Euler-Lagrange Equation: Constraints and Multiple Dependent Variables

[@johnmccrae52]

Oh my God. I actually understand it! Every other derivation I have seen is absolutely dwarfed by the simplicity and elegance of your explanation.

[@FacultyofKhan]

Thank you!

[@hudsonbarth5641]

This video along with the second video of this series are the best derivations of the EL equations I’ve seen. My textbook makes giant leaps and is super hard to follow. This is great. Thanks

[@shobhapakhare942]

Your videos are very good. I would request you to kindly also make a video on how to take second variation of a functional. This will help all of us to generalize the procedure of taking variation of a functional.

[@p.z.8355]

Maybe you can do a video about the application of what we learned to Lagrangian mechanics ?

[@Rodolfoalvescarvalho]

Thank you! I don't know what do you do to make it so simple!

[@laautaro11]

Best content in all KZbin, thanks a lot this is amazinglly well explained

[@kunzabatool6663]

Thanks for making amazing content.More power to you and your faculty.

[@GSecer]

excellent explanation without going into calculus of variations

[@andrijauhari8566]

Thank you man, u're awesome

[@FacultyofKhan]

No problem!

[@presidentevil9951]

Hey can you include a mix of multiple independent variables and dependant variables and multiple derivatives? also include function shift problems too and mix of solving for functions and a variable too

[@ozzyfromspace]

This is awesome! Here’s a natural application I’m dealing with right now: In electrostatics, the electric field of a charge distribution depends on negative the gradient of the electric potential, where the electric potential is a function of said charge density. Assume the charge is confined to some region, the surface of a spherical conductor, say. The neat thing is that you can set the charge density to be any function on the surface that you want, and you’ll get the corresponding electric field. But in practice, we know that the charge is evenly distributed across the surface, because an even distribution of charge on the surface of the sphere minimizes the potential energy stored in the continuous charge distribution. The energy stored in this charge distribution is given by one half epsilon (a constant) times the volume integral of the square of the electric field (which depends on your choice of phi, the charge density function). By the principle of least action, our aim is to find a charge density function on the sphere’s surface that minimizes the energy stored in the charge distribution. But we already know that the charge distribution is evenly distributed on the surface of our sphere, so why would we do this? Because if we can do it for the surface of a sphere, then we can do it for the surface of an arbitrary object. So solving a slightly modified version of the Euler Lagrange equations for the work functional lets us find the charge that a) solves the Maxwell’s equations, and b) actually exists according to the principle of least action. If you can do this with electrostatics, you can do it with electrodynamics. Thus, you can write numerical code to estimate the electromagnetic response of arbitrary charges and currents in a system. Build some Lorentz Force law into this and boom, you now have a really powerful way to simulate the approximate mechanical and electromagnetic responses of systems with little or no lab work (assuming you’re poor and can’t build a stand to test things, like me lol). Ofc if you don’t wanna use EL, you could just try a ton of different functions by trial and error, but if you’ve seen Faculty of Khan’s video on the geodesic equation on a sphere, you know that the math can get unruly very quick. So quick, in fact, that for real world systems, you probably won’t guess the result (or more timely, your ML approach will probably fail to predict the function that goes into the F() ). Faculty of Khan, your series on Calculus of Variations was the perfect refresher! Thank you for being a mathematical Chad! 😊

[@ohjoshrules]

Thank you so much for this video.

[@SamLaseter]

When you begin the derivation, you first move the derivative operator inside of the integral and change it to a partial. Why is this allowed and why does it become a partial?

[@FacultyofKhan]

It's part of something called differentiation inside the integral. See: en.wikipedia.org/wiki/Leibniz_integral_rule

[@user16391]

I cant wait the next video

[@jinwoongpark6408]

9:31 Where did the function K=I+(lambda)J come from? What is the exact theory behind that?

