Not only the mechanics, almost everything in physics can be expressed in lagrangian terms; the variational principles are a very powerful tool when you construct physical models. Also, it is interesting see that the symmetry principles and conservation laws in physics can be analized by this approach (Noether theorem)
@@david-joeklotz9558 My Bad jejejejejeje, but i hope you understand where was going that parte, using the lagrangian to study the symmetries of a physical system.
There is a great discussion of Least Action in Feynman's Lectures on Physics, Volume II. Teaching a graduate course on Calculus of Variations was one of my favorite courses to teach.
Hi Dear maths_505. This introduction to the Lagrangian mechanics and the derivation of the EL (Euler-Lagrange) equations, and using Langrange's method to minimize the action functional (this action integral is called a functional, a special case of linear operator), and this cute example that leads to the Newton's 2nd law is awesome. I would also be very glad if you do series of videos, a mini lectures on the calculus of variations, a really fascinating math analysis theme of its own. Actually I have started some research on Bergman spaces, which are Lp spaces of analytic functions on the unit disc in the complex plane with special norm and metric, and there is some investigation of functions (called extremal functions) which minimize some special norms on these spaces, or equivalently unit vectors (i.e functions of norm 1) which have zeros on prescribed points in the unit disc. These are like a generalization of calculus of variations to the complex plane. Fascinating ( for me)!
Great, many thanks. Just I wish to have seen this 30 year ago... You did this much easily to follow than the traditional courses of theoretical mechanics do:-)
Yes 🙌 this has made my day! I am on a Lagrangian binge lately, so it's an absolute pleasure to see one of my favourite TH-cam mathematicians explain the concept . Thank you, keep up the good work mate 👍
Great video, exactly what i needed! Glad more analytical mechanics are coming up, could you perhaps emphasize a bit more on the general coordinates? Maybe explicit examples?
Years back, I watched Leonard Susskinds MITOCW series on classical mechanics. That took about a week. Somehow you made the same point in 20 minutes. Fantastic.
Yeah I'm gonna be adding more physics videos this week and in the long term as well. I've been wanting to for a long time but my screen real estate wasn't allowing me to do so.
In 2:57, i belive you didnt mention something quite important which is an assumption about the constrain function, as i recall it MUST be a function of the coordinates and only the coordinates in order to be correct. for instance think of some G(x,y,t)=x+y+sin(wt). please let me know what you think :)
Dude such breeze of fresh air trying out new content type! this is one heck of an explanation of lagrangian which I keep hearing about in the bunch of QM books here and there, I'm so down with more random explanations videos from time to time.
BROOOOO I WAS JUST STARTING WITH TENSOR ANALYSIS!!! l got The Bishop and Goldberg book and thought the topology section was a hassle, plus not enough exercises. @@maths_505
@@thewolverine7516 oh and I also have fit in tensors sometime....I should probably upload a course on it after complex analysis before I start messing around with special relativity here.....
As a math major, I bow down to you sir, also statistical methods in finance are baby steps compared to statistical mechanics and such, I AM LOOKING AT U MACHINE LEARNING
Jennifer Coopersmith wrote a fascinating book titled: "The Lazy Universe - An Introduction to the Principle of Least Action" that explores the evolution of the history and the mathematics of the Principle of Stationary Action from humble beginnings right up through the topic of this video. Well worth the read. Also, a big well done for this video.
Crazy timing. I just had my Arfkin open rereading near the end, Calculus of Variations. Arfkin has a nice discussion of CoV ending up with the Euler equation.
omg are you starting a math course, like MathMajor by Michael Penn? I'm starting undergrad soon but that would be absolutely amazing. I'd binge watch all of that lol
Already started it....link in the description of this video. As far as physics is concerned, that's still part of the math for fun thing going on for this channel. But hey, you never know.....
Hello, thank you for this video! Just a question, can you please explain further what this sentence of yours means? "all paths including the chosen path start and end at exactly the same value"
okay I understand it now, I thought you meant the initial and starting points are the same (that is it's closed). what you meant really is that the initial points for each path is the same as well as the final point.
Derive the zeta Functional equation. zeta(x) = 2^x*%pi^(x-1)*sin(%pi*x/2)*gamma(1-x)*zeta(1-x) I once made a demonstration that uses complex analysis. From the integral zeta(x)*gamma(x) = integral(t^(x-1)/(exp(t)-1),t,0,Infinity) Create a keyhole integral path on complex plane of integral-line(t^(z-1)/(exp(t)-1),t,C) with contour C= line(r,R) and circle(R) and line(R,r) and circle(r) where r=0 and R=Infinity. Using the Theorem of Residues gives the result. Notice that the poles of exp(t)-1 are infinite but periodic, and gives t=2*%pi*%i*k for all integers k. The infinite sum of residues are equal to the infinite sum of the zeta function, and this gives part of the formula. By other hand, the arc circles are bounded by a quantity that drops to zero with a restricted domain for z. The periodic exponential t^(z-1) over the branch cut on the positive real axis separate two branches with 2*%pi*%i wide split. The original integral are recovered and make equal to the sim of residues. Once made the algebraic manipulations the result are obtained. You need to use the Euler Reflection Formula (which use the same keyhole contour but with the Function t^(z-1)/(1+t), and the pole are just t=-1), given by gamma(z)*gamma(1-z)=%pi/sin(%pi*z), to finish the Proof.
sir; Great video as usual. However, please include free body diagrams when you are doing mechanics problems. It makes it easier for simpletons like me to understand. Thank you!
