Hi everyone, thanks so much for your support! If you'd like to check out more Physics videos, here's one explaining the First Law of Thermodynamics: th-cam.com/video/3QCXVKUi7K8/w-d-xo.html Edit: to answer a question I've seen a few times now, the "q" in the Euler-Lagrange equation can be thought of as a generalised coordinate. So in this instance, we replace q with x, and q(dot) with x(dot). In a system showing motion in multiple different directions, we would get multiple equations for each of the relevant coordinates. So for example a system varying in both the x and y directions, would give us an equation with x and x(dot) in it, as well as another equation with y and y(dot) in it.
Elizabeth meghana Inside my Physics & Applied Maths, I insert loose notes (size 8" x 6"). On them, I jot names of video titles and verbatim copy out problems and solutions from tutorials. I use notes to bookmark vital pages. Whatever chapters I am studying or revising from, I have my notes there. That makes studying a lot easier.
not really; just write dL/dq - d/dt(dL/d \dot q) - f(t,q,dot q) = 0 and you have your lossy term f. It obviously breaks conservation of energy and momentum and may be a bit more complex to solve, but the Lagrangian method still outperforms Newtons forces in this regard.
I got my BSc 22 years ago, but I’m still watching these videos, reading books etc. 😃 I had about two or three years away from it, but if you love Physics, you’ll always love physics. 😊 I found Uni very rushed and there are loads of subtleties, connections and historical contexts I’ve learnt since. I’ll probably still be watching these videos in another 22 years. 😊
I got my M.Sc. in physics in 2007, and an M.Phil. a year after. I also cleared the NET equivalent of my state (TN SET) and am working as an Assistant Professor of Physics for the past 11 years... and here I am... watching this video... It just fun... and rekindles my love for physics... also, I believe I have something to learn from everyone, no matter how small it is... Best wishes...
Surely this is one of the best explanations of the Lagrangian on TH-cam. Although it’s not detailed it’s it’s coherent and it’s a great overview of what is really going on. I’ve tried for years to understand it now I feel like I’m actually getting it. Thank you!
Id like to add, 1:15 "The Lagrangian is indeed defined as the kinetic energy minus potential energy" This isn't actually true General Definition of a Lagrangian For a given mechanical system with generalized coordinates q=q(q1,q2,...qn), a Lagrangian L is a function L(q1,...,qn,q1(dot),...,qn(dot),t) of the coordinates and velocities, such that the correct equations of motion for the system are the Lagrange equations dL/dqi = d/dt(dL/dqi(dot)) for [i=1,...,n] This definition is given in Classical Mechanics by John R. Taylor page 272. Notice that it does NOT define a unique Lagrangian. Of course the definition provided in this video for this case fits this definition, and for most cases T-V will satisfy this definition. The video may have been hinting at this for point number 2 but something I would also like to add is that one of the advantages of this REformulation of Newtonian mechanics is that it can bypass constraining forces. For example consider a block on a table connected by an inextensible rope and pulley to a block hanging over the edge of the table. To work out the equation of motion using Newtonian mechanics you'd have to consider the tension in the rope while looking at the forces on the individual blocks, and that is a constraining force. As for lagrangian mechanics you don't. Which as an aside means qualitatively you'd be missing out on the physics of the problem ( and other problems) so if you've already learned how to do this problem using Newtonian mechanics then by all means use Lagrangian mechanics. You can of course apply Lagrange multipliers to find the constraining force if you want but then you'd need to include a constraint equation. 1:38 The Hamiltonian is defined by that IF you have time independence it is NOT in general defined that way. As for deriving Lagranian mechanics, incase anyone is interested where this comes from, here are two ways you can do this. First is the 'differential method' of D'Alembert's principle where the principle of virtual work is used. the second would be an 'integral method' whereby you look at various line integrals. Lastly, some further reading if you're interested I don't talk about it in my comment however this is a crucial concept. The principle of stationary action. en.wikipedia.org/wiki/Principle_of_least_action For more on Lagrange mulitpliers see page 275 of Classical Mechanics by John R. Taylor "D'Alembert's principle where the principle of virtual work is used" One resource for this would be page 16 Classical Mechanics Third Edition by Goldstein, Poole & Safko This is a more advanced textbook though. 3:52 As a side point, I'd just like to also point out that the dot notation is not specifically for time derivative and its a notation that you might want defined before hand. For example, see page 36 Classical Mechanics Third Edition by Goldstein, Poole & Safko, being used to mean dy/dx=y(dot). dL/dqi - Generalized force dL/dqi(dot) - Generalized momentum q - Generalized coordinates q(dot) - generalized velocity Overall an excellent video
Yeah, I was just about to say. I'm of a mind to introduce the Hamiltonian _first_ just because it's EoM are symplectically related to eachother, making it kinda special, and then understand the Lagrangian as the Legendre transformed Hamiltonian - basically the same thing but half the coordinates are changed from momenta to velocities.
Bruh he didn't even tell us what q was... Don't get me wrong I appreciate this very quick intro to the subject, but professor's tend to give much more thorough explanations. The real issue is lectures aren't a good way to learn complicated concepts for the first time.
q is the generalized positional coordinate in question (this corresponds with x in his one dimensional example). In general there is one of these equations for each independent spatial coordinate in the system. One of the outstanding (and convenient) features of the Langragian approach is that all of these equations take the same form regardless of the coordinate system used (e.g. Cartesian, spherical, cylindrical, etc). There is obviously a lot more to this than that which can be presented in a ten minute video, but this is a an excellent short explanation and introduction.
Did he satisfactorily qualify his use of the word 'better', and why 'better' in all-caps is justified beyond the requirements of bait, and that LM can be derived from first principles without any NM? That kind of 'better'? Or to be more clear, could Lagrange have developed LM had he been contemporaneous with Newton?
