This is the most elegant description of Lagrangian mechanics I have yet seen. Mind blowing stuff.

Never saw such a good explenation of Lagrangian Mechanics. Wish that my professors would have just a tenth of your teaching skills.

FINALLY someone who actually explained where the lagrangian came from!!!! Thank you!!!

Best visualization I've seen of the Lagrangian mechanics! thank you

I've been wondering about that minus sign for many years. Thank you very much for this very insightful video.

Need more videos in this channel Perfect explanation

This is the ONLY explanation I've seen on KZfaq. I dont understand why others, including physics luminaries, just dump the formula on the screen and then explain algebra. This is BRILLIANT!

One of the best explanation of Lagrangian.....Thank you ❤️

I had been waiting for that video for almost 30 years ... First time ever I see an explanation for why the Lagrangian is T-V. Awesome work, thank you! The only obscure part left as far as I am concerned is the one justifying that the time variation of the mean value of L should be 0, and how it relates to determinism ...

I look forward to seeing a mathematical connection between spacetime geometry and the quantum mechanics states for momentum and position. Something that is natural and common sense.

Excellent demonstration of the thought behind how Lagrangian is created. Thanks so much! This is particularly useful for people who wants to pursue a more rigorous mathematical thinking behind Physics.

This is a masterpiece. It would be nice to compare and contrast with Hamiltonian mechanics which uses the sum of kinetic and potential energies.

Until I watched this video, I knew why the derivative of the functional S with respect to epsilon should at 0 should be 0 and this improve me to understand more the Euler-Lagrang Equation, so thank you veryyyyyyyy much.

This has to be the most precise and clear explanation of Lagrangian Mechanics that I've ever come across. Brilliant.

I watched this 3 times! This is by far the most beautiful and clear graphical illustration of Lagrangian Mechanics fundamentals I have seen! Thank you so much for ur work and really look forward to more videos of urs!

13:16 "In most treatments of Lagrangian Mechanics, students are simply handed this formula, and left wondering why on earth one would take an interest in the difference between the kinetic and potential energies" YES!! This has been me for the past 20 years since I was handed the formula in *my* classical mechanics course! Thank you for this wonderful explanation! Can you tell me whether the Lagrangian for a charged particle can be derived in a similar way, by considering a constraint relation between the various energies? Also, did you get this insight from a particular textbook, or is it original to you? Either way I hope people spread the word far and wide because it is a very beautiful explanation!

Agree with others ... the (very intuitive) explanation where L=T-V comes from is reason enough to watch the video and to merit a like and subscribe to make sure I watch future videos ... absolutely brilliant

Flipping the potential energy axis in the 3D-view tied a knot in my brain.

I´m stuying Physics and I love this video. I read the Goldstein and is the same facts but this explanation is exponentialy better

Awesome work! Looking forward to part 2

Beautiful, Elegant & Robust. Awesome explanation!!! Thank you. The use of vector calculus very clever and showing where Lagrangian (T-V) appears. Looking forward to further videos.

Wonderful! I have no "proper" math/physics education (it's only a kind of hobby for me) but so far this video (in my experience at least) tried to show the same thing from the most number of aspects to really grasp the idea behind. I am aware what lagrangian mechanics is about, but this video is really a great help to deepen my knowledge about the core idea behind.

Fantastic video. Thank you so much. At 19:48, roughly, you probably mean "integration by parts" .. would be systematic .. not trial and error.

Really glad I was recommended this, I'm a novice in physics but your video brought a lot of clarity to something that's mystified me until now. Looking forward to your future works!

i'm really looking forward for the continuation of these videos you are great!

Simply beautiful derivation of the Euler-Lagrange equation! Thank you. And your suggestion that the extremum may represent some stability point for our universe reminds me of the Archimedes' law of the lever when it is construed as a special case of the conservation of angular momentum. It's as if every conservation law can be understood in analogy to the equation of equilibrium for an Archimedean lever. The universe always seeks balance.

Extremely good description. I took "classical mechanics" last semester which actually just turned out to be an engineering dynamics course, so of course we didn't cover any of the interesting physics like gravitation into keplers laws, or lagrangian mechanics at all. We just did a bunch of stupid rube-goldberg math. Anyways, apart from that rant, now that the semester is over and grad school applications are all sent off I've been trying to teach myself the things I wasn't taught in class. I've acquired a couple classical mechanics textbooks, watched many videos, and this is the ONLY ONE that's ever motivated WHY the lagrangian is T-V. The energy space and allowed coordinates only being a single line, which can be represented by a single equation combining the two unit vectors is crazy. It make so much sense and only requires knowing you can make different coordinated spaces and knowing total energy is conserved! As well, I don't fully understand yet why the single trajectory restriction implies that the variation in the _time average_ of the lagrangian is 0. Shouldn't it just be the variation in the Lagrangian, or even just the integral of it over time? Why time average? Anyways, I've subscribed and will be watching this entire series. You have done amazing work here and I will be rewatching and taking notes on every video you make! I would love if you made more videos some day but definitely only do what you want to. Your passion for this material has made an incredible lecture and I'd be saddened to see another amazing creator run down by the algorithm. Thank you so much for your work, you have impressively contributed to the sum of human knowledge for all of us to enjoy.

