Відео

Fascinating.

I watched this video a while back and understood nothing So I took after studying multivariable calculus, vectorial geometry and linear algebra, I came back with full confidence Now I understand up to 2:15

This is a very good explanation. good enough that i have it book marked so i can watch it whenever i need a refresh on lagrangian mechanics.

Yo it’s Pharrell!

Reminds me of the lopitowel rule!

Love you bro:

Commenting to boost your engagement bro❤️🫦

Wow

Great example. Why exactly is it that v=lθ' (7:13)?

I love your honesty here and your pursuit of developing an intuition. I too am trying to develop a deeper understanding of Lagrangian Mechanics, hope to see you there. Thanks for the great video.

This was an amazing video, thank you so much for making it!! As a math major, the most intuitive proof for the euler-lagrange equations is writing d'alamberts principle in terms of generalized coordinates. :) The concept of generalized coordinates is a bit tricky, but pretty intuitive after doing a few basic examples. Just means every coordinate is independent of the others (i.e. for a pendalum, the x-position depends on the y-position, so they are not "generalized coordinates", but the angle is (since its a single coordinate). D'alamberts principle is a bit trickier to understand, but really its F=ma but also considering constraint forces. Again, do a few examples (e.g. double pendulum) and it starts making sense! Put the 2 together, and you get the Euler-Langrange equation.

Amazed, Thanks. This is a fairly new intuition for me.

7:30 thanks a lot for NOT using the stupid approximation in case of pendulum.

this is mindblowing bro, excellent video intuition makes math so much more beautiful

About the visualization that you start at around 2:00 The type of visualization you implemented there is key: you plot the kinetic energy and the potential energy and you demonstrate how they change as you sweep out variation. At that point you are so close to cracking it! When I implemented my own visualizations I did the following: I mirrored the plot of the potential energy, so that it was a plot of the minus potential energy. Returning to your visualization: At 2:53 you demonstrate what the two plots are like for the physical trajectory. There the plots for the two energies mirror each other; the plots have *matching slope* everywhere (with opposite sign). The matching slope is expression of the work-energy theorem; the rate of change of kinetic energy must match the rate of change of potential energy. I will use the expression 'trial trajectory' for the hypothetical trajectory that is evaluated at any point in variation space. When you sweep out variation: the integral of the kinetic energy has its rate of change (as a function of variation sweep), and the integral of the potential energy has its rate of change (as a function of variation sweep). At the point (in variation space) where the trial trajectory coincides with the physical trajectory the two rates of change (as a function of variation sweep) are matching.

A tip for your animation: if you use \left( and ight) instead of simple brackets () in Latex it looks much nicer.

The major fail of the classifier tech is that it does not have a default class of "other" or "unknown". It always tries to fit the object into some class which is really stupid, because it's one thing to misclassify something and another to not be able to admit that one does not know what one is seeing. Smart people know when to say they "don't know" and smart classifiers should also have this ability. Good concept, though the classifier misses A LOT of clear indications of a gun in the frame and misclassifies person as knife a lot of the times. The execution of the concept needs improvement or it will misbehave in real-life making it unreliable.

The example @ 2:09 is somewhat nonsensical in that it shows that random paths can take negative values for height, which is somewhat impossible. Besides, the first example was about dropping a ball, and the second one is about throwing the ball up, which is not really mentioned and adds to the confusion. I do however like the video as a whole - nice visuals and clear explanation of the subject. If only this was thought in school that way.

I'd encountered the concept of "functions as vectors" before but not the "functions as infinitely long column vectors." That tickled my brain

awesome!

background music completely unnecessary and extremely annoying

Corrections: - While many people say "Least action", "Stationary action" is correct. - With the pendulum example, omega should be sqrt(g/l) (and that's what should be inside our sin function also) - The reason we can treat y and y' as independent has to do with the derivation of the Euler-Lagrange equation and is not related to us not knowing the whole path. (If I understand right) If you see anything else, feel free to let me know and I'll add it to the list. I'm really impressed with the comments I've seen!

Universe isn't lazy. Lame math.

This is a great explanation, thanks!

It minimizes the area according to thermal dynamics

At 8:04 there was a mistake integrating the differential equation. The frequency should be sqrt(g/l) and not sqrt(l/g). The longer the pendulum the lower the frequency and the stronger the gravity the higher the frequency.