@@-danR Langrangian, and Hamiltonian, are better in the sense that if the system can be solved with 2 variables, you can more easily end up with 2 variables. Imagine 2 weights attached with a string. The string passes through a hole in a table, where one weight is hanging, and the other is spinning in a circle on the table. This looks like a 3d problem, but it's not. It's a 2d problem. You can perfectly represent it with 2 variables(length of string from one weight to the hole, and angle of the weight on the top of the table with respect to some 0 angle).
Great video. I want to point out that definition of 'q' and 'q dot' is missing in the Euler-Lagrange equation. These are placeholders for 'position' and 'momentum' respectively for those wondering.
Thank you, I was puzyxled - nay, ANNOYED - by the introduction of the E-L equation with a term "q" that was completely ignored, without any explanation. For this reason _alone_, the video deserves a FAILED and a thumbs down.
This brings me back 50 years ago when first being introduced to the subject and walking back to the dorm knowing I must be the dumbest guy in the world. Thanks for bringing me back to those memories.
I find it fascinating that although the L doesn't represent anything physical - at least not obviously so - it sort of hints at a much deeper underlying structure to what we perceive and analyse. Brilliant video Parth. Thanks for your work.
@@RiyadhElalami Agreed! I love this discussion, and that it includes applications. It would be interesting to see an experiment comparing the two in some sort of physiological manner.
Great video for those who wish to have a primer/overview on Lagrangian mechanics! However, I would note that the title is a bit off. Lacking the appropriate context, saying LM is better than NM is short sighted. Don't get me wrong, having learned the topic myself in Uni I was wide-eyed in disbelief why this wasn't taught to me sooner. You alluded to the reason in your video so much props, and that is variational calculus. From a pedagogical standpoint, most people a physics professor will teach will be non-physics students. Newtonian mechanics can be summed up fairly "easily" with algebraic techniques (the much maligned Algebraic Physics), and extended quite significantly with the addition of basic uni-variate calculus (F = dp/dt for example). With these relatively low level mathematical techniques, one can solve a wide variety of problems, even challenging ones. Contrast this with the workhorse of LM, the E-L equation. Right out of the gates, we have partial derivatives (multivariate calculus), and, in the gorier forms, with respect to the "generalized coordinates" and "generalized momenta." This of course opens up the universe of possibilities to doing calculus on potentially horrendous coordinate systems (chaos/multi pendulum as a simple example), but hardly the highest priority for people who don't plan on doing physics in their eventual career. Needless to say, the mathematical overhead required to explain why this machinery works, is no trivial matter. Minimization of integrands, finding the variation about fixed points are fairly high level concepts that involve a pretty broad understanding of the topic of calculus. Usually this FOLLOWS a course in Real or even Complex Analysis. Maths majors know this isn't for the faint of heart. All this being said, which is better LM or NM? That is like asking which is better, a spoon fed GUI that allows point and click, or a command-line interface which a litany of abstract and esoteric commands. Better how? The GUI allows a much broader swath of the population access to the power of the computer, whereas the pro's find the command-line much more efficient and powerful (though not all and preference does play a role, imperfect analogy being what it is). LM is definitely more powerful, as the number of systems which can be analyzed drastically increases over NM. However NM has great utility in the problem solving domain, still even for pros, but has significantly less overhead for all your typical/simple problems. Generally it doesn't usually even come up until you have gone through a process of ever increasing difficulty culminating in, from my anecdotal experience, moving reference frames where the simple F=ma gives way to all sorts of additional "imaginary forces" that come about from the rotation, for instance, of a reference frame. This is where the topic can be introduced as a way to short circuit the otherwise gory mess of equations you would end up with using simple NM. Just my two cents. All this being said though, still like the video only had an issue with the title. Keep spreading the word and your passion for physics!
sir you said that lagrangian doesn't have a physical significance but can we say it is just the excess amount of energy within the system to perform work , synonymous to the concept of gibbs free energy in thermodynamics .....
Interesting connection. My intuition is no, since in thermodynamics one cares about the change in (Gibbs free) energy, whereas the Lagrangian is a total, sign sensitive quantity of energy, and hence is usually equivalent up to an arbitrary constant. It is my understanding that the Lagrangian's significance is in all the equation it features in (i.e. the Euler Lagrange equation), which is a rate of change equation--hence killing the arbitrary constant if it were ever included. I suspect that neither the Lagrangian nor the Action (hitherto undiscussed) have any direct physical significance to the system--instead, they can be interpreted as tools used to arrive at the correct equations of motion (which are the things which themselves obviously have a ton of direct significance).
Great video. One of my favourite modules in my physics degree. It's so refreshing after years of writing F=ma that they turn round to you in second year of uni and say 'well actually there's a better way'
Sometimes TH-cam's algorithms recommend videos from content creators that are actually quite good, such as this one by Parth G. Quick and concise , highlighting the most important questions that a student might ask, without dumbing anything down. Right up my alley, Mr. G.
5:52 This a simple second order differential equation with solutions of either sine, cosine, or an exponential (power of e). This results in a cyclic sine or cosine curve (depending on where you place the origin) when position is graphed as a function of time. The fact that the acceleration has sign opposite to position makes this a restoring force, i.e., motion is constrained within boundaries.
I don't know about better, but an additional viewpoint is almost always informative. And yes, scalar quantities like energy are simpler than vectors. But it's also interesting to think directly in terms of forces, even though it's messier, and perhaps more error prone. On the other hand, one could argue that Hamilton's principle, or least action principles in general, are "best" in the sense of elegance and simplicity. Ultimately though, Feynman told us that it's useful (and interesting) to have a variety of different mathematical formulations available for any given theory. Maybe that is the approach that is "better."