[@amalguptan6716]

Beltrami identity. Watch the previous videos

[@ravimani8179]

@@amalguptan6716 no it's not - it comes from the method of Lagrange multipliers. You can google it, or this is where I learned it: dec41.user.srcf.net/notes/IB_E/optimisation.pdf

[@zhongyuanchen8424]

How does simply combining the original functional I with the constraint functional G make a functional K that if the EL equation is appplied, the functional I is maximized or minimized? I mean the y that maximize or minimized K could have G taken on any value. How do you make sure that G take on certain constant while applying the EL equation? This is what the Lagrange multiplier is all about right? By the way, I am wondering if there is something similar to the EL equation that not only applied to a line in space but rather a plane or solid in 2-3D？The functional that involed integrating over a high dimensional domain involving Multiple intgral.Would these equations be too advanced to cover?

[@FacultyofKhan]

> How do you make sure that G take on certain constant while applying the EL equation? This is what the Lagrange multiplier is all about right? Yes, that is what the Lagrange multiplier is all about. If you solve a constrained variation problem (assuming one constraint and one dependent variable), you'll end up with 3 unknown constants (2 'integration constants' from solving the EL equation on K and 1 Lagrange multiplier). The values of those three constants are found from 3 equations (2 equations for the boundary condition and the third equation from fact that the constraint functional must equal a particular constant). Thus, by computing the constants, you basically ensure that J takes on the certain constant you want it to take on, while your solution y satisfies the desired boundary conditions AND makes I stationary given the constraints. You'll see this principle in action (pun intended) when I derive the catenary equation in the next video. > By the way, I am wondering if there is something similar to the EL equation that not only applied to a line in space but rather a plane or solid in 2-3D？The functional that involed integrating over a high dimensional domain involving Multiple intgral.Would these equations be too advanced to cover? Yes there is. This is basically when you have a functional with multiple INDEPENDENT variables (i.e. multiple x's), and multiple partial derivatives with respect to each of those independent variables. In that case, you have double/triple/multiple integrals describing your functional. It's still possible to find an Euler-Lagrange equation for them, and I might even cover it in a separate 'other extensions to the E-L equation' video, but right now, I'm going to hold that idea. EL equations arising from multiple-integral functionals don't come up very often in Analytical Mechanics for typical systems, which is what I'm going to cover soon. Hope that helps!

[@zhongyuanchen8424]

How, that is wonderful. I can't wait to see more of your videos on these topic. Your videos have always been helpful for me. Many of your videos have made important but hard to understand concepts that people usually see in a book extremly easy understand. You introduce these important concepts more throughout than a book does by mentioning more points and telling us what to pay attention to. Your videos allow people to grab onto the intuition of new concept easily. Thank you agian for making the start of learning a new concept easier.

[@vishalsinghdhamiak47]

Best series on youtube.., Thanks Man🍾

[@OtoKemo]

What do you mean in "makes our functional stationary"? Finding the extremal of the functional?

[@FacultyofKhan]

Think of a regular function f(x). For a point x_0 to make f(x) stationary, f'(x0) must be zero (stationary = function is not changing at that point). It's a similar idea with calculus of variations, but we're now dealing with functions of functions or functionals (so we have to find a function that makes a functional stationary). In other words, we have to find the function where the 'derivative' of the functional is zero. This could be an extremal (which make the functional a maximum or minimum) as you said or a saddle point.

[@ianbrown6639]

How does one extract the dynamics of the system from the equations of motion that result from this? Does one have to solve for the lagrange multiplier somehow? Confused how this helps

[@lucagagliano5118]

I am not fully convinced the lagrange multipliers are constants, I think in general they're supposed to be functions somewhat (in calculus of variations at least). I am not an expert but it seems that in physics for example they define them as a function of time.

[@Rex-xj4dj]

Question, would this type of method work to find the function that best approximates a real life phenomenon given only the points? What I mean is regression but like, only with accurate points that we measure from a real life phenomenon?

[@justalittlebitoflove6520]

Maybe a stupid question, but when we construct K, why do we need lagrange multiplier? Why is not K=I+J enough for the purpose of killing two birds with a stone?

