Calculus of Variations
ฝัง
- เผยแพร่เมื่อ 15 ก.พ. 2018
- 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!
Here’s the link to the pictures about minimal surfaces:
en.wikipedia.org/wiki/Minimal_surface#Examples
Click on each name to see a pretty picture :)
Enjoy!
Maybe I just haven't seen enough math abbreviations but I typically don't associate "WTF" at 20:40 with "Want to Find" haha
One of the most charming and charismatic math teachers I have ever seen.
❤️
I would say he is definitely the most charming
What does not vary is the excellence of these videos!
when you wrote 0 at 20:00 my mind clicked, that was amazing!
If this is my prof, I would love to attend 5 hours of class everyday.
Love your pace Dr Peyam!
Very nice work, as always Peyam!
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
I really enjoy this video. You explains this at the easiest way, for me. :)
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!
Thank you very much for the great explanation, Sir!
Awesome Lecture! thank you sir :)
wooooooooow, such an elegant and easy to follow explanation
20:45 WTF!
I laughed so hard xD
My favourite subject. Not much on youtube covering 2nd variation and conjugate points.
Thanks Dr Peyam, this is very usefull in physics.
Excellent video! Thank you sir
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.
That’s a really good point, thank you! :) But the cool thing is that even for local minimizers (even just critical points), we still get the PDE
Also, the cool thing is that under strict convexity, local minimizers are global minimizers, but as you said, in general they’re different!
That’s a really good point, thank you! :) But the cool thing is that even for local minimizers (even just critical points), we still get the PDE
Also, the cool thing is that under strict convexity, local minimizers are global minimizers, but as you said, in general they’re different!
OMG, Go Bears!!! :D I bet it must have been very different in the sixties 😝
To which I can only reply, I bet it must be very different now.
When I lived there, Berkeley was full of hippies, many of whom did not attend the university. The effects were pervasive. A few scenes might set the stage, but there were many more:
On my first walk down Telegraph Avenue, I heard a stage whisper: "Grass... acid... psilocybin." Such sales were in no way legal, and my 17 year old East coast self suddenly felt startled, amazed and scared, all at once.
The supermarket near my apartment was populated with individuals sporting the strangest hair and clothing you might imagine. Some were barefoot. On my first entry I had to stop and stare for several minutes before venturing to shop. Fifty years later I can still feel a bit dazed by it all.
At some point my girlfriend and I signed up for an extension massage class, held in the Student Union. You had to bring a partner and a beach towel so you and the other person could take turns lying on the floor and massaging or being massaged. I guess there were about 100 couples at the first class, with the result that Pauley Ballroom was full of entirely naked people on towels being massaged by their clothed partners. Halfway through, the instructor announced a switch, and the previously clothed were naked and the once-naked clothed. I recognized a small and very pretty girl in that room who was in my ODE class, and I could not quite look her in the face for the rest of the quarter.
I realize now that the local and global minima might turn out to be the same, but the first time I saw all this that was not at all clear -- with the result that I got seriously confused and even outraged!
Very nice video!
Thanks once again, Doc.
Thanks, nice explanation!
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!
There’s a functional analysis overview on my channel
Nice video, Dr. π🍠
Thank you very much. Very helpful.
Great video thanks so much! Could you explain at 28:17 how solving the minimization problem would allow us to solve the summation?
Is it really necessary to keep the minus in -f"(x)=0?
Yes, it’s the correct way of setting up a PDE :P The reason being is that when you multiply the equation by any function g (which is very common) and you integrate by parts, that minus sign disappears!
Wow, now at last I have a hope that I'll understand something in this cool topic
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, >= ^2 ( = mean of a function), hence I[f] = ^2 =(integral of f')^2 = f(1) - f(0) = 1. So h(x)=x minimizes I
Hey, i came across a lecture about Kurzweil Integral.Can you do video about it, since you have one about Lebesgue Integral?
It can integrate some functions that aren't Lebesgue integrable, but is much easier to learn.
Thank you, love your videos .
Interesting! I’ll think about it!
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*
Yay calculus of variations!
In the first part of this video there is a similar idea with differentiate a non-differenciable function. The following recall me a bit the calculus of the minimum in statistics series of 2 variables. The end with the example is hard to understand. Thanks.
9:06 smiley face with a straw hat.
If you are a fan of functional analysis, could you do a video one day on Path Integrals?
'..and it minimizes the Dirich.., this energy!.." Just say it damn it!
When it comes to minimization problems is it necessary to keep track of the 1/2 at the front tof the integral?
In this case no, but there are more general functionals like 1/2 |Du|^2 - uf where it matters
Was it through the theory of distributions that you calculate the derivative of a nondifferentiable function in order to calculate energy?
Yep, that’s how to do it, and it leads to more general minimizers of energy functionals, especially when the Euler-Lagrange PDE doesn’t have a smooth solution!