I agree that Lagrangian mechanics is great, especially if you are dealing with systems consisting of many variables. But what Newtons formulation handles way better is friction, just add a model of friction (eg. -v or -v^2), doing this with lagrangians is an absolute pain.
You nailed it, you delivered exactly what I was looking for. If all your videos get to the point and are as clear as this one, I have here plenty of things to enjoy.
This video really helped push back my ignorance - mainly to show there is so much more I am ignorant of than I realised. A great video that helped make complex concepts approachable.
Didn't understand a thing he said, but I'm still transfixed like a deer in headlights ... Here, take my money ... like taking all the potential from my kinetic ... and I'm wobbling my head up and down like the doll on the dashboard!!!
It is probably understood, but just to state it explicitly, Lagrangian Mechanics and its successor, Hamiltonian Mechanics are both directly derived from principles of Newtonian Mechanics. For anyone interested in the details of this derivation, I recommend *Goldstein, “Classical Mechanics”, 3rd edition* (or the older 2nd edition) published by Addison Wesley. In the first chapter, sections 1-3 give a crash-course basic Newtonian Mechanics (this is only for those already reasonably versed in Newtonian Mechanics as a brief refresher). Sections 4-6 derive Lagrangian Mechanics starting from D'Alembert's Principle in section 4. Chapter 2 introduces Variational Calculus or Hamilton's Principle applied to the Lagrangian. For the more advanced curiosities among us, Hamiltonian Mechanics is introduced in chapters 8-10, Classical Chaos in chapter 11, and the foundations of Field Theory (including Noether's Theorem) in chapter 12.
This is very interesting, but I long for the hour long video that actually makes the case posed by the title instead of acting as an introduction so that a person could understand the title.
Great video! By the way, often the “curly” L represents the so called “density of Lagrangian” which is Lagrangian per unit of volume. The Lagrangian itself is represented by the capital L. Just a tiny detail!
But . . . but . . . but . . . What is q in the E-L equation? And exactly how do you plug your Lagrangian into the E-L equation to obtain the result you claimed?
At 6:51 the term on left side is not the total force on the system but describes the acceleration of the system. In other words, it is Newton's Second law which relates acceleration to the total force on the system which appears on the right hand side. That was a great journey. Thank you
Awesome! I learned that the Hamiltonian is the sum of KE and PE, I got more xposure to dot notation, and I should go back and rewatch the derivation of F = -kx. A question: why is "the difference between KE and PE" not physical? The difference between its actual energy and what it can do? I like my math to be physical
Hi Parth. I just found your channel and watched this very informative video on Lagrangian Mechanics. I dig your approach to physics and have just subscribed! I'm trying to catch up on math and physics since I'm now retired. I look forward to learning from you!
6:57 true, the Lagrangian approach doesn't explicitly mention "force", but to come up with potential V, wouldn't forces (and it's integral over dx) be needed in the background anyway? So it seems we can't really abandon the force concept
Luv the way you tought sir .......extremely impressive .......if a person luv physics, then they surely start liking you to fr ur creative teaching😊 thnkuuu
Very nice explanation. I do find it interesting that you stress so often that the Lagrangian isn't a physical quantity but rather a mathematically useful quantity when that is equally true of energy as well. We typically say that things 'have' energy, but energy is just as much a mathematically constructed quantity as the Lagrangian, useful only for its apparent conservation. Like the Lagrangian, energy cannot be measured; only calculated.
Excellent presentation and explanation. I have read and listened to number of presentations by others but none as understandable as yours. Thank you and keep it up.
How much time do you think, Self studying would take- If one starts from undergraduate Classical Mechanics and Electrodynamics to Quantum Mechanics and good level GR stuff and so on? Being in India I dont think enrolling into a Physics course is a good one, but I am just too much interested in Physics to leave it off for my Electrical Engineering B Tech. Please do guide as I guess it maybe useful for others too 😀
i think that, assuming you have sufficient discipline, it would take 3-4 years to get to this standard (which I believe is 2nd or 3rd year knowledge). I'm a first year so take this with a pinch of salt aha
It depends on how efficient your study method is and how much time you spend studying per day. At its most hardcore, 1,5 weeks should be enough to learn one semester module of 12credits worth of work, but it may be very exhausting. So maybe 2,5 to 3 weeks for one module. Assuming there are an average of 6 modules per semester, it can take 54 up to 108 weeks (equal to 1 to 2 years) to complete an undergrad course. This may not be sufficient to master your work but it should be enough to work through some problems and understand the concepts
Great video sir I just completed my course in classical mechanics but Lagrangian and Hamiltonian mechanics were not included.. Now I will learn this from u❤️
This was really interesting. I'm currently taking a class on Newtonian mechanics and I am curious to understand the connections between Lagrangian, Hamiltonian, and Newtonian mechanics.