[@micayahritchie7158]

I'm sorry, just a notational question here. Isn't this actually a multivariate probably where what we're saying is that I is a function of all the epsilon i and that it's stationary precisely when it's stationary with respect to each of them? So shouldn't we really be doing partial I partial epsilon i here?

[@blzKrg]

Can you make a video on isoperimetric problems.

[@FacultyofKhan]

I've made a video on a constrained variation problem, which follows the same process as an isoperimetric problem but is not exactly the same. Maybe it'll help? kzbin.info/www/bejne/pKHXZ3yhrrGSnJY

[@blzKrg]

@@FacultyofKhan thank u so much. It was really helpful.

[@FacultyofKhan]

@@blzKrg No problem! I'll also work on a video on isoperimetric problems (since they're rather important in Calculus of Variations) to be released over the next few weeks.

[@blzKrg]

@@FacultyofKhan oh thanks, i will make sure to watch it.

[@hzkzg1614]

Hi much appreciate your work. Could you please make a video on the essential calculus must knows like chain rule, total differentials. If not can you advise me where to completly understand it like a pro

[@ArduousNature]

This seems analogous to differentiation from first principles to me, so why don't we have to talk about the limit of dI/dƐ as Ɛ approaches 0 instead of just setting Ɛ=0?

[@haggaisimon7748]

like it; very simple and clear. maybe not 100% rigorous, but then it would be harder to follow. for example the derivative with respect to epsilon i should be zero, given that all other y_i's are not y_j bar.

[@matthewjames7513]

Thanks so much for your video! I'm using this to solve my own optimization problem! At 9:36 what would you do if you had an inequality constraint, J, which was just simply y(x)^2 < 1 for ALL x? You could write this as -y(x) < 0 [first constraint] y(x)-1 < 0 [second constraint] And then you'll have K = I + mu_1 * (-y) + mu_2 * (y-1) where mu_1 and mu_2 are constants. But I can't figure out what to do from here. Thanks for your help :)

[@alexboche1349]

Inequality constraints are handled similarly to how they would be in the finite dimensional case. So if the constraint binds, it's treated as an in equality constraint, and if it doesn't bind, it is ignored (λ=0 and so the term it's multiplying drops out). See en.wikipedia.org/wiki/Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions For more rigorous approach, see Luengerber Vector Space methods textbook.

[@yarooborkowski5999]

I don't undersand why You simply put there lambda coefficient (Lagrange multiplier) instead of just sum of functionals? Could You try to prove that we can use Lagrange multipliers for functionals similarly like for functions? Best regards.

[@jamesmonteroso824]

Is the beltrami identity still feasible in this?

[@conceptualmathacademy4619]

What about more than one independent variables.

[@VictorHugo-xn9jz]

Omg, I confused Euler-Poisson (higher derivatives of same y) with this one (first derivatives of multiple y_i).

[@joaovanderven5029]

2:52 : ""What I want to do, is find the particular y_i in this family, that makes our function y stationary"". While you are saying y_i, you are writing y_i-bar. I guess you meant to say y_i-bar, or am I wrong?

[@willyou2199]

11:00 what is lambda?

[@FacultyofKhan]

It's the 'Lagrange multiplier': an unknown constant that you have to solve for when solving the problem. If you'd like to see an example, see my Catenary Problem video (I think you already have since you commented on it)!

[@AdiCherryson]

Could someone help me? What is "i"? I thought i = 1 ... n. So how F(x, y_1, y_1', y_2, y_2', ... , y_n, y_n') became F(x, ybar_1, ybar_1', ... , ybar_i, ybar_i', ... , ybar_n, ybar_n') and what is "i" in this case?

[@FacultyofKhan]

'i' is an index representing a generic term. It can take on any value from 1 to n, which is what I meant by the statement i | 1 -> n. I'm just using 'i' in the computation to compute a generic answer. Hope that helps!

[@AdiCherryson]

Yes, this helped. Thank you for your time. I was just little baffled by the notation 1, 2, ... i ..., n since i is again from 1 to n.