The Math of Bubbles // Minimal Surfaces & the Calculus of Variations #SoME3

This is my entry to the #SoME3 competition run by @3blue1brown and @LeiosLabs. Use the hashtag to check out the many other great entries!

0:00 Fun with bubbles!
0:46 Minimal Surfaces
2:35 Calculus of Variations
6:27 Derivation of Euler-Lagrange Equation
11:31 The Euler-Lagrange Equation
13:10 Deriving the Catenoid
15:25 Boundary Conditions

Bubbles naturally try to minimize surface area, and so if we make a wire-frame boundary for the bubble to attach to, the big question is what is that minimal surface going to be? And how can we compute it out mathematically? In this video I am going to approach this question from the perspective of the Calculus of Variations. We will see that the surface area for one of the simplest shapes, the catenoid formed between two parallel circles, results in what is called a functional - a surface area integral in terms of a function f(x) and its derivative, so our question will be what function minimizes that surface area integral. We will derive the 1 dimensional Euler-Lagrange equation via the Calculus of Variations and apply it in our case to deduce the f(x).

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[hvok]

You take expert care to explain the intuition and reasoning for every calculation step, thank you for the presentation. Really helps to keep the big and beautiful picture in mind while wading through all of the minutiae of studying.

[davidhill]

I took calculus a generations or so too early. Had you, 3B1B and everyone else been there when I was an engineering student, I would have enjoyed it so much more. Your enthusiasm is wonderful, as are your explanations.

[quantheory]

I'm a mathematician myself now, but when I was an undergraduate physics major, I always felt uncomfortable with the way that the Euler-Lagrange equation was presented, without explaining in any intuitive way where it came from. I wish I'd been able to see this video back then!

[burakki]

by far fhe best explanation of this subject on youtube

[HeroDarkStorn]

There is something deeply philosophical in the realization, that the universe uses the language of bubbles to calculate 3D multivariable integrals in real time.

[johnchessant]

Great video! The square bubble at 0:39 reminds me of the famous "Steiner problem": given four towns arranged as vertices of a square, minimize the total length of roads you build between them such that each town is connected to every other. (Hint: The answer is _not_ the two diagonals; you can go even shorter.)

[tobywang]

Great video about Calculus of Variation! This is always, I think, one of the most important topics in Lagrangian and Hamiltonian mechanics - the principle of least action.

[benhsu]

we were just at a museum looking at soap bubbles!

[micayahritchie]

Very good video. Takes me back a couple years to when I was 17 trying to figure out how to characterise a geodesic on a come because my younger brother said the liquid ice running down his cone was in a straight line and my parents said no it wasn't because it curved around the cone.

I tried to find this using all the methods I know and gave up and looked it up and thats where I saw calculus of variations for the first time. I did examples and understood but never did quite get that cone thing to defend my brother. I finally got it a few months ago in the last year of my degree, when I came across pictures of the scrap book i did all my working in back then. Its good to see growth.

Anyway I was rambling. Point is, 6 years later and I think this is the cleanest introduction I've ever witnessed, this is including both from my micro economics and classical mechanics courses. Great work

[StratosFair]

This was a wonderful introduction to the calculus of variations. And congrats on winning the SOME3 contest by the way !

[gibson]

I did this exact concept as my senior thesis (minimal surfaces) using calc of variations. Super cool to see you employ the same tools (bubbles!) and explain it in a great way. Love the way you approached it. Great video.

[uwuifyingransomware]

Really good video! I studied minimal surfaces a while ago from a more algebraic perspective (in terms of symmetry groups in particular, and extending to infinite minimal surfaces) so it’s great to see a more analytical approach. This sort of topic and its nature as a minimisation problem makes it awesome to explore with calculus of variations, though the algebraic approach is beautiful in its own way. Good luck with your entry!

[agrajyadav]

Possibly the best explanation of calc of variations on youtube. Amazing as usual.

[BrainOfAPenguin]

HI, here from the SoME3 voting! Just wanted to reiterate what I said, but there are so many things that make this video good as an explainer and for youtube. There are quick cuts at the beginning and satisfying visuals, and you also incorporate good explainers and animations. Good job! 😁😁

[NuclearMex]

Congratulations for winning the contest.

[ProgressiveEconomicsSupporter]

Amazing! Knew about this subject earlier, but learned about you via 3Brown1Blue 🇩🇪😎🙏

[aayushbajaj]

haha well played with the submission! this is one of my favourite videos of yours. thank you for the sophisticated math, and thank you even more for making it look so easy :D

[joemcz]

I've been watching several calculus of variations videos recently and none have put it as intuitively as you have. For the other texts/videos, I've followed along with the steps of derivation for the E.L. equation, but I had to go over it multiple times to make sense of what every step truly meant. Though I wish I had seen this one sooner, perhaps even first, I at least appreciate having seen it now.

[kruksog]

This was so good Dr. Bazett. Well-done. I'm stoked every time I see that you've uploaded.

[johnnelcantor]

this is probably the best explanation i have seen on this topic so far, great work!