It would have been helpful to have abour 1 more sentence in the intro towards which target audience this video is directed. To be fair u said that the video is about basics but imo thats kind of an unclear definition. Would be nice if u could add that in ur future videos. For me personally it still was a nice video yet not what i was looking for as this rather seems to be an introducion to this topic and not an actual explanation of why the things are happening the way they are. Ty for your videos :)
There is a very beautiful connection between the "physical properties" and the Lagrangian. By performing a Legendre Transform from the variable "velocity" to its slope, called momentum p, we get the symmetry condition of the Legendre transform as \dot q = \dfrac{\partial H}{\partial p} just as the original defintion of the canonical momentum reads p := \dfrac{\partial L}{\partial \dot q}. Now comes the breakthough: With this "second" equation we can write the total time evolution of the Hamiltonian as \dot H = \dfrac{\partial H}{\partial t}+\dfrac{\partial H}{\partial q}\dot q+\dfrac{\partial H}{\partial p}\dot p and take the transformed Version of the Euler-Lagrange-equation of motion for \dot p and the Legendre-Transform for \dot q and have a closed form where q, p and t are the only variables, and even more: They appear in an anti-symmetric ararrangeemnt, commonly denoted by Poissons' bracket, a special case of the Lie-brackets (commutator of two operators) commonly used in Quantum mechanics. The point is: You cannot achieve this anti-symmetric closed arrangement with the Lagangian as by the very same calculus \dot L = \dfrac{\partial L}{\partial t}+\dfrac{\partial L}{\partial q}\dot q+\dfrac{\partial L}{\partial \dot q}\ddot q and the acceleration \ddot q does not appear in the general Euler-Lagrange equation (just take any coordinate frame other than carthesian and you will see that the acceleration in a coordinate is not necessarily easily extracted/isolated), so the only meaningful way we can make predictions on the time evolution of the Lagrangian (and therfore its physical meaning) is by using the Legendre Transform again, writing L = H - \dot q p and reasoning \dot L = \dfrac{\partial L}{\partial t} + \{H,\dot q p\}. In general, this is not an easy thing to do, but if 1) time symmetry holds and 2) the momentum is linear in velocity with some constant term p=\dot q/a, then the Lagrangian (plus a constant) is simply int \dot L dt = \int \{H,\dot q p\} dt = \int \{H, a p\} p + \{H, p\}\dot q dt = \int (a*\dot p+\dot p)\dot q dt which is, if you squint you eyes, the total change in momentum, called a force, integrated over a path of motion ds = \dot q dt, which is the classical Newtonian definition of Work. The classical Lagrangian is a multiple of the total work done in a physical process, and the principle of least action states that the total amount of work done within a certain time frame must be extreme (mostly minimized). There you go, classical mechanics is really just "The universe is lazy". And also, most of the facy commutators of quantum operators you learn in QM can be solved by calculating corresponting Poisson brackets; the underlying anti-symmetry of its arguments is transferred from one theory to the other, or as we call it: Algebra remains. :) ps sorry for typos :/
Fantastic video, really interesting because as an alevel physics student have never dealt with lagrangian only newtonian mechanical physics. Also, you have incredible head hair sir!
As an engineering student, I have put this into practice, but I have no clue why xdot and x are assumed to be totally independent of each other when you are doing the partial derivatives.
Because they represent completely independent quantities. x-dot is velocity, which is typically only relevant when finding kinetic energy, whereas x is position, which is typically only important when finding potential energy. The velocity of an object moving at x-dot meters per second is the same whether it's at x = 0 or x = infinity. The only link meaningful link between x and x-dot in this type of problem is that they're happening on the same axis.
@@dwangnoderbora In the case where x (position) is increasing, you can always parameterize the velocity to be a function of position. Then they are literally not independent of each other. Riddle me that.
Deriving the equations of motion for a double pendulum from a Lagrangian and Newtonian perspective is enlightening: it's pretty straight-forward from a Lagrangian perspective, but more challenging from a Newtonian perspective.
Hey buddy your ability of explanation is highly lucid and alluring. Could you tell me from which country you belong and from which university have you pursued your undergrad postgrad and PhD
I was looking forward to hear an argument "Why Lagrangian Mechanics is BETTER than Newtonian Mechanics F=ma." The video states @7:38 three reasons about complicated system with multiple forces and coordinates. BUT the example is a 1-d case that doesn't need the Langrangian. Maybe a complicated system with cylinder coordinates or other deviation from the simplest cases would have been a better example. . . . I'm looking forward to your Noether theorem video, but please show also what you tell us.
Lagrangians made me crazy I started seeing the energy of the curves on buildings and objects around me, and I still have to think about other stuff, because maaaaaaan
What a coincidence! I was just looking for LangragIan just a few minUtes before you uploaded this and there you go ,I am loving this and it's my first time watching your videos . Please keep posting more good mathematical and physics stuffs 👍
Hi! I have been your follower time immemorial. I got a question. If Lagrangian mechanics gives the equations of motion of a particle without being so concerned with forces, how is it we still need to get the potential energy using a force (conservative force to be specific, which is derivable from the potential)? In your example F=-kx is a spring force. That's how we come up with the potential energy U = 1/2 kx^2., plugged into the Lagrangian.
I think what the Lagrangian is representing is the quantity of kinetic energy actively being used at any one time, assuming that potential energy is potential kinetic energy
At 7:12, in point-1, as L=T-V, to determine V, shouldn't we know all the forces in advance. Like in spring mass system, we derive it's potential energy as 1/2 kx^2 as we know the form of the force as F=-kx. In that sense isn't Newtonian mechanics superior than Lagrangian Mechanics?
Hey parth! I love your videos! Can you make a video on Lagrange multipliers and how to deal with constraints using Lagrange mechanics? I want a video from you because I like your style of teaching
To be fair, Euler-Lagrange (EL) is 'better' than Newton-Euler (NE) in certain situation. Indeed, defining constraint force can be tricky for the NE, but there a few situations where NE outshines EL. If you're dealing with a large complex dynamical systems (i.e., large number of degrees--of-freedom), the EL-equations become very lengthy. The reason for this, is that EL equations require analytical expressions for time-derivates of the generalized coordinates. If the coordinates are highly dynamically coupled, this can becomes exponentially complex quickly... In these cases, its better to use a numerical approach to compute the dynamics through the NE-equations (this is often done in robotics). I also believe most physics engines use NE-equations over EL-equations since its more suitable for numerical computation.