3:39 Asking why nature exremizes the action is the same as asking why Newton's laws are obeyed in non relativistic classical mech

One thing that should be mentioned when talking about Lagrangian mechanics, specifically Lagrange's equations of the second kind is the use of generalized coordinates. This is important because the number of Euler-Lagrange equations equals the number of generalized coordinates you need to describe your system or rather its degrees of freedom. From this you can immediately spot conserved quantities such as generalized momenta, which is when the Lagrangian doesn't depend explicitly on the corresponding generalized coordinate (yet another example where you literally need to differentiate between partial and total derivative !). At 8:57 I'd also like to mention that another way you can arrive at the expression for T is by treating your position vector as a multivariable function of the new coordinates. Then finding the total time derivative of the position vector r just becomes a matter of making use of the multivariable chain rule, setting up r_dot in the new basis and then forming the dot product of r_dot with itself in this new basis to get r_dot^2. At 4:34 you talk about the importance of knowing how to take derivatives. To get a better understanding of derivatives it might be worth taking a look at multivariable calculus. Also bear in mind the need for two constants of integration for a second order differential equation at 8:07.

Excellent video. I never learned this in my physics course.

Good, but some slides are difficult to see on a mobile phone screen.

There is a good book called Calculus of Variations by Weinstock where it talks about some of this. In any case, the idea is that of Hamiltonian's principle which is the idea that the "system" tries to minimize the Lagrangian. Why? Well, you have to realize what the minimization is(it is in the sense of functional variation in the sense of calculus of variations). So the Lagrangian is simply the difference in potential and kinetic energy(so in a system with constant total energy it tracks how the energy flows from from potential into kinetic and vice versa). Hamiltonian says that out of all the ways the system could transfer energy from one to the other it always chooses the "optimal/shortest/quickest" way. E.g., the Energies depend on functions(such as position, velocity, acceleration) and these functions are generally not or only partially known(the point of physics is to determine them). The Lagrangian encodes the "dynamics" of the general system(specified in mathematical relationships corresponding to the system) representing the motion. L(q, q') where the q's are "generalized"(abstracted) variables/functions from before. The point here is that L(q,q') is a functional, that is, it depends on functions(and it doesn't explicitly depend on time) What the integral does is sum over the entire time frame the total change in energy of the system's dynamics. Hamiltonian's principle says that out of all possible dynamics(q and q's) *reality* chooses the q and q' that minimize the integral, or minimizes that change over the entire time. E.g., the will be some q's that will produce some dynamics that is not correct. E.g., think of some physical world which is vastly different than ours, it may work there but it won't make sense in ours. Some q's will produce some dynamics that are correct in some sense in that they will have the same "action" but they will still be wrong because they are not "optimal" in some sense. Out of those q's there will be one that is optimal. E.g., we have a "surface" of from QxQ' but there is only a single curve of (q,q') where q' = (q)'. Out of those (q,q') there is only one that will minimize the *functional* w.r.t to the integral(it won't minimize the integral as the integral is already minimized by all the (q,q')'s that worked). E.g., you can surely imagine that there is some q such that it increases T and V both by the same amount which then cancels in the Lagrangian. But that is not necessarily a valid q. Remember, that there are many q's that give the same q'. E.g., q = ax+b, q' = a. There is an entire space of q's to choose from. But there is at least one that would give the minimal "action". In this case what makes them different is simply their position so there is some point in phase space that represents our space. In some sense there are two parts to the integral. One is the minimization of it(minimizing the action). Some possible dynamics of the system simply are not going to produce a minimal action. But there are many systems that can produce a minimal action... but out of all those there is only one that will be the minimal(well, given certain differentiability constraints) in the sense that it will "*minimize the minimal action*". E.g., think of a system that wastes a lot of energy at the start then somehow compensates for it at the end. Overall, it may produce the same minimal action but it isn't correct because somehow it lost energy and then gained it back. That suggests our system is incomplete(something else going on) or is not what we observe in reality. But there is one system that will behave in a *uniform* way(sorta like a uniform approximation) as to always work to minimize loss of energy(or keep it zero). After all, we only have V and T and we can only transfer one to the other(conservation of energy) so if we are losing or gaining energy this would suggest some other term we didn't write down(some "hidden variable"). So by ensuring we find the "least action" we are essentially ensuring that we have the system that isn't doing anything funky. Also, we expect this to be a local behavior, not a global behavior, because why would the interval of time matter. I.e., the system should be "time invariant" and "linear". So ultimately, then, this results in the Euler-Lagrangian differential equation which tells us how the system has to "flow" to always conserve energy(T - V can change but it can't change arbitrarily). The dual system is the Hamiltonian which is working in coordination and has to essentially Obey similar equations(since it is related to the Lagrangian). The main point is that these ideas express the conservation of energy in them. A universe does not get a free lunch. As something is happening in the universe, as long as all the potential and kinetic energy is accounted for, the total energy will always be constant(the universe is completely isolated from whatever is outside it) and inside it can only flow in a very specific way. Remember this too, Int(T - V) is very specific. The kinetic energy is always positive. This puts a very specific type of constraint on the system. It doesn't allow the integral to take on any values. The potential sets a sort of "lower bound". What this means is that there are many systems with, say, very large kinetic energies that will produce nonsense. E.g., just imagine that V = 0, then one has Int(|T|) and then the minimal solution is when T = 0. Say V = 1, then int(T - 1) in this cause T being 100 would produce a large positive value which clearly is not close to the minimum. T = 1 would give a minimum of 0 but it might not describe our system. E.g., T = 1 + a*cos(wt) might describe our system where a is very small. here the integral would be a*Int(cos(wt)) and one could imagine that the integral happens to be near zero and a << 1 giving ~= 0. The point here is that 1 + a*cos(wt) is not at all the same as 1. They are very different functions describing very different systems. In both cases though the "action" is near zero. [T would have to be a kinetic energy though and so it likely would not be able to actually have such a form] Ultimately if one thinks of the potential energy as a sort of "predetermined reservoir" of energy that the system can pull energy from to do something with(to give motion) then it might make more sense. But this reservoir always has to be filled back up at the "end"(whatever that end is). e.g., it's like a bank. You can take out a loan(K.E.) but you have to pay it back(unless you are a politician or billionaire) at the end of the day. But you can't just do anything with energy. When you pull it out you can only do certain types of motions(e.g., those in accordance with Newtons laws). E.g., there cannot be any discontinuity. In fact, it essentially has to be piecewise differentiable and, in fact, it seems to have to be infinitely differentiable. E.g., this is like a bank requiring you to spend the money only in specific ways(e.g., they give you a loan for a mortgage). It is the Euler-Lagrange equations that precisely tell you how you can "convert" the energy from "position" to "motion" and these constraints make it so that you can convert it back exactly(there are no "losses". E.g., think of friction. The equation for friction is different than non-friction. These equations are the equations for no "frictional losses").

