Lagrangian Mechanics I: Introducing the fundamentals
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- Опубліковано 23 вер 2024
- In this video, we discover the classical Lagrangian, the principle of stationary action and the Euler-Lagrange equation. For the best viewing experience, make sure to watch in full-screen and with 4K (2160p) resolution.
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#Physics #Lagrangian #Mechanics
This is the most elegant description of Lagrangian mechanics I have yet seen. Mind blowing stuff.
Should be preserved for eternity
Never saw such a good explenation of Lagrangian Mechanics. Wish that my professors would have just a tenth of your teaching skills.
Thank you for your kind words!
This is the ONLY explanation I've seen on UA-cam. I dont understand why others, including physics luminaries, just dump the formula on the screen and then explain algebra. This is BRILLIANT!
Agreed! I took high school physics 70 years ago, and the Newton was all we got. I've been figgering out this Lagrangian thingie bit by bit as I go along, but it's a huge relief to see it done simply and competently like this.
Well done and thank you, Physics Fuency folks!
FINALLY someone who actually explained where the lagrangian came from!!!! 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 ...
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.
It never made sense to me why L=T-V until I saw this video! this is the best explanation I've found!
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!
Thanks for watching, glad you found the explanation useful! Like you I've been frustrated by the lack of reasonable interpretations of T-V, and have been looking for such an interpretation for some time, as a sort of hobby project. I eventually realized that since the Hamiltonian T+V and the Lagrangian T-V are the symmetric and anti-symmetric combinations of T and V, they should in some sense be orthogonal, which led me to consider an energy space, and the rest sort of just unfolded from there. I have not found anything similar in any textbook or online, which is the main reason why I wanted to make this video and share what I'd found.
As for the Lagrangian of a charged particle, I haven't really gone there yet, but it would be interesting to think about, for sure! It would make sense for it to work the same way.
@@physicsfluency5541 well in that case you might like to put a short paper on arxiv about it, so that you can claim this discovery as your own! ;-)
@@physicsfluency5541 Insta-subscribed after that!
@@MessedUpSystem Yeah, me too haha
@@physicsfluency5541
_"Like you I've been frustrated by the lack of reasonable interpretations of T-V, and have been looking for such an interpretation for some time, as a sort of hobby project. "_
Quite right.
Unfortunately, I see a shortcoming of your video in the following:
(11:55): _"One way to translate it into math is by postulating that no variation of the time average of L is allowed."_
And this "postulate" seems to come right out of nothing, from outer space or somewhere, abracadabra. Am I missing something? It claims that it has something to do with the "deterministic nature of the universe", but in no way is there an explanation how you would come from determinism to this postulate. It's just "handed down".
Best visualization I've seen of the Lagrangian mechanics! thank you
This has to be the most precise and clear explanation of Lagrangian Mechanics that I've ever come across. Brilliant.
This means a lot, thank you so much!
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This is a masterpiece. It would be nice to compare and contrast with Hamiltonian mechanics which uses the sum of kinetic and potential energies.
Thank you for the compliment! I agree, unfortunately my knowledge of the Hamiltonian version isn't good enough for me to consider making a video about it yet, but perhaps sometime in the future. What is interesting (perhaps you noticed) is that the constant vector which gets us onto the Lagrangian line in this video is essentially the Hamiltonian (E_t/sqrt(2)) times a unit vector in the direction (T hat) + (V hat), which is of course orthogonal to (L hat). This means that the energy trajectory can be written as the vector sum E = H*(H hat) + L*(L hat) where H is the total constant energy. I considered going into this in the video but felt that it was a bit outside its scope.
I totally agree. A Hamiltonian treatise would be incredible.
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!
That's awesome, thank you! More videos are on the way =)
Man you are great .
your intuition is concrete.
Thank you for sharing it
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
Awesome work! Looking forward to part 2
Thanks! Hopefully I'll get it done reasonably soon :)
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.
I´m stuying Physics and I love this video. I read the Goldstein and is the same facts but this explanation is exponentialy better
goldstein doesnt get at the nitty gritty of the negative sign like this one!
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.
Except he makes a mistake here. See my comment.
i'm really looking forward for the continuation of these videos you are great!
Beautiful, beautiful, beautiful! Thank you so much!!!
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.
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.
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One of the best explanation of Lagrangian.....Thank you ❤️
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’ve never seen the idea L = T - V explained in any logical way. It makes the generalized coordinates make much better sense.
excellent stuff
soo underrated !!!