I appreciate your effort for putting out a video on the solutions of the Schr(accent)odinger Equation. My new request is a video on Kaluza-Klein Theory , I am sure that video will be out soon too.
5:07 okay, The "L" was obvious, but what is q here?? 5:10 Euler Lagrange Equation is consistent with Newtonian Mechanics, and mention of resources to study about that
q is just notation, it's just an arbitrary position of a particle, or in the case of q dot, it's speed. For your second question, this is actually obvious once you differentiate the Lagrangian. x with two dots represents acceleration, so the left side is ma, and by Hooke's law, -kx is the spring force. Therefore F = ma. That's not a complete proof however. You can also remember that on the left hand side we are differentiation the left hand side by time, since kinetic energy is usually defined in terms of v, this will gives us acceleration. We also can remember that the change in potential energy over space is force. And so the right hand side will be in terms of a force, and the left will be in terms of acceleration and mass. Once again there's more to it than that, however if you're curious I recommend you search up the Euler-Lagrange Equation, and Calculus of Variations.
@@therealjezzyc6209 hi, thanks for the detailed input. The second point was not a question - i just wrote it to save for myself - like as a reminder that this was mentioned here. but thanks a lot for explanation. I liked it and it saved me time :) otherwise i'll have to search it later.
Then T=(H+L)/2 and V=(H-L)/2: the kinetic energy is the hamiltonian plus the lagrangian divided by 2, and the potential energy is the hamiltonian minus the lagrangian divided by 2. I don't know if it has a meaning or not :-)
Is the potential energy signed? As written x can be positive and negative. With a positive potential when the displacement is positive (stretched spring) and a negative potential when the displacement is negative (compressed spring). That right? Feels counterintuitive since the kinetic energy is unsigned.
Hi everyone, thanks so much for your support! If you'd like to check out more Physics videos, here's one explaining the First Law of Thermodynamics: th-cam.com/video/3QCXVKUi7K8/w-d-xo.html
Edit: to answer a question I've seen a few times now, the "q" in the Euler-Lagrange equation can be thought of as a generalised coordinate. So in this instance, we replace q with x, and q(dot) with x(dot). In a system showing motion in multiple different directions, we would get multiple equations for each of the relevant coordinates. So for example a system varying in both the x and y directions, would give us an equation with x and x(dot) in it, as well as another equation with y and y(dot) in it.
Hie Parth can you make video on conservation topic. Means conservation of energy, conservation of momentum please
Can you please make a video on variational principle for newtonian mechanics. 😊
hey parth, how r u doing ? i need a textbook session in which plz tell us about the textbooks that must be read by all physics students.
Elizabeth meghana Inside my Physics & Applied Maths, I insert loose notes (size 8" x 6"). On them, I jot names of video titles and verbatim copy out problems and solutions from tutorials. I use notes to bookmark vital pages. Whatever chapters I am studying or revising from, I have my notes there. That makes studying a lot easier.
That was awesome!
An even more interesting conversation is why this popped up in my recommended
So you dont watch physics videos?
I had a mechanics exam today lol
Currently taking Calculus!
@@d.charmony6698 i love calculus.....you should watch. 3blue1brown's series on calculus.
@@addy7464 Ok! Thanks for the recommendation!
Everybody gangsta until friction comes around
No friction in fundamental physics 😎
I just like doing the problems. Makes math more like a puzzle game
Daniel: Force
Cooler Daniel: Generalised Force
not really; just write dL/dq - d/dt(dL/d \dot q) - f(t,q,dot q) = 0 and you have your lossy term f. It obviously breaks conservation of energy and momentum and may be a bit more complex to solve, but the Lagrangian method still outperforms Newtons forces in this regard.
Lagrangian is derived from variational principle of energy. "The path of least action"... so friction, atleast Coulomb, ain't gonna be a huge problem.
Parth Congratulations, your video has been added to MIT open Courser ware along with Walter Lewin lectures
Last week, I got my M.Sc in physics. I wonder why I'm here after all the hard work :D Great content btw.
Congrats
congratulations
Now stop watching youtube and get a phd
I got my BSc 22 years ago, but I’m still watching these videos, reading books etc. 😃 I had about two or three years away from it, but if you love Physics, you’ll always love physics. 😊 I found Uni very rushed and there are loads of subtleties, connections and historical contexts I’ve learnt since. I’ll probably still be watching these videos in another 22 years. 😊
I got my M.Sc. in physics in 2007, and an M.Phil. a year after. I also cleared the NET equivalent of my state (TN SET) and am working as an Assistant Professor of Physics for the past 11 years... and here I am... watching this video... It just fun... and rekindles my love for physics... also, I believe I have something to learn from everyone, no matter how small it is... Best wishes...
Surely this is one of the best explanations of the Lagrangian on TH-cam. Although it’s not detailed it’s it’s coherent and it’s a great overview of what is really going on. I’ve tried for years to understand it now I feel like I’m actually getting it. Thank you!
Id like to add,
1:15 "The Lagrangian is indeed defined as the kinetic energy minus potential energy"
This isn't actually true
General Definition of a Lagrangian
For a given mechanical system with generalized coordinates q=q(q1,q2,...qn), a Lagrangian L is a function L(q1,...,qn,q1(dot),...,qn(dot),t) of the coordinates and velocities, such that the correct equations of motion for the system are the Lagrange equations
dL/dqi = d/dt(dL/dqi(dot)) for [i=1,...,n]
This definition is given in Classical Mechanics by John R. Taylor page 272. Notice that it does NOT define a unique Lagrangian. Of course the definition provided in this video for this case fits this definition, and for most cases T-V will satisfy this definition.