The rough intuition is if you apply the Euler-Lagrange Equation to a general potential (U' = F) and general kinetic energy (T = [mv^2]/2) and work through the algebra, it reduces to F = ma, meaning EL is equivalent to Newton's laws of motion

Ah yes, the priciple of the least explained principle in physics

It is possible to have a transparent understanding of how the stationary action concept connects to Newtonian mechanics. In preparation I must first say some things about calculus of variations. As example I take a case that is often solved with calculus of variations: what is the shape of a hanging chain? For short: the catenery problem. The usual presentation of the catenary problem is with the two points of suspension at equal height, but we can generalize to allowing any ratio of horizontal displacement and vertical displacement. Also, when the two points of suspension are not of equal height then the chain can be so short that it from the lowest of the two suspension points the chain points upward. The catenery problem is scale invariant; let's say you have a particular ratio of horizontal and vertical displacement. If you make that half the size, and you make the chain half the length also, then you get the same shape. This scale invariance is a key factor. When you subdivide a catenary in subsections each subsection is an instance of the catenary problem. This divisibility property has no lower limit; all the way to infinitesimally small subsections it remains valid. The way to solve the catenary problem (using calculus of variations) is to obtain a differential equation that accomodates any ratio of horizontal and vertical displacment. You then impose the demand that that differential equation must be satisfied over the whole domain concurrently. The equation that does that is the Euler-Lagrange equation (when populated with the appropriate Lagrangian). Another crucial aspect of calculus of variations: with the catenary problem what we are looking to find is a curve that is a function of the horizontal coordinate. The variation that is applied, on the other hand, is variation of the vertical coordinate. For emphasis: the variation that is applied is exclusively variation of the vertical coordinate. To find the point where the derivative with respect to variation is zero you differentiate with respect to the applied variation. Since the variation is exclusively variation of the vertial coordinate: setting up to take the derivative with respect to variation boils down to taking the derivative with respect to the vertical coordinate. With the above in place I turn to Hamilton's stationary action. Many authors assume that it is about minimization, but there are also classes of cases such that the true trajectory corresponds to a maximum of Hamilton's action. That is why the physics community has shifted to the name 'stationary action' 'Stationary action' is a compact way of stating: 'the true trajectory is the point in variation space such that the derivative of Hamilton's action is zero'. The derivative doesn't have to be a minimum or a maximum, that is immaterial. The criterion that counts is: that the derivative of Hamilton's action (with respect to variation) is zero. As we know: the true trajectory has the property that the sum of potential energy and kinetic energy is conserved. From that follows the property: at every point along the trajectory the rate of change of kinetic energy *matches* the rate of change of potential energy. It is sufficient to establish that in every infinitesimally small subsection of the trajectory those rates of change are matching. When that matching- rate-of-change demand is concurrently satisfied everywhere then that property propagates out to the integral of Hamilton's action. That is: as you sweep out variation: the derivative of Hamilton's action is zero if and only if on every infinitisimally small subsection the derivative is zero. Hamilton's action consists of two elements: kinetic energy and potential energy. To take the derivative of Hamilton's action means you are evaluating two derivatives: derivative of the kinetic energy and derivative of the potential energy. The true trajectory has the property that those two derivatives are matching. In mechanics we are accustomed to taking time derivatives. In the case of mechanics (Hamilton's stationary action): the derivative that you take is *perpendicular* to the time axis. You take the derivative of Hamilton's action with respect to the *position* coordinate. When you populate the Euler-Lagrange equation with Hamilton's action (T - U), you see that the equation takes the derivative of the potential energy with respect to the position coordinate. And while the operation that is performed on the kinetic energy looks different we know that *dimensionally* it must be the same. Sure enough: it is mathematially the same; in effect the Euler-Lagrange equation takes the derivative of the kinetic energy with respect to the position coordinate. On why in Hamilton's action the potential energy is *subtracted* from the kinetic energy: That has to to with the fact that the differentiation is differentiation with respect to position coordinate,; it's not time differentiation It is when we follow change over time that the potential energy and the kinetic energy are counter-changing (from which it follows that the sum is a conserved quantity.) On the other hand: when you sweep out variation (a hypothentical change) then with respect to that change the potential energy and the kinetic energy are co-changing; to find the point in variation space where the two rates of change match you subtract one from the other.

