The Math of Bubbles // Minimal Surfaces & the Calculus of Variations
ฝัง
- เผยแพร่เมื่อ 2 มิ.ย. 2024
- 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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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.
Glad it was helpful!
There is something deeply philosophical in the realization, that the universe uses the language of bubbles to calculate 3D multivariable integrals in real time.
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!
by far fhe best explanation of this subject on youtube
Thank you so much!
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.)
Oh ya that’s a great comparison!
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.
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.
Thank you! And I agree:D
@DrTrefor Yes thanks for sharing.
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.
Very cool!
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! 😁😁
Hey thank you so much!
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
Congratulations on the SoME3 win! 👍
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!
Thank you! People also study minimal surfaces using more differential geometry approaches, quite a number of ways
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
Glad you enjoyed it!
Congratulations with #SoME3 winning!
This was a wonderful introduction to the calculus of variations. And congrats on winning the SOME3 contest by the way !
Amazing! Knew about this subject earlier, but learned about you via 3Brown1Blue 🇩🇪😎🙏
Congratulations for winning the contest.
Amazing ! Thank you very much for this great video and explanation !
This was so good Dr. Bazett. Well-done. I'm stoked every time I see that you've uploaded.
Thanks!!
I only really briefly encountered the calculus of variations in my engineering physics program, but I really enjoyed it! I wish I'd had the opportunity to learn more about it. It's so cool to me that by taking the ideas of calculus and extending them up a level, as it were, we can achieve a mathematical model of mechanics that makes it much easier to solve many problems by considering only energy (a scalar), rather than forces (vectors). Clearly it has other applications, but as an EP, this is the one I learned about.
congrats on winning. You deserved it . Fantastic video
Thanks so much!!
#SoME3 is so cool. The amount of mathematical knowledge it produces is mindbogglingly large. Great Video man!
It really is an amazing competition:)
The generalization of this to higher dimensions (minimaly hypersurfaces) is the branch of math that I look forward to learning about the most. Thanks for this video as a great introduction into the topic.
It’s a really cool topic!
imma save this in my folder for when i do classical m&e
Minimal surfaces is the research area of my calculus professor, Tony Tromba (UC Santa Cruz, back in the 20th Century). He would usually end his Friday lectures with something for us to percolate on, such as "The Barber's Paradox" and "consider a function that is 1 when the argument is rational and 0 when irrational.".
That sounds like a great prof!
Amazing..the beauty of math is unparalleled..❤
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.
Thank you so much for this video. I actually was super interested in this topic a while ago, but no videos explained it well so I gave up trying to learn it. You're an amazing teacher and these videos are always phenomenal, I'd love to see more calculus of variations in the future maybe with more complicated shapes
Thank you so much!
Great video as always!!
Possibly the best explanation of calc of variations on youtube. Amazing as usual.
Awsome video and explanation
Very nice!
It's worth adding that many so-called 'minimal surfaces' don't actually minimize area: its just that their variational derivative is zero. (They're the equivalent of "saddle points".) However, for any point on a minimal surface, its possible to choose a small enough region around that point so that the surface _does_ minimize surface area with respect to the boundary of that region. (That is, large regions are not necessarily minimizing, but small enough regions are.)
Absolutely, that's a great clarification, thank you.
Finally helped to understand lagragnges equation of motion
this is probably the best explanation i have seen on this topic so far, great work!
Wow, thanks!
Calculus of variations is so beautiful! Thank you so much for this video!
You're so welcome!
Awesome, learned a lot!!!
The last one is so cool! I would never have guessed.
This was so fun to watch! One of my regrets in life is taking a career/hobby path where I haven't had much need for the really fun calculations from Calculus 1-3. I always enjoyed the puzzling out of problems like what you have here, but you've gotta keep flexing those muscles or they tend to wither. I guess a bit like understanding someone speaking another language vs. speaking it yourself.
I'm a B.S. in Mathematics and M.S. in Actuarial Science. However, I found the lecture in this video very intriguing, insightful and also hard to grasp. I feel like I'm always an infant in the math Kingdom and can never touch a giant's knee lol
WOAH THE CUBE!!!
Isn't that one crazy? So surprising
@@DrTreforIt almost looks like a 3D projection of a tesseract which I think is really cool 🤓.
I learned this in my masters in a module called non-linear continuum mechanics. This is problem has very nice history. It was inspired from the famous Brachistochrone Problem. I really enjoyed the animations.
Good luck in the competition!
Thank you!
Absolutely fantastic video ! I am glad to have discovered your channel, thanks to SoME3
Thanks and welcome!
You are a fantastic Maths educator - thank you.
One of the best explanation of the Lagrangian Equation I have ever seen.
The final solution of the final differential equation with two constants was beautiful.
Thank you - I hope you are going to win the competition.
Wow, thank you!
A true full pleasure. Thanks for this great vidéo ❤
Glad you enjoyed it!
Congrats!
Great video explaining the calculus of variations using an very interesting example. Lagrangian mechanics is abother cool, but more physics-related example of the power of the calculus of variations. Good luck with your SoME submission.
Thank you! I really enjoyed lagrangian mechanics WAY back in my physics undergrad
Great video! One comment I'll make is that there are configurations where the catenoid is still a local minimum but not a global minimum for the surface area. So if you pull apart the two rings as you did at 16:18, there is a short period of time where the surface area of two disks is smaller than the surface area of the catenoid. But since the soap can only sense local variations, it does not immediately jump to the optimal shape. However, once you pull the rings apart far enough, the catenoid is no longer a local minimizer and the film collapses to two disks.
Cool observation!
If the surface area is smaller then why does it only last a short time?