The video may have been hinting at this for point number 2 but something I would also like to add is that one of the advantages of this REformulation of Newtonian mechanics is that it can bypass constraining forces. For example consider a block on a table connected by an inextensible rope and pulley to a block hanging over the edge of the table. To work out the equation of motion using Newtonian mechanics you'd have to consider the tension in the rope while looking at the forces on the individual blocks, and that is a constraining force. As for lagrangian mechanics you don't. Which as an aside means qualitatively you'd be missing out on the physics of the problem ( and other problems) so if you've already learned how to do this problem using Newtonian mechanics then by all means use Lagrangian mechanics. You can of course apply Lagrange multipliers to find the constraining force if you want but then you'd need to include a constraint equation.
1:38 The Hamiltonian is defined by that IF you have time independence it is NOT in general defined that way.
As for deriving Lagranian mechanics, incase anyone is interested where this comes from, here are two ways you can do this. First is the 'differential method' of D'Alembert's principle where the principle of virtual work is used. the second would be an 'integral method' whereby you look at various line integrals.
Lastly, some further reading if you're interested
I don't talk about it in my comment however this is a crucial concept.
The principle of stationary action.
en.wikipedia.org/wiki/Principle_of_least_action
For more on Lagrange mulitpliers see page 275 of Classical Mechanics by John R. Taylor
"D'Alembert's principle where the principle of virtual work is used" One resource for this would be
page 16 Classical Mechanics Third Edition by Goldstein, Poole & Safko This is a more advanced textbook though.
3:52 As a side point, I'd just like to also point out that the dot notation is not specifically for time derivative and its a notation that you might want defined before hand. For example, see page 36 Classical Mechanics Third Edition by Goldstein, Poole & Safko, being used to mean dy/dx=y(dot).
dL/dqi - Generalized force
dL/dqi(dot) - Generalized momentum
q - Generalized coordinates
q(dot) - generalized velocity
Overall an excellent video
crickets from @parth G
Yeah, I was just about to say.
I'm of a mind to introduce the Hamiltonian _first_ just because it's EoM are symplectically related to eachother, making it kinda special, and then understand the Lagrangian as the Legendre transformed Hamiltonian - basically the same thing but half the coordinates are changed from momenta to velocities.
This is what a master looks like when explaining something. Took you 10 minutes to explain what my professors took hours.
Bruh he didn't even tell us what q was... Don't get me wrong I appreciate this very quick intro to the subject, but professor's tend to give much more thorough explanations. The real issue is lectures aren't a good way to learn complicated concepts for the first time.
@@nahometesfay1112 excellently put
q is the generalized positional coordinate in question (this corresponds with x in his one dimensional example). In general there is one of these equations for each independent spatial coordinate in the system. One of the outstanding (and convenient) features of the Langragian approach is that all of these equations take the same form regardless of the coordinate system used (e.g. Cartesian, spherical, cylindrical, etc). There is obviously a lot more to this than that which can be presented in a ten minute video, but this is a an excellent short explanation and introduction.
Did he satisfactorily qualify his use of the word 'better', and why 'better' in all-caps is justified beyond the requirements of bait, and that LM can be derived from first principles without any NM? That kind of 'better'?
Or to be more clear, could Lagrange have developed LM had he been contemporaneous with Newton?
@@-danR
Langrangian, and Hamiltonian, are better in the sense that if the system can be solved with 2 variables, you can more easily end up with 2 variables. Imagine 2 weights attached with a string. The string passes through a hole in a table, where one weight is hanging, and the other is spinning in a circle on the table. This looks like a 3d problem, but it's not. It's a 2d problem. You can perfectly represent it with 2 variables(length of string from one weight to the hole, and angle of the weight on the top of the table with respect to some 0 angle).
currently in a robotics major and lagrangian mechanics is probably the coolest thing i have learned
Great video. I want to point out that definition of 'q' and 'q dot' is missing in the Euler-Lagrange equation. These are placeholders for 'position' and 'momentum' respectively for those wondering.
Thnx buddy, I was wondering the same.
q is for generalized position, and q dot is generalized VELOCITY.
Thank you, I was puzyxled - nay, ANNOYED - by the introduction of the E-L equation with a term "q" that was completely ignored, without any explanation. For this reason _alone_, the video deserves a FAILED and a thumbs down.
This brings me back 50 years ago when first being introduced to the subject and walking back to the dorm knowing I must be the dumbest guy in the world. Thanks for bringing me back to those memories.
Currently going through this now. Glad to know people are the same regardless of time frame.
I find it fascinating that although the L doesn't represent anything physical - at least not obviously so - it sort of hints at a much deeper underlying structure to what we perceive and analyse. Brilliant video Parth. Thanks for your work.
I remember how amazed I was at how usefull Lagrangian mechanics are dealing with complicated mechanics problems, when I learnt about them.
This is so absolutely mind-blowing and well explained. This is incredibly well explained! Bravo. Thanks for sharing this with us.
Yes I have never learned about the Lagrangian in relation to Mechanics. Very cool indeed.
@@RiyadhElalami Agreed! I love this discussion, and that it includes applications. It would be interesting to see an experiment comparing the two in some sort of physiological manner.
Great video for those who wish to have a primer/overview on Lagrangian mechanics! However, I would note that the title is a bit off.
Lacking the appropriate context, saying LM is better than NM is short sighted. Don't get me wrong, having learned the topic myself in Uni I was wide-eyed in disbelief why this wasn't taught to me sooner. You alluded to the reason in your video so much props, and that is variational calculus. From a pedagogical standpoint, most people a physics professor will teach will be non-physics students. Newtonian mechanics can be summed up fairly "easily" with algebraic techniques (the much maligned Algebraic Physics), and extended quite significantly with the addition of basic uni-variate calculus (F = dp/dt for example). With these relatively low level mathematical techniques, one can solve a wide variety of problems, even challenging ones.
