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!
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!
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!
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!
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
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!
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!
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?
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
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!
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
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.
@@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.
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 .
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.
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.... 🤣🤣🤣
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!!!
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
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...
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!
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!
when you wrote 0 at 20:00 my mind clicked, that was amazing!
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!
wooooooooow, such an elegant and easy to follow explanation
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
Excellent video! Thank you sir
If this is my prof, I would love to attend 5 hours of class everyday.
Love your pace Dr Peyam!
20:45 WTF!
I laughed so hard xD
Nice video, Dr. π🍠
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
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!
Thanks Dr Peyam, this is very usefull in physics.
I really enjoy this video. You explains this at the easiest way, for me. :)
My favourite subject. Not much on youtube covering 2nd variation and conjugate points.
Very nice work, as always Peyam!
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
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*
Thank you very much. Very helpful.
Thanks once again, Doc.
Thanks, nice explanation!
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.
'..and it minimizes the Dirich.., this energy!.." Just say it damn it!
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?
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!
Very nice video!
Awesome Lecture! thank you sir :)
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!
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.
Great video thanks so much! Could you explain at 28:17 how solving the minimization problem would allow us to solve the summation?
9:06 smiley face with a straw hat.
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.
If you are a fan of functional analysis, could you do a video one day on Path Integrals?
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!
always the shortest past its a straight line
Wow, now at last I have a hope that I'll understand something in this cool topic
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
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
Can you continue with this topic and make it a series?
Seconded.
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.
- "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
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
Yay calculus of variations!
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.... 🤣🤣🤣
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
20:40 my reaction to all this
More stuff on PDEs please!!!
Will do!
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
ua-cam.com/video/3YARPNZrcIY/v-deo.html
Also, Lagrangian mechanics is the basis for quantum field theory, which leads to Feynman diagrams and all that cool stuff!
Imagining the voice of Gilbert Gottfried here
18:04 "Mr. or Mrs. Integration by Parts" 🤔
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...
I smell stone weierstrass to prove the first fact.
pls try not to block whiteboard while writing.
I learned a lot about the calculus of variations from this playlist: ua-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 :)
first
Calculus literally means study of change.