Hello my friend. Always wonderful to read your comment ❤️ I'd love to include diagrams but this particular topic didn't allow for free body diagrams. I might need different kinds of diagrams moving forward but for now they are simply not applicable. I will try to include diagrams whenever possible.
Is it reasonable to write qi(t, α)=qi(t,0)+αλi(t)? Although this way of derivation could avoid the mess from the calculus of variation and is pretty enlightening, I think it lacks mathematical rigor...
As a Physics Schlor.....I can say you have done a great job. Better then my professors🤐🤐🙂🙂🙂🙂
This comment made my day ❤️
Thank you
Not only the mechanics, almost everything in physics can be expressed in lagrangian terms; the variational principles are a very powerful tool when you construct physical models. Also, it is interesting see that the symmetry principles and conservation laws in physics can be analized by this approach (Noether theorem)
analYzed or it means something else 😁
@@david-joeklotz9558 My Bad jejejejejeje, but i hope you understand where was going that parte, using the lagrangian to study the symmetries of a physical system.
There is a great discussion of Least Action in Feynman's Lectures on Physics, Volume II. Teaching a graduate course on Calculus of Variations was one of my favorite courses to teach.
I can't believe this, just started studying lagrangian methods. Thanks
Hi Dear maths_505. This introduction to the Lagrangian mechanics and the derivation of the EL (Euler-Lagrange) equations, and using Langrange's method to minimize the action functional (this action integral is called a functional, a special case of linear operator), and this cute example that leads to the Newton's 2nd law is awesome. I would also be very glad if you do series of videos, a mini lectures on the calculus of variations, a really fascinating math analysis theme of its own. Actually I have started some research on Bergman spaces, which are Lp spaces of analytic functions on the unit disc in the complex plane with special norm and metric, and there is some investigation of functions (called extremal functions) which minimize some special norms on these spaces, or equivalently unit vectors (i.e functions of norm 1) which have zeros on prescribed points in the unit disc. These are like a generalization of calculus of variations to the complex plane. Fascinating ( for me)!
Great, many thanks. Just I wish to have seen this 30 year ago... You did this much easily to follow than the traditional courses of theoretical mechanics do:-)
Yes 🙌 this has made my day! I am on a Lagrangian binge lately, so it's an absolute pleasure to see one of my favourite TH-cam mathematicians explain the concept . Thank you, keep up the good work mate 👍
As an engineering graduate (who doesn’t do engineering anymore), this was wildly refreshing
Great video, exactly what i needed! Glad more analytical mechanics are coming up, could you perhaps emphasize a bit more on the general coordinates? Maybe explicit examples?
Years back, I watched Leonard Susskinds MITOCW series on classical mechanics. That took about a week. Somehow you made the same point in 20 minutes.
Fantastic.
Best 20 minutes of my day. Fascinating topic, please keep'em coming!
Yeah I'm gonna be adding more physics videos this week and in the long term as well. I've been wanting to for a long time but my screen real estate wasn't allowing me to do so.
In 2:57, i belive you didnt mention something quite important which is an assumption about the constrain function, as i recall it MUST be a function of the coordinates and only the coordinates in order to be correct. for instance think of some G(x,y,t)=x+y+sin(wt). please let me know what you think :)
Dude such breeze of fresh air trying out new content type! this is one heck of an explanation of lagrangian which I keep hearing about in the bunch of QM books here and there, I'm so down with more random explanations videos from time to time.
I'm gonna add physics to the regular mix of calculus and DEs
Gonna do some tensor analysis too
BROOOOO I WAS JUST STARTING WITH TENSOR ANALYSIS!!! l got The Bishop and Goldberg book and thought the topology section was a hassle, plus not enough exercises. @@maths_505
Amazing! I always learn something new by watching your videos. Hats off!
I'll be adding more physics on a regular basis
@@maths_505 That's my boi benzi(Jamaal)!
@@thewolverine7516 oh and I also have fit in tensors sometime....I should probably upload a course on it after complex analysis before I start messing around with special relativity here.....
@@maths_505 Eagerly waiting for the goodies then.....
