A Taste of Calculus of Variations

Dirichlet's Principle

In this video, I give you a taste of calculus of variations by illustrating Dirichlet's principle, which says that a function u is a minimizer of a certain Dirichlet energy (kinetic + potential energy) if and only if u solves Poisson's equation. This is a neat way of relating minimizers of energies with PDEs.

[realhumphreyappleby]

Dr peyam laughing at FU made my day lol

[thedoublehelix]

The proof for the Euler Lagrange equation has always struck me as unreasonably clever like I can't even imagine I'd be able to come up with something like that. It basically splits up this huge space of functions by choosing an arbitrary function as an "axis" and minimizing along that axis to ghrn extrapolate to the pde

[albertemcstein]

I love how its all coming together in the end.

[sundaybutane]

Love how he says Laplace with a french accent

[drandrewsanchez]

Call me a physicist/engineer but I would love to see more applications. Thank you!

[Chariotuber]

*chefs kiss* beautiful video Dr. Peyam!

[koenth]

"For *every* τ?" - "Yes, Terence, every τ"

[mcqueen]

Thanks for this! Would love to see more as I’m thinking about working on optimization problems in grad school and this lets me know what’s out there.

[akadirizm]

Hi Dr. Peyam. Thank for the video. It is very helpful. Which software do you use for this?

[frozenmoon]

-fU :D and people who are emotional couldn't get over this probably. Hilarious and very educational video as usual :)

[Tomaplen]

Could you maybe do a video on why the Euler density integrated is proportional to the Euler characteristic? or wheere to find info about that

[brianez]

I was thinking that the pudding is actually in the proof because the proof is where all the delicious substance is

[Stelios.Posantzis]

This is much better than the shiny whiteboard. Hope it's here to stay - the reflections on the whiteboard were a bit harsh on the eyes. The image quality is much better also and it's far easier to read what's written.

[Hobbit]

Could variational calculus solve differenital equations? i've heard that it has some connection to the finite element method

[scose]

Would you consider doing a proof of Pontryagin's principle?