There’s a video on that, btw, in case you haven’t seen it :)
Thanks!
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?
Thanks for asking but I doubt it, it’s too advanced. Check Evans’ book for viscosity solutions
@@drpeyam Prof. Peyam, Thank you for your reply. I am aware of the book.
Haben sie schon einmal etwas vom Satz von Hamilton (Satz der kleinsten Wirkung) gehört?
Der kommt aus den Energiemethoden der Mechanik und arbeitet auch mit Variationen die an den Zeitgrenzen verschwinden und stellt am Ende sowohl die (partielle) DGL als auch die Randbedingungen auf.
So werden zum Beispiel die
Lagrange-gleichungen 1. und 2. Art hergeleitet.
Wikipedia:
de.m.wikipedia.org/wiki/Hamiltonsches_Prinzip
Ja, ich hab das ein bisschen studiert :) Stimmt, es gibt viele Ähnlichkeiten hier!
I recognize ∫ √(1+∇f)² dx as the arc length formula for a two dimensional function, which is basically the same thing as surface area in three dimensional functions
By two dimensional function do you mean function of one variable? And three dimensional function you mean function of two variables?
Can you continue with this topic and make it a series?
Seconded.
More stuff on PDEs please!!!
Will do!
I know if a functional is extremal by euler lagrange equation but how do i know if that extremal is minima or maxima?
Second derivatives
@@drpeyam is there any way in the calculus of variations? I heard that there is a second variation method and some say that it can be done by weistrass function but i can't seem to find it anywhere.
always the shortest past its a straight line
why we multiply by one half to find the energy? is it a convention?
Yeah, just a convention. It’s because the derivative of 1/2 x^2 is x, which is nicer than 2x
1:11 can anyone tell me how to get this expression? Thank you.
It’s given, like a given ode or pde. Physically it is the kinetic energy
Me recordó las clases de Matemáticas donde los más nerds de la carrera de matematicas exponian temas... los gestos, la manera de hablar... como que todos ellos pertenecen al mismo conjunto.... 🤣🤣🤣
Im lost all over the place. 11:37 Is g only zero and endpoints 0 and 1 and not in between? And if int _0 ^ 1 {fg} = 0, for any g such that at boundary point g is zero, f has to be 0 everywhere in the interval?
g could be zero in between as well, but what we know for sure is that g is zero at the endpoints.
And correct, if the integral is zero for *every* g, then f must be zero everywhere in the interval
To see that, suppose f were positive in some interval between 0 and 1. We could choose a g that is positive on that interval and zero everywhere else, and get a nonzero integral. Likewise if f is negative in some interval. So f can't be positive or negative anywhere, therefore it must be zero everywhere.
20:40 my reaction to all this
What does it mean to square the gradient? Isn't the gradientn a vector?
We’re squaring the norm
of the gradient, that is taking the sum of squares of the components
Dr. Peyam's Show oh, yeah, I kissed that. Thanks
I watched till 2:30 for now ( have some more stuff to do ) But,
OMG that seems so familiar I mean isn't that a minimum principle or something like that?
u think it's a. way of calculating how gravity would work or something like that, what I want is for someone to clarify wether I'm right or wrong!!!
Yep, it’s definitely motivated by physics, I’m not surprised if it shows up there as well :)
Variation principles are fundamental in hamiltonian and lagrangian mechanics, they are one of the way to find Euler-Lagrange's equations, Hamilton's equations, and most of all Hamilton-Jacobi's equation (witch is a first order non-linear partial differential equation) and the Hamilton's principal function!
Calculus of variations is greatly used in quantum mechanics, and yes it is used in the minimum action principle, as a proof for snell's law and the brachistocrone problem! So lot's of stuff
en.wikipedia.org/wiki/Principle_of_least_action
th-cam.com/video/3YARPNZrcIY/w-d-xo.html
Also, Lagrangian mechanics is the basis for quantum field theory, which leads to Feynman diagrams and all that cool stuff!
18:04 "Mr. or Mrs. Integration by Parts" 🤔
- "but what we know about U?, we know that f is positive in U and also g is positive in U"
- oooh you stop it
I learned a lot about the calculus of variations from this playlist: th-cam.com/play/PLdgVBOaXkb9CD8igcUr9Fmn5WXLpE8ZE_.html
Oh man, and here I thought I was being original 😝
Dr. Peyam's Show it's okay you are doing it in A new way we can see your face and all that exciting motion of yours is helping alot :)
I smell stone weierstrass to prove the first fact.
Imagining the voice of Gilbert Gottfried here
I have only one thing to say to this video: I hate function space calculus!
not so one thing but: 26:05 there are a lot of different letters you can use as functions' name...
pls try not to block whiteboard while writing.
first
Calculus literally means study of change.