Functions arnt just vectors. Functions are values.

This is arguably the starting point for functional analysis…

This is a subtle point, but the physical path is not necessarily the one that minimizes the action. It is more accurate to say that the physical path is one with stationary action. This just means that for small enough perturbations to the physical path, the action does not change. In fact, the left-hand-side of the Euler-Lagrange equation can be thought of as the "derivative" of the action. The statement of the Euler-Lagrange equation, then, is that the physical paths are the "critical points" of the action.

but where does the euler lagrange formula come from? would love a video showing the derivation

Awww man i feel like you didn't do the fourrier transform justice. It's sooo mindblowing when you start thinking about a projection of a function onto rotational space. It would have been more mindblowing if you didn't spoiler the fourrier transform before actually projecting it. But anyways, happy to see that it's not being gatekept

this video was very insightful but please slow down your voice and animations a bit to let people a fair chance of think as they are listening to you just like 3b1b 👌👌

The principle of stationary action actually follows directly from the path integral formulation of quantum mechanics. There, the wave function Psi(x, t) that describes the probability of finding the object at x at time t is proportional to the sum of e^{iS} over all paths x_0 to x, where S is the action. Now imagine a path that doesn't have stationary action. On the quantum scale the change in S might be small, so (thinking of discretizing the integral into an infinite Riemann sum) most of these complex arrows will go into the same direction, resulting in a nonzero value for the wave function, resulting in a nonzero probability of finding the quantum particle at any other location than those its path of stationary action takes. But in the classical limit, big objects have large actions. So if we imagine a path of non-stationary action and look at the sum of arrows, we find that e^{iS} is all over the place, as S varies drastically over any path with non-stationary S. So it is just as likely for any vector to be in one direction or its opposite direction, and the integral for that path averages out to zero, meaning no contribution from that path for almost all paths in space from x_0 to . The only paths that contribute to the probability of finding a macroscopic object at (x, t) are those where S is (almost) stationary so the vectors share direction, and those paths are therefore the only paths on which we could ever observe it.. So macroscopic objects are only likely to be found on paths of (nearly) stationary action, and classical mechanics is then what we observe.

Mind blowing! Can you make a video that goes more in depth about the topic?

Functions are infinite dimension vectors

This is something you learn in physics by intuition because no one annotates their maths properly. And then multiplies a function by a matrix without explaining it

Happy to see another video after your previous banger! 😁 I feel you: I am also often missing intuition on why the objective functionals which we want to minimize look how they look in variational problems. Often the choice of the objective functional seems arbitrary and only after deriving the Euler-Lagrange equations it makes sense ... 🤔 Keep going with the videos!

गुरुजी आपला आवाज आणि शिकवण कानाला आराम देऊन जातात! आभार 🙏🏼 This Indian kid is waiting for your next video.

Why your videos are too good ❤

Great video 🎉

valuable intuition

Numbers are sets