Euler-Lagrange Equation: Constraints and Multiple Dependent Variables

In this video, I begin by deriving the Euler-Lagrange Equation for multiple dependent variables. I show that in order to make a functional involving multiple y's stationary, it is necessary to solve an Euler-Lagrange equation for each of those y's. This is going to be useful when we work in 2-D or 3-D coordinate systems to solve Action Problems in Classical Mechanics.

In the second part of the video, I show how to approach variational problems when there are one or more constraints involved. The technique described comes from Lagrange multipliers and is a relatively simple one. This will also come in handy for my classical mechanics videos where there are constraints imposed on the particle's motion.

Questions/requests? Let me know in the comments!

Prereqs: Just these two videos (you could probably watch the rest of the playlist too, which is what I would recommend):

Special thanks to my Patrons for supporting me at the $5 level or higher:
- Jose Lockhart
- Yuan Gao
- Justin Hill
- Marcin Maciejewski
- Jacob Soares
- Yenyo Pal
- Chi
- Lisa Bouchard

[johnmccrae]

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

[hudsonbarth]

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

[laautaro]

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

[shobhapakhare]

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.

[Rodolfoalvescarvalho]

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

[p.z.]

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

[kunzabatool]

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

[vishalsinghdhamiak]

Best series on youtube.., Thanks Man🍾

[GSecer]

excellent explanation without going into calculus of variations

[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! 😊

[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?

[jinwoongpark]

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

[lucagagliano]

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.

[ianbrown]

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

[zhongyuanchen]

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?

[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?

[hzkzg]

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

[blzKrg]

Can you make a video on isoperimetric problems.

[justalittlebitoflove]

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?

[Rex-xjdj]

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?