@@maths_505 😀
Good Proof. Thank you
Holy! Thank you so much, made Lagrange seen alot easier
I've understood nothing, but your voice makes me continue watching 🙂
Thanks bro
As a math major, I bow down to you sir, also statistical methods in finance are baby steps compared to statistical mechanics and such, I AM LOOKING AT U MACHINE LEARNING
Jennifer Coopersmith wrote a fascinating book titled: "The Lazy Universe - An Introduction to the Principle of Least Action" that explores the evolution of the history and the mathematics of the Principle of Stationary Action from humble beginnings right up through the topic of this video. Well worth the read. Also, a big well done for this video.
Very Cool, thanks!
Crazy timing. I just had my Arfkin open rereading near the end, Calculus of Variations. Arfkin has a nice discussion of CoV ending up with the Euler equation.
I love it
I did this axam a few months ago and it was amazing
have you considered doing a video on path integrals given you seem to be approaching the topic?
Yeah I'm gonna be adding alot more physics to the mix
@@maths_505 oh awesome im excited for them :)
omg are you starting a math course, like MathMajor by Michael Penn?
I'm starting undergrad soon but that would be absolutely amazing. I'd binge watch all of that lol
Already started it....link in the description of this video.
As far as physics is concerned, that's still part of the math for fun thing going on for this channel. But hey, you never know.....
Plssss explain hamiltonians as well
(Sees the thumbnail)
Hey, we've got some action going on here!
... I'll see myself out.
Hello, thank you for this video! Just a question, can you please explain further what this sentence of yours means? "all paths including the chosen path start and end at exactly the same value"
okay I understand it now, I thought you meant the initial and starting points are the same (that is it's closed). what you meant really is that the initial points for each path is the same as well as the final point.
Yes exactly
hi maths 505: we know that there is definition for zeta(s) = sum n>=1 1/n^s for Re(s)>1 . is there a definition for Re(s)
Derive the zeta Functional equation.
zeta(x) = 2^x*%pi^(x-1)*sin(%pi*x/2)*gamma(1-x)*zeta(1-x)
I once made a demonstration that uses complex analysis.
From the integral zeta(x)*gamma(x) = integral(t^(x-1)/(exp(t)-1),t,0,Infinity)
Create a keyhole integral path on complex plane of integral-line(t^(z-1)/(exp(t)-1),t,C) with contour C= line(r,R) and circle(R) and line(R,r) and circle(r) where r=0 and R=Infinity.
Using the Theorem of Residues gives the result. Notice that the poles of exp(t)-1 are infinite but periodic, and gives t=2*%pi*%i*k for all integers k.
The infinite sum of residues are equal to the infinite sum of the zeta function, and this gives part of the formula.
By other hand, the arc circles are bounded by a quantity that drops to zero with a restricted domain for z. The periodic exponential t^(z-1) over the branch cut on the positive real axis separate two branches with 2*%pi*%i wide split. The original integral are recovered and make equal to the sim of residues.
Once made the algebraic manipulations the result are obtained. You need to use the Euler Reflection Formula (which use the same keyhole contour but with the Function t^(z-1)/(1+t), and the pole are just t=-1), given by gamma(z)*gamma(1-z)=%pi/sin(%pi*z), to finish the Proof.
10:42 bro is this the calculus of variations
Great video bro you are the landau lifshitz of youtube ❤❤
Yup....it's calculus of variations. Thanks bro❤️
@@maths_505 I don't understand Calculus of variation. Can you do some videos on it too?
16:10 Why couldn’t the sum of (partial q-i to alpha) over all i be zero as well? The derivations of the generalized coordinates could be negative, no?
the equations are called Euler-Lagrangean equations. Euler first got it in the field of maths and lagrange rediscovered it! 😆
It's Euler's world....we just happen to live in it
sir;
Great video as usual. However, please include free body diagrams when you are doing mechanics problems. It makes it easier for simpletons like me to understand. Thank you!
Hello my friend.
Always wonderful to read your comment ❤️
I'd love to include diagrams but this particular topic didn't allow for free body diagrams. I might need different kinds of diagrams moving forward but for now they are simply not applicable. I will try to include diagrams whenever possible.
Is it reasonable to write qi(t, α)=qi(t,0)+αλi(t)? Although this way of derivation could avoid the mess from the calculus of variation and is pretty enlightening, I think it lacks mathematical rigor...
Maybe in general but for our context it's perfectly fine as there are no loopholes to interfere with our analysis of mechanical systems.
It is very reasonable. It is an example of a homotopy.
when are we getting statistics content
Nah bro not exactly my piece of cake
sweet
This was literally way out of my league 😂😂😂
Hey big fan!
Question: do u have coding knowledge? If so pls upload related videos too!
Step 1 subscribe gpt 4
Step 2 enter prompt for desired code
Step 3 copy code
Step 4 paste code
Step 5 execute code
Step 6 bask in the glory of efficiently written perfect code
@@maths_505 🤐🤐👍👍
Mfw ChatGPT writes shitty code:
@@maths_505 dayum I'm broke
@@A2431A oh it's cool I'm using gpt 3.5 too 😂😂😂
Coding there is op too.....makes you wonder about the future of programming and data science
Salaberga que basado.