Beautiful
Okay so I’ve watched this video finally! It’s funny, my classical mechanics professor mentioned how calculus of variations was a powerful tool in their physics toolbelt, and now I can see why.
The desmos animation really helped with this, by the way. I can’t imagine explaining this stuff to students without visualizing how the perturbations work… good job!
Thank you!
Oh damn you derived the Euler Lagrange equation niceeee
haha had too!
Love your videos!
Thank you!
Excellence in nerding. Thanks uploader. A nice and clear explanation.
Glad you enjoyed it!
@@DrTrefor I'm going to have to admit that I was strongly swayed in my appreciation by your π and topology tee-shirts.
Outstanding!
Thank you!
I love watching videos like this, but I fear my experience with advanced mathematics is much like Douglas Adams with deadlines (about the only comparison I could ever honestly make with the great DNA❣): I love the sound as they woosh by 🤣#
They tend to stick in my head for a matter of minutes but it does mean that I can watch each video with a fresh anticipation 👍
Great!
Really cool Video!!!!
Thanks!
Yay! You made a submission!
Amazing! Thank you!
Now you're making me want to create a Costa's minimal surface bubble. I'm thinking you could do it by constructing both halves separately and touching their legs to a disk shaped soap film.
Ya it was hard but I’ve seen another person on TH-cam who got it
I really like your T-Shirt!
Thank you!
we were just at a museum looking at soap bubbles!
Cool! And thank you so much:)
Thanks for this video! Could you explain why it is important to consider the Lagrangian as a function of both f, and f' it's derivative? Naively, the derivative is completely determined by the function f, so it is not obvious why f' can be thought of as a free parameter to the Lagrangian, and therefore one can take partial derivatives with respect to it.
This video was published at my birthday
Thanks for the great explanations! Quick question/comment: at 8:11, you apply the Leibniz integral rule. However, the partial derivative of L wrt epsilon is always 0, because there's no direct dependency of L on epsilon. I think writing a total derivative (the indirect dependencies on epsilon through g_epsilon and its derivative, then broken down by the chain rule) would be more appropriate. Any thoughts?
I don’t particularly mind either way, my thinking here was that L has two independent variables in it, and y, as well as dependent variables like f
I dont like maths, but I apreciate the effort. Nice shapes, rad!
Glad you like them!
The concept of minimal surfaces is clear from now on will never forget due to that small experiment and to find these we have a tool called variation of calculus, but my question is where do we actually use these minimal surfaces concept? Area of application , why do we need to know about these surfaces, anything which makes me inspire to know about this subject where I can use it quite often being a mathematician and a engineer?
I like how nature has auto-optimizing capabilities.
I think an architect used soap bubble surfaces to design a high hanging roof of a stadion. The frame was constructed in a scale model, dipped in soap water and produced the optimal roof to be '3D'-photographed.
I will make some experiments with frames and soap {(.-D={
I've seen a photo of that too!
How does the maximum x separation vary with with the sizes of the discs? It'd be interesting if that could be derived from the physics of surface tension too.
Paused at 3:00 to try to solve it.
ff'' = 1+f'f'
Solution is r0 cosh(x/r0 + phi) with r0 and phi chosen to match boundary conditions.
So the big idea is that we introduce a modifier that we then show to be irrelevant - ahh yes the mathematical equivalent of a judo throw where the thrower uses the solution against itself.
So do I upvote for the graphics, the soap film models or the explanation of the calculus of variations - tough choice.
Haha if only one could triple upvote:D
I once blew a bubble and it flew straight into our dog's eye and his eye turned red
I've heard that the method of calculus of variations fails if you are trying to determine the minimal surface for a bubble on bubble. It has something to do with the singularities that exist at the intersections of boundaries.... or something.
That (for all P fP = 0) implies f = 0 principle gets abused so hard when working out the basics of Statistical Mechanics. Arbitrariness is powerful when it comes to minimization
Math is cool
Can you please elaborate as to why f(x) being minima implies phi'(0)=0 and not f'(x)=0 and f"(x)>0? Thanks!
When epsilon=0, then g_e=f. So we are evaluating at zero to make sure we get f which is what we are claiming is a minimum. And then that the derivative equals zero is just the usual requirement that for any (differentiable) function to be a minimum needs to be zero. If not, you could change the epsilon and get something smaller.
@@DrTrefor Ah thank you! Great video BTW.
but whats the parametric surface equation of the hexa-hexaflexagon
You need a better TH-camr than I for that!
@@DrTrefor smh,..- where are the Vi-Hart collabs when we need them. its topological like a mobius strip with multitudes of twist, which is 'trivial', but those darn cusp/fold has led me to catastrophe theory, like more then chaos?really ,..really
As far as I can remember a strict (in the modern sense) proof requires calculus in infinite-dimensional Banach spaces.
Your proof is Eulerian or good enough for physicists 🙃
ha yes, in effect I claimed a particular method and argued for its reasonableness but I didn't set out to prove the method
@@DrTrefor Of course, the "modern" proof is a bit difficult for TH-cam.
I remember well that it appeared in the appendix of Analysis III by Reiffen/Trapp almost 50 years ago.
The year before Analysis was taught based on the books by Dieudonné. That must have been really hard.
The argument is fine, the only detail that wasn't explained is why you can put the derivative inside the integral sign, but that is easily justified using the dominated convergence theorem. What do you mean with infinite dimensional calculus in banach spaces? Do you mean the direct method?
It’d be epic if your shirt expanded the 🍩 into a straw
I don't understand. Why do bubbles minimize surface area?
I thought bubbles are just fun. Now, they also do math. You ruined bubbles for me. \j
this is so complicated
Gawd, I hate the fake upbeat energy.