ALL OF MECHANICS depends on this one integral
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9 ай бұрынMy new channel for formal math courses:
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@artham6565 9 ай бұрын
As a Physics Schlor.....I can say you have done a great job. Better then my professors🤐🤐🙂🙂🙂🙂
@maths_505 9 ай бұрын
This comment made my day ❤️ Thank you
@MrFtriana 9 ай бұрын
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 9 ай бұрын
analYzed or it means something else 😁
@MrFtriana 9 ай бұрын
@@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.
@cblpu5575 9 ай бұрын
I can't believe this, just started studying lagrangian methods. Thanks
@Calcprof 9 ай бұрын
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.
@user-fz9go8pj4t 9 ай бұрын
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)!
@willalsagoff4355 9 ай бұрын
As an engineering graduate (who doesn’t do engineering anymore), this was wildly refreshing
@martinstyblo6355 9 ай бұрын
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:-)
@joshuawalsh6968 9 ай бұрын
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 KZfaq mathematicians explain the concept . Thank you, keep up the good work mate 👍
@spaceman4286 9 ай бұрын
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?
@sergten 9 ай бұрын
Best 20 minutes of my day. Fascinating topic, please keep'em coming!
@maths_505 9 ай бұрын
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.
@nickbarrett1968 9 ай бұрын
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.
@naturallyinterested7569 9 ай бұрын
Very Cool, thanks!
@MrWael1970 9 ай бұрын
Good Proof. Thank you
@manstuckinabox3679 9 ай бұрын
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.
@maths_505 9 ай бұрын
I'm gonna add physics to the regular mix of calculus and DEs
@maths_505 9 ай бұрын
Gonna do some tensor analysis too
@manstuckinabox3679 9 ай бұрын
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
@PhysicsNg 9 ай бұрын
I love it
@Boteruna 3 ай бұрын
Holy! Thank you so much, made Lagrange seen alot easier
@thewolverine7516 9 ай бұрын
Amazing! I always learn something new by watching your videos. Hats off!
@maths_505 9 ай бұрын
I'll be adding more physics on a regular basis
@thewolverine7516 9 ай бұрын
@@maths_505 That's my boi benzi(Jamaal)!
@maths_505 9 ай бұрын
@@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.....
@thewolverine7516 9 ай бұрын
@@maths_505 Eagerly waiting for the goodies then.....
@Bernhardseckm 5 ай бұрын
@@maths_505 😀
@AB-nu5we 9 ай бұрын
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.
@michelenunziata6948 9 ай бұрын
I did this axam a few months ago and it was amazing
@nitayweksler3051 6 ай бұрын
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 :)
@wagsman9999 9 ай бұрын
sweet
@jalepezo 9 ай бұрын
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
@garymemetoo2238 9 ай бұрын
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.
@fwgaming3838 9 ай бұрын
(Sees the thumbnail) Hey, we've got some action going on here! ... I'll see myself out.
@g.o.a.t4674 9 ай бұрын
10:42 bro is this the calculus of variations Great video bro you are the landau lifshitz of youtube ❤❤
@maths_505 9 ай бұрын
Yup....it's calculus of variations. Thanks bro❤️
@g.o.a.t4674 9 ай бұрын
@@maths_505 I don't understand Calculus of variation. Can you do some videos on it too?
@asdfasdf-dd9lk 9 ай бұрын
have you considered doing a video on path integrals given you seem to be approaching the topic?
@maths_505 9 ай бұрын
Yeah I'm gonna be adding alot more physics to the mix
@asdfasdf-dd9lk 9 ай бұрын
@@maths_505 oh awesome im excited for them :)
@monikaherath7505 9 ай бұрын
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
@maths_505 9 ай бұрын
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.....
@jesusangulosolano8375 9 ай бұрын
Plssss explain hamiltonians as well
@thomasblackwell9507 9 ай бұрын
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!
@maths_505 9 ай бұрын
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.
@erivaldolopes632 9 ай бұрын
the equations are called Euler-Lagrangean equations. Euler first got it in the field of maths and lagrange rediscovered it! 😆
@maths_505 9 ай бұрын
It's Euler's world....we just happen to live in it
@matthias7335 9 ай бұрын
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?
@milessodejana2754 3 ай бұрын
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"
@milessodejana2754 3 ай бұрын
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.
@maths_505 3 ай бұрын
Yes exactly
@onusiddartha1641 9 ай бұрын
This was literally way out of my league 😂😂😂
@anupamamehra6068 9 ай бұрын
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)
@vascomanteigas9433 9 ай бұрын
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.
@MatthisDayer 9 ай бұрын
when are we getting statistics content
@maths_505 9 ай бұрын
Nah bro not exactly my piece of cake
@danielyurodidon8312 9 ай бұрын
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...
@maths_505 9 ай бұрын
Maybe in general but for our context it's perfectly fine as there are no loopholes to interfere with our analysis of mechanical systems.
@angelmendez-rivera351 9 ай бұрын
It is very reasonable. It is an example of a homotopy.
@erickherrerapena8981 9 ай бұрын
Salaberga que basado.
@daddy_myers 9 ай бұрын
WOOOOOOOOO BABAYYYYYYYYY
@maths_505 9 ай бұрын
WOOOOOOOOOOOO
@daddy_myers 9 ай бұрын
Absolute gem of a video, I'll be the first to binge if this becomes a series.
@A2431A 9 ай бұрын
Hey big fan! Question: do u have coding knowledge? If so pls upload related videos too!
@maths_505 9 ай бұрын
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
@artham6565 9 ай бұрын
@@maths_505 🤐🤐👍👍
@daddy_myers 9 ай бұрын
Mfw ChatGPT writes shitty code:
@A2431A 9 ай бұрын
@@maths_505 dayum I'm broke
@maths_505 9 ай бұрын
@@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