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I was able to derive the first equation of motion at 12:06, however, I don't understand how you derived the second equation of motion. Could you explain this?

wow

This has given me hope in understanding why T-V. Can this also be deduced by adding the constraint g(x,dx/dt)=T+V -Etotal to represent the conservation of energy?

Great stuff here! Thank you.

I'm sorry, but at 16:48 you incorrectly imply that η(t)=c*y(t). (This is based on your curves.) Because you don't parameterize η(t) into a general function of t, the video is rather incomplete.

So... let me get this straight, You're arguing the principle of least action is equivalent to saying the time average of the lagrangian over any time interval is uniquely determined since the trajectory in energy space is also uniquely determined as a result of deterministic classical mechanics. Therefore, the path that the particle takes must be such that dL = 0, i.e. a local extremum of L, as if this were not the case, a slight variation in path would produce a variation in the lagrangian, thereby producing a path infinitesimally different to the original, but with a different time average of the lagrangian? Thanks 4 the video! Awesome stuff :) Critique welcome of above phrasing!!! It was conceived very quickly.

The unpleasant noise, the supposed "music," played over the voice here is very unpleasant. Looking on the good side, it has shown you who you have in your organization who is so stupid they need to be fired. First, fire the moron. Then re-make the video, OK?

Amazing video, man! 👏

This series is great!! Did you stop making videos?

Amazing video! Thank you very much

these series on lagrangian mechanics are seriously the best videos on lagrangian mechanics I have ever seen! Funnily enough I live about 800 meters away from where Lagrange used to live here in Turin

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?

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

Him: "And that is how we found the Kinetic Energy by making many restrictions~" Me: "... That is one big equation for a super simple problem." Also, Amazing series btw. Can I ask how you create these beautiful animations? Thank you again!

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

Why did we stop for tea?

The best Lagrangian video series I have seen.

But what is the delta δL means? 12:28

PLEASE CONTINUE THE CHANNEL! YOUR WORK IS AMAZING, MAN.

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

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

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.

@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

Excellent content. Hate the music.

For the rotational kinetic energy you used the moment of inertia about the cylinders center of mass. Why didn't you then apply parallel axis theorem to get the inertia about the pivot point? Since this cylinder is not rotating about its center of mass.

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

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

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

Such a satisfying payoff to visualize the eigendecomposition at the end!

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

What a deeper explanation in all 3 video on Classical Mechanics! Thank you sir to make Lagrangian approach a part of my understanding the Nature. Regards!

Thank you! I finished all three videos. The last one I had to rewind in a couple of places since there are quite a bit of calculations. These are by far the best series on Lagrangian Mechanics I can find on the internet. I understand this may be just a hobby project but the production is just unbelievable. Can't imagine how many hours you have put in creating these videos...

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

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. <3

Very great video 😊

Hey, where is the new one😢

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

We really Miss you sir 😢

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

you gotta make more videos man.

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

Amazing lectures. What software do you use to make these graphical geometries?

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

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

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

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

Great video, wonderful explanations. 🎉

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 is such a great video thank you !!

Helped me grasp what I couldn't in classroom. Thank you!