Please make more videos, we can’t get enough of thissss
Will do! =)
I've been wondering about that minus sign for many years. Thank you very much for this very insightful video.
this is so good !!!!! actually explains things in a satisfying way.
The Best Explanation 👌, sir thanks
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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.
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!
Need more videos in this channel
Perfect explanation
Fantastic video. Thank you so much. At 19:48, roughly, you probably mean "integration by parts" .. would be systematic .. not trial and error.
I guess that might be a better word for it - though it for sure feels like trial and error a lot of the time (at least when I do it). ;)
That was truely fascinating. Briliant work, well done.
Thank you!
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.
Flipping the potential energy axis in the 3D-view tied a knot in my brain.
Sir, this is a pristine video about a really hard topic in classical and quantum mechanics. Great great great! Thank you very much!
Hi. First, thanks a f* lot for your wonderful video.
Now, did LAGRANGE have a coherent explanation for (L - V)?
LOTS OF RESPECT, APPRECIATION AND GRATITUDE .... (from Kashmir)
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.
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.
This is a very good and interesting question. I've considered how friction (which for a mechanical system increases the thermal energy of the parts) would fit into this model and my first instinct was like you suggest to add a third axis (say, U) for the internal energy. For a "closed system", the Lagrangian would then be confined to a plane and we would have two degrees of freedom for the trajectory in energy space, rather than one. I'm not sure how to find the corresponding Lagrangian then, but it would have to be a 2-variable function as far as I can tell, involving T, V and U.
Then again, internal energy is essentially a measure of the kinetic energy of the particles which make up the system. So perhaps the correct way to account for it would be to model the system on an atomic scale. We would then still only have two energy dimensions, T and V.
Amazing video! Thank you very much
Amazing work and extremely helpful! Thank you!
wow, this is amazingly well made and helped me a tonne! amazing job!
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Magnificent !
Amazing video, man! 👏
Wow that was mind blowing. Thank you for your work!
Cool! Thank you!
Very great video 😊
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
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.
Great video, wonderful explanations. 🎉
Excellent video! Thank you!
Nice that you have changed least action to stationary action...
This series is great!! Did you stop making videos?
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!
The idea is that both L(t) and y(t) should be unique (as long as we specify initial velocity and position), since anything besides that would violate determinism, which also seems to be what you're suggesting. Hope it didn't seem like I was implying the opposite!
Glad you liked the video! =)
@@physicsfluency5541 Thanks for the clarification! Your knowledge is much appreciated :)
Wonderful. Do you mind if I pleade ask you which software you are using to create the animations. Thank you.
Thanks a lot! I'm using MATLAB. It doesn't really have much support for making this type of animation though, I've created a whole bunch of my own functions for manipulating equations and stuff.
@@physicsfluency5541 Nice and thank you.
Thank you so much!
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?
Good questions, thank you!
You are correct that dS = 0 indeed forces S to be an extremum, but it may not be very obvious to most people (at least it wasn't to me, at first), so I wanted to emphasize that point a bit.
As for dL = 0, as far as I know, it is an equivalent constraint, as you suggest. The reason we'd rather use the average is that it allows us to deal with a single number rather than a curve by taking the integral of L, and this conveniently leads to the Euler-Lagrange equations through calculus of variations. Now that I think about it, it probably would've been more intuitive to start with dL = 0 and then suggest taking the integral from there.
@@physicsfluency5541 Thanks a lot for the explanation!
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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?
@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?
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.
wow
Subbed!
What textbook did you use to learn the derivation of the Lagrangian
Or did u figure it our yourself.
Pls respond 🙏🏼.
As a 16 year old who got bored of ways of newton , this new approach for solving complex systems is like better
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.
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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
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.
But what is the delta δL means? 12:28
Wondering this as well, does it mean that the L function will always remain on the "L-axis"? There's two energy and one time direction in which the curve could be seemingly "perturbed"
17:44 oh f(x+b) ~ f(x)
when f'(x) = 0
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?
This is the path integral formulation of quantum mechanics.
@@ryan-cole Military and civilians are observing UFOs. We are encouraged to take a closer look at the underlying quantum mechanics that make the laws of motion work. Conservation laws, gravity, etc.
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.
Since, as you say, energy is not a vector quantity, the vector sum in kinetic & potential phase space is not meaningful. 45⁰ line in phase space expresses the energy conservation constraint.
Why did we stop for tea?
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!