Calculus of Variations

In this video, I give you a glimpse of the field calculus of variations, which is a nice way of transforming a minimization problem into a differential equation and vice-versa. And the nice thing is that I'm not using much more than single-variable calculus, enjoy!

[drpeyam]

Here’s the link to the pictures about minimal surfaces:


Click on each name to see a pretty picture :)

Enjoy!

[mitchkovacs]

Maybe I just haven't seen enough math abbreviations but I typically don't associate "WTF" at 20:40 with "Want to Find" haha

[Peter_]

One of the most charming and charismatic math teachers I have ever seen.

[punditgi]

What does not vary is the excellence of these videos!

[TheRedfire]

when you wrote 0 at 20:00 my mind clicked, that was amazing!

[yugzed]

If this is my prof, I would love to attend 5 hours of class everyday.

[RalphDratman]

Dr. Peyam, thank you for an excellent talk, the best introduction I have seen to the calculus of variations.

I have a constructively critical suggestion. Every time I have watched or read an introductory presentation about the calculus of variations, it has bothered me that the problem has been described as if this method could find a global minimum functional result, rather than just a local minimum. In the simplest version of the problem, as you present here, it turns out that the result is also a global minimum. But in a slightly more complicated version of the problem there is also a potential function over which the functional result has to be minimized.

If in that version of the problem the method were required to find a global rather than merely a local minimum, finding the answer would in general be much harder. In fact the problem might be NP-complete in the case of certain potential functions, because it would be equivalent to a traveling salesman problem.

I suggest it would be much better to mention early on that the result to be found is only required to be a local minimum rather than necessarily a global minimum. Otherwise someone new to all this can get the impression that the D.E. method is performing some kind of magic.

What do you think?

P.S. I was an undergraduate at Berkeley in the late 1960s.

[marsag]

Dear Dr. Peyam, great lecture. I would suggest a follow-up lecture on the Pontryagin's minimum principle. That would be well connected with this one. Great job as usual!

[aleksybalazinski]

I think that in this particular example, you could also do it like this: let h(x)=x, then I[h] = 1. For any arbitrary function f, <f^2> >= <f>^2 (<.> = mean of a function), hence I[f] = <f'^2> >= <f'>^2 =(integral of f')^2 = f(1) - f(0) = 1. So h(x)=x minimizes I

[fcvgarcia]

Thank you very much for the great explanation, Sir!

[skeletonrowdie]

wow this was so clear thank you!! i've trying to get to the bottom of lagrangian physics and this really hit the last nail in the trainrails :p

[thatkindcoder]

Is left with the DE: -f''(x) = 0
Me: "Oh that's simple enough."
Dr Peyam: "This is known as the Euler-Lagrange PDE"
*Boss music plays*

[MusicKnowte]

such an elegant and easy to follow explanation

[genaromarino]

I really enjoy this video. You explains this at the easiest way, for me. :)

[RossMcgowanMaths]

My favourite subject. Not much on youtube covering 2nd variation and conjugate points.

[frankym]

Dear Peyam! Thank you for your amazing lectures and I really like the ways you break things down and detail every step. It would be awesome if we can learn some abstract stuff like functional analysis, measure-theoretic probability theory from you!

[bermudezfelipe]

Thanks Dr Peyam, this is very usefull in physics.

[xy]

Is it really necessary to keep the minus in -f"(x)=0?

[tamimyousefi]

'..and it minimizes the Dirich.., this energy!.." Just say it damn it!

[gareebmanus]

Thanks for another excellent video lecture. Can I request you to cover a bit more advanced topics like Viscosity Solutions and Gamma Convergence, etc. please?