Contrast this with the workhorse of LM, the E-L equation. Right out of the gates, we have partial derivatives (multivariate calculus), and, in the gorier forms, with respect to the "generalized coordinates" and "generalized momenta." This of course opens up the universe of possibilities to doing calculus on potentially horrendous coordinate systems (chaos/multi pendulum as a simple example), but hardly the highest priority for people who don't plan on doing physics in their eventual career. Needless to say, the mathematical overhead required to explain why this machinery works, is no trivial matter. Minimization of integrands, finding the variation about fixed points are fairly high level concepts that involve a pretty broad understanding of the topic of calculus. Usually this FOLLOWS a course in Real or even Complex Analysis. Maths majors know this isn't for the faint of heart.
All this being said, which is better LM or NM? That is like asking which is better, a spoon fed GUI that allows point and click, or a command-line interface which a litany of abstract and esoteric commands. Better how? The GUI allows a much broader swath of the population access to the power of the computer, whereas the pro's find the command-line much more efficient and powerful (though not all and preference does play a role, imperfect analogy being what it is). LM is definitely more powerful, as the number of systems which can be analyzed drastically increases over NM. However NM has great utility in the problem solving domain, still even for pros, but has significantly less overhead for all your typical/simple problems. Generally it doesn't usually even come up until you have gone through a process of ever increasing difficulty culminating in, from my anecdotal experience, moving reference frames where the simple F=ma gives way to all sorts of additional "imaginary forces" that come about from the rotation, for instance, of a reference frame. This is where the topic can be introduced as a way to short circuit the otherwise gory mess of equations you would end up with using simple NM.
Just my two cents. All this being said though, still like the video only had an issue with the title. Keep spreading the word and your passion for physics!
Wow being an msc student this is easily the best introductory explanation i have heard . Keep going forward u r a great teacher 👍
sir you said that lagrangian doesn't have a physical significance but can we say it is just the excess amount of energy within the system to perform work , synonymous to the concept of gibbs free energy in thermodynamics .....
Interesting connection. My intuition is no, since in thermodynamics one cares about the change in (Gibbs free) energy, whereas the Lagrangian is a total, sign sensitive quantity of energy, and hence is usually equivalent up to an arbitrary constant. It is my understanding that the Lagrangian's significance is in all the equation it features in (i.e. the Euler Lagrange equation), which is a rate of change equation--hence killing the arbitrary constant if it were ever included.
I suspect that neither the Lagrangian nor the Action (hitherto undiscussed) have any direct physical significance to the system--instead, they can be interpreted as tools used to arrive at the correct equations of motion (which are the things which themselves obviously have a ton of direct significance).
It’s a scalar representation of the phase of the system in the phase space
Great video. One of my favourite modules in my physics degree. It's so refreshing after years of writing F=ma that they turn round to you in second year of uni and say 'well actually there's a better way'
You have explained this very well, I understood it without having had very advanced calculus, only integration and derivatives. So good job!
Sometimes TH-cam's algorithms recommend videos from content creators that are actually quite good, such as this one by Parth G. Quick and concise , highlighting the most important questions that a student might ask, without dumbing anything down. Right up my alley, Mr. G.
Hey parth Walter Lewin put your this video in his 8.01 playlist
Waited so long for this one! Can you do some problems from Lagrangian Mechanics?
5:52 This a simple second order differential equation with solutions of either sine, cosine, or an exponential (power of e). This results in a cyclic sine or cosine curve (depending on where you place the origin) when position is graphed as a function of time. The fact that the acceleration has sign opposite to position makes this a restoring force, i.e., motion is constrained within boundaries.
Another gem found in youtube.
I don't know about better, but an additional viewpoint is almost always informative. And yes, scalar quantities like energy are simpler than vectors. But it's also interesting to think directly in terms of forces, even though it's messier, and perhaps more error prone. On the other hand, one could argue that Hamilton's principle, or least action principles in general, are "best" in the sense of elegance and simplicity. Ultimately though, Feynman told us that it's useful (and interesting) to have a variety of different mathematical formulations available for any given theory. Maybe that is the approach that is "better."
I agree that Lagrangian mechanics is great, especially if you are dealing with systems consisting of many variables. But what Newtons formulation handles way better is friction, just add a model of friction (eg. -v or -v^2), doing this with lagrangians is an absolute pain.
It really depends on the scenario. They're certain times when thinking of stuff vectorally allows you to make quick approximations
You nailed it, you delivered exactly what I was looking for. If all your videos get to the point and are as clear as this one, I have here plenty of things to enjoy.
Squiggly L and H are usually used for Lagrangian and Hamiltonian densities which are slightly different from Lagrangians and Hamiltonians.
This video really helped push back my ignorance - mainly to show there is so much more I am ignorant of than I realised.
A great video that helped make complex concepts approachable.
Didn't understand a thing he said, but I'm still transfixed like a deer in headlights ... Here, take my money ... like taking all the potential from my kinetic ... and I'm wobbling my head up and down like the doll on the dashboard!!!
It is probably understood, but just to state it explicitly, Lagrangian Mechanics and its successor, Hamiltonian Mechanics are both directly derived from principles of Newtonian Mechanics. For anyone interested in the details of this derivation, I recommend *Goldstein, “Classical Mechanics”, 3rd edition* (or the older 2nd edition) published by Addison Wesley.
In the first chapter, sections 1-3 give a crash-course basic Newtonian Mechanics (this is only for those already reasonably versed in Newtonian Mechanics as a brief refresher). Sections 4-6 derive Lagrangian Mechanics starting from D'Alembert's Principle in section 4. Chapter 2 introduces Variational Calculus or Hamilton's Principle applied to the Lagrangian. For the more advanced curiosities among us, Hamiltonian Mechanics is introduced in chapters 8-10, Classical Chaos in chapter 11, and the foundations of Field Theory (including Noether's Theorem) in chapter 12.
