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

Most excellent! Both rigorous and intuitive.

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

OMG! U re the only person on Youtube that actually understands Langrangian mechanics! please make a video about Hamiltonian mechanics too!

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 ...

THE BEST VIDEO ABOUT LAGRANGIAN MECHANICS IM ACTUALLY CRYING
Edit: Can you make a separate video about Hamiltonian mechanics?

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

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

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

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

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.

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

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!

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.

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

Incredible, you made everything click for me so beautifully!

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

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

I love how you discovered subspace, so to speak, by vectorizing energy space the way you did. Energy is a scalar in physical space but a vector in this space beneath the fold. Thus, finding the L scalar in the direction of constant energy feels foundational; a mirror between the spaces, to use your analogy. As others have said, I'm surprised that I've never seen this done before. It's been a long time since my days officially poking around in phase and configuration spaces, back when I was a physics undergrad. But your approach feels deeply satisfying and intuitively correct to me. Understandably, you may be spending time learning more and/or writing papers, either for researchers or educators. But I hope you pick it up again after your hiatus and continue your clear and thoughtful investigations.