I'd be very interested to see how Lagrangian mechanics translate into a numerical context, if that's something you'd be able to cover in a later part of this series. I've read of people representing e.g. orbital trajectories in terms of Lagrangian state variables to ensure that energy is conserved even with large timesteps and large integration error (where integrating the Newtonian equations directly bleeds off energy to that integration error over time). But I've never found any good resources on how to actually use Lagrangian mechanics in in a way that makes it amicable to numerical rather than symbolic methods.

Beautiful, beautiful, beautiful! Thank you so much!!!

excellent stuff

Man you are great . your intuition is concrete. Thank you for sharing it

Please make more videos, we can’t get enough of thissss

It never made sense to me why L=T-V until I saw this video! this is the best explanation I've found!

soo underrated !!!

That was truely fascinating. Briliant work, well done.

Sir, this is a pristine video about a really hard topic in classical and quantum mechanics. Great great great! Thank you very much!

this is so good !!!!! actually explains things in a satisfying way.

I certainly hope you continue to post videos about more advanced ideas in classical mechanics - and other areas of physics too of course - but there is currently a lack of excellent material on mechanics at a high level and these are excellent.

Absolutely amazing!

Amazing work and extremely helpful! Thank you!

Wow that was mind blowing. Thank you for your work!

The Best Explanation 👌, sir thanks

Excellent video! Thank you!

Hi. First, thanks a f* lot for your wonderful video. Now, did LAGRANGE have a coherent explanation for (L - V)?

Great video, wonderful explanations. 🎉

Cool! Thank you!

Magnificent !

I want to know more about the art of forming Lagrangians, especially when other forms of energy are involved, say for example, thermal energy, alongside kinetic and potential energies. In such cases, we have three unit vectors for the energy axes, and we are perhaps dealing with Lagrangian planes. The confusing/ambiguity is coming from the subtraction. Your thoughts? I’d appreciate if you could share some resources towards forming Lagrangian when more than two forms of energy are present in the physical equilibrium.

Very great video 😊

LOTS OF RESPECT, APPRECIATION AND GRATITUDE .... (from Kashmir)

This is such a beautiful and unique explanation of Lagrangian mechanics. I am curious on how you thought up/discovered such an explanation and what you used to make the video

wow, this is amazingly well made and helped me a tonne! amazing job!

I’ve never seen the idea L = T - V explained in any logical way. It makes the generalized coordinates make much better sense.

Thank you so much!

Nice that you have changed least action to stationary action...

@14:24 I couldn’t visualize taking the signed integral of the L function and how it gives positive and negative areas wrt time axis. It looks all positive area

Thank you so much for this work. ¿There is a bibliographic reference or an achademic resource that we can use to cite this proposal to understand Lagrangian? or ¿How can we cite your work?

I love this, Lagrangian mechanics may not necessarily be new, but this provides excellent alternative insight to the knowledge I was taught in Newtonian mechanics. I struggle to understand exactly why we know that L(t) and y(t) are unique with respect to the initial conditions, and why that further violates determinism. Are you saying that energy and position are independent of one another, and thus it's not possible to determine both functions with only initial conditions? I cannot wait for the second video to this!

Subbed!

Wonderful. Do you mind if I pleade ask you which software you are using to create the animations. Thank you.

In qm, the Lagrangian is not the most probably trajectory, it's the average trajectory. If you have 50% chance of +1 and 50% chance of -1, average is 0 but you will never get 0.

you gotta make more videos man.

What textbook did you use to learn the derivation of the Lagrangian Or did u figure it our yourself. Pls respond 🙏🏼.

subbed

Why do you treat it like a surprise when the S turns out to be a minimum? Doesn't your postulate dS = 0 basically force S to be a an extremum point with respect to the trajectory? Also when using determinism to set a constraint for L, is there a reason we can't just have dL = 0 as our postulate instead of using the average?

Why cant we say dT or dV = 0 I guess because L contains both kinetic and potential energy so average height of curve is affected by each

This is the best visualized introduction to Principle of Least Action. I'm sorry Eugene, your vid was a bit confusing and couldn't understand the same way.

But what is the delta δL means? 12:28

17:44 oh f(x+b) ~ f(x) when f'(x) = 0

Why did we stop for tea?

Newtonian mechanics is great. But wouldn't it be better if physicists could find a connection between Lagrangian mechanics and quantum states for momentum and position?

As a 16 year old who got bored of ways of newton , this new approach for solving complex systems is like better

You assumed potential and kinetic energies as vectors, but they are not. If they were, as you put it, the square of the total energy would be the sum of the squares of potential and kinetic energies, which we know is not true. Furthermore, even making this consideration, there would still be other errors. During your calculations, you assumed that the unit vector for L would be at -45°, which would only occur if kinetic and potential energies were always equal in magnitude all the time.

Behavior of alpha, beta and gamma within a medium......

Stupid musical background! How can one think adding noise enhances the signal?

Ultraviolet catastrophe and about phonons are very pleasant to see yours vids on these.

Великолепно!!! Огромное вам спасибо! Вот бы аналогичное для гамильтониана!