This is very interesting, but I long for the hour long video that actually makes the case posed by the title instead of acting as an introduction so that a person could understand the title.
Great video! By the way, often the “curly” L represents the so called “density of Lagrangian” which is Lagrangian per unit of volume.
The Lagrangian itself is represented by the capital L.
Just a tiny detail!
Excellent video. Very interesting, informative and worthwhile video. Parth is a brilliant explainer.
But . . . but . . . but . . . What is q in the E-L equation? And exactly how do you plug your Lagrangian into the E-L equation to obtain the result you claimed?
Congratulations on the clarity of your presentation! You have natural teaching skills.
At 6:51 the term on left side is not the total force on the system but describes the acceleration of the system. In other words, it is Newton's Second law which relates acceleration to the total force on the system which appears on the right hand side.
That was a great journey.
Thank you
Awesome! I learned that the Hamiltonian is the sum of KE and PE, I got more xposure to dot notation, and I should go back and rewatch the derivation of F = -kx. A question: why is "the difference between KE and PE" not physical? The difference between its actual energy and what it can do? I like my math to be physical
For larger rigid body systems, iterative newtonian methods are actually numerically "BETTER" (require less computational time)
Thank you sooooooo much, good point for programmers
Hmm... do I detect a game engine programmer?
Hi Parth. I just found your channel and watched this very informative video on Lagrangian Mechanics. I dig your approach to physics and have just subscribed! I'm trying to catch up on math and physics since I'm now retired. I look forward to learning from you!
6:57 true, the Lagrangian approach doesn't explicitly mention "force", but to come up with potential V, wouldn't forces (and it's integral over dx) be needed in the background anyway? So it seems we can't really abandon the force concept
Luv the way you tought sir .......extremely impressive .......if a person luv physics, then they surely start liking you to fr ur creative teaching😊 thnkuuu
Popped up in my recommendation and changed my life..thank you yt!
Just make it a goddam 40 min long video ill watch it in one go because of how interesting you made it
Very nice explanation. I do find it interesting that you stress so often that the Lagrangian isn't a physical quantity but rather a mathematically useful quantity when that is equally true of energy as well. We typically say that things 'have' energy, but energy is just as much a mathematically constructed quantity as the Lagrangian, useful only for its apparent conservation. Like the Lagrangian, energy cannot be measured; only calculated.
Excellent presentation and explanation. I have read and listened to number of presentations by others but none as understandable as yours. Thank you and keep it up.
Thank you for this video. Bringing in the Hamiltonian explanation helps forming the picture in my "trying to catch up" head.
First time I've seen any of your videos Parth, and it's a straight up subscribe for me. I like people who can "really" explain, and enjoy what they do
How much time do you think, Self studying would take- If one starts from undergraduate Classical Mechanics and Electrodynamics to Quantum Mechanics and good level GR stuff and so on?
Being in India I dont think enrolling into a Physics course is a good one, but I am just too much interested in Physics to leave it off for my Electrical Engineering B Tech.
Please do guide as I guess it maybe useful for others too 😀
i think that, assuming you have sufficient discipline, it would take 3-4 years to get to this standard (which I believe is 2nd or 3rd year knowledge). I'm a first year so take this with a pinch of salt aha
It depends on how efficient your study method is and how much time you spend studying per day. At its most hardcore, 1,5 weeks should be enough to learn one semester module of 12credits worth of work, but it may be very exhausting. So maybe 2,5 to 3 weeks for one module. Assuming there are an average of 6 modules per semester, it can take 54 up to 108 weeks (equal to 1 to 2 years) to complete an undergrad course. This may not be sufficient to master your work but it should be enough to work through some problems and understand the concepts
Great video sir
I just completed my course in classical mechanics but Lagrangian and Hamiltonian mechanics were not included..
Now I will learn this from u❤️
L = difference between Kinetic and Potential energy. I assume this means L is related to the potential for change.
Glorious explanation. I can only dream of having professors this effective at my uni...
Great Explanation. The point is you kept everything simple while still useful and let us see its potential, definitely subcribed
As a mathematician and a image processing specialist, Euler Lagrange equation is very important in minimazing energy functionals
This was really interesting. I'm currently taking a class on Newtonian mechanics and I am curious to understand the connections between Lagrangian, Hamiltonian, and Newtonian mechanics.
It would have been helpful to have abour 1 more sentence in the intro towards which target audience this video is directed. To be fair u said that the video is about basics but imo thats kind of an unclear definition. Would be nice if u could add that in ur future videos. For me personally it still was a nice video yet not what i was looking for as this rather seems to be an introducion to this topic and not an actual explanation of why the things are happening the way they are. Ty for your videos :)
There is a very beautiful connection between the "physical properties" and the Lagrangian. By performing a Legendre Transform from the variable "velocity" to its slope, called momentum p, we get the symmetry condition of the Legendre transform as \dot q = \dfrac{\partial H}{\partial p} just as the original defintion of the canonical momentum reads p := \dfrac{\partial L}{\partial \dot q}. Now comes the breakthough: With this "second" equation we can write the total time evolution of the Hamiltonian as \dot H = \dfrac{\partial H}{\partial t}+\dfrac{\partial H}{\partial q}\dot q+\dfrac{\partial H}{\partial p}\dot p and take the transformed Version of the Euler-Lagrange-equation of motion for \dot p and the Legendre-Transform for \dot q and have a closed form where q, p and t are the only variables, and even more: They appear in an anti-symmetric ararrangeemnt, commonly denoted by Poissons' bracket, a special case of