Calculus of Variations ft. Flammable Maths

This video is an introduction to the calculus of variations. We go over what variational calculus is trying to solve, and derive the Euler-Lagrange equation, the key partial differential equation to all this.

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0:00 Intro to Variational Calculus
3:38 Derivation of Euler-Lagrange equation
19:20 Application of Euler-Lagrange equation
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Music by Chillhop:

[vcubingx]

Just a quick clarification why we set the integrand equal to 0 by lucabla:
"
The key is that delta_q is an arbitrary continuous function (I don't know if the continuity of delta_q is mentioned in the video, but it should be) and a theorem from calculus states: If f: [a, b] -> IR is continuous and the integral of f*g from a to b is equal to zero for all continuous functions g: [a, b] -> IR, then f=0. Applying this with g=delta_q and f being the Lagrangian, then you get the result.
"
Another explanation by Bram Lentjes:
"
This is not trivial. It is called the "vanishing lemma" in one of my analysis books. This is true when the integrand is a continous function. The proof of this is quite nicely. Let the integrand be denoted f as a function of x. Suppose by contradiction f > 0, then there exist a point in the domain, let's say x0 such that f(x0) > 0. Since f is continous, for a given epsilon > 0, there exist a ball around x0 of radius delta > 0 such that |f(x)-f(x0)| < epsilon. Now choose epsilon like f(x0)/2 > 0, hence. 0 < integral over the ball of f(x0)/2 < integral over the ball of f(x) < integral over domain of f. A contradiction, since we assumed the integral over the domain was zero. 😀
"

[BJCaasenbrood]

For those interested in mathematics, the approach discussed in the video is also called Hamilton's variational principle.

[pattiknuth]

This has got to be THE most enthusiastic presentation of the Euler-Langrange equations ever made.

[leonid]

A less known way to derive the Euler Lagrange equations is the way Euler did it originally: He took a discrete version of the functional (a sum of functions in n variables and discrete difference quotients representing slopes), then differentiated this discrete version of the functional with respect to n variables and took the limit as n (number of variables) goes to infinity, so the array of n variables converge to a function, the sum converges to the integral and the difference quotients converge to the derivative of the function. At the time though, Euler did not prove rigorously the convergence but it turned out to be correct.
That way you can also visually see the terms in the euler lagrange equations and where they come from. Euler's argument is quite intuitive.
The way it is usually derived (due to Lagrange) is more efficient from a computational point of view.

[jakobr_]

I’d heard the calculus of variations mentioned before in passing, but before this video I had absolutely no clue how they got it to work. I was imagining all the different ways to change a function, all the uncountably infinitely many variables (one for each x value) approaching zero you’d need to take into account, instead of just the one variable (h) we use in regular differentiation. How on earth is that manageable? A stroke of genius, that’s how. Just multiply all that variation by a single variable (s), and let that approach zero! It’s so simple!

Beautiful video, thanks for sharing it. I learned a lot

[pubgplayer]

Since I'm a physics boi; I recognised it as the Lagrange Equation with different variable names! I always associate this with a mechanical system.

[GundamnWing]

This is the first time I've understood the derivation, thanks so much!

[joem]

You're the first person I ever trusted who draws their integrals from bottom to top.

[YashMRSawant]

I would like to tell for one of the tricky parts in the proof, at 19:00. To make integral to zero, \delta q can be made a dirac delta with centre at point x which is b/w x_1 and x_2 (which simply means perturbing f only at point x) and therefore right-hand side of Euler-Lagrange equation evaluated at x is zero. This random point therefore can be chosen in any of points b/w x_1 and x_2 which gives us Euler-Lagrange equation.

[devtech]

Your videos help my through 3rd year mathematics and field theory. Very intuitive.

[nunetoyamato]

Thanks a lot for this presentation which helped me understand such important principle. Not that it changes the final result, but just for the sake of being thorough, I think the derivative result should actually be f´´(x) / ( (1 + f´(x)^2)^3/2 ). IIt's easier to get it right when you use the "MATH INPUT" feature.

[caio]

Both are amazing, but I can really see the difference between the extremely didact approach of v3x that really is able to foresee any kind of doubt someone can have when following the lecture, basically making the learning process as smooth as possible. Flammable also does it - and he is really really good (maybe a bit nervous in this video), but for some reason, it makes me think much more to grasp any point of the lecture. But thank you, it was amazing! This is also used in graduate level Economics (Macroeconomics), for example, to find the optimal savings decisions of a household in the economy, btw!

[sebastiangudino]

Im astonished at how good your explainations were, and how bad that other dude's were omg

Like: "Ah, yes, to derive this equation we are going go define the equation in terms of q-hat, and then use Leibniz rule, if you dont know that go watch a tutorial"

My sweet brother, why did you think i came here looking for???

[ianprado]

12:55 flammable looks like he is questioning if too much weird came out for this collaboration

[jacoboribilik]

I'm new to this channel and this video is fantastic. Great, great piece of work.

[kebman]

Within programming we call those callback functions, meaning a function that uses another function as input, and then makes a callback to it. They can also be used in recursive functions, or recursive programming, meaning a function that calls itself.

[trigon]

Wow, manim looks so good but I can’t code... if only

[gergodenes]

So we don't need to take "The shortest distance between two points in Euclidian space is a line" as an axiom anymore?
FINALLY, AFTER ALL THESE YEARS, ONE AXIOM LESS!

[maye]

Papa's becoming international tho.
Daaaamn ;)

[jimklm]

In 19:02 instead of saying that the function in the integral must be equal to zero, we could better say that the expression [θq(L)-d(θq'(L))/dt] must be equal to zero so that the integral of δq*[θq(L)-d(θq'(L))/dt] be equal to zero for EVERY noise-function δq ?