grear work, i read ur channel discription ,u said u r a University student, if u r ok ,can u tell me what's ur degree/current education title

[@mark.p9824]

Great video but still I have a question: Why does x get integrated out at 2:48 ? Could someone help ? thanks :)

[@abdulgineer]

Ive been wondering this too and I think I just figured it out, i think by integrating the function and then evaluating it at x1 and x2 since the integral is between x1 and x2(and since we are given the values y(x1) and y(x2)) then we are left with nothing in terms of x and only epsilon. (that was my best guess by what he means hope that helps)

[@mark.p9824]

@@abdulgineer Thanks it actually makes sense :)

[@hypernova4334]

Question: you mention that the EL equations are a necessary condition for the functional to be stationary, and that the EL equations are not a sufficient condition for its being stationary. I am confused by this. Aren’t the equations a sufficient condition for the stationary condition but only a necessary condition for extrema (max, min)?

[@andsussen]

I've seen one derivation using taylor expansion: f(x,y*,(y*)') = f(x,y,y') + epsilon( n*(df/dy) + n'(df/dy')) + O(epsilon^2) where d is meant as partial derivative, but maybe I'm misunderstanding something but wouldn't the partial derivatives be with respect to y* and (y*)' ? Thankful for any help!

[@danielc4267]

Awesome lecture! By the way, I kept thinking whether the voice recording was real or machine-generated.

[@FacultyofKhan]

Hey now, I'm supposed to trick the humans into thinking that I'm one of them; I can't have you going around and blowing my cover. (It's real btw, and I'm glad you liked the lecture)

[@FirstNameLastName-kn9dg]

Amazing video!

[@hajramughal11]

great job ! awsome explanation. each and every point is clear. thankyou

[@matteopeterlongo5007]

This is glorious. Thank u

[@mariamrazmadze9730]

could you please tell me ,at 4:10, why can we move the derivative inside?

[@FacultyofKhan]

Because we're allowed to since the limits of integration are constants. This is a special property of differentiating inside the integral and is a well-established formula. See here: en.wikipedia.org/wiki/Leibniz_integral_rule

[@mariamrazmadze9730]

exactly what i needed , thank you

[@jerrysmcnuggets]

Fantastic video

[@GaudioWind]

I love the videos of Kahn Academy. I started watching this video and this first question came up. Well, if y is function of x and it is continuous then I think y'=dy/dx is also a function of x, therefore F could be written as a function of only x, right? So, is there really a particular detail of F that would make it different from ordinary functions f(x).

[@syamkrisna]

Thanks for the lucid explanation on how to derive E-L equation. But i have a query. while applying chain rule for the second term -- integral(eta) was set to zero in uv term, where as limits were not applied to eta term in integral(vdu), that is eta term was retained. why is it so?

[@FacultyofKhan]

Thank you for the kind feedback! For your questions: it's because integration by parts for definite integrals looks like this: (a to b)|integral(u*dv) = u*v|(evaluate at a and b) - (a to b)|integral(v*du). So in the first term on the right-hand side, we're actually applying the limits to eta itself (and we know it's zero at the boundaries, so it cancels out). In the second-term on the right, we're applying the limits to the WHOLE integral, instead of just applying them to eta. In other words, we're not doing integral(v|(a,b)*du), we're doing (a to b)|integral(v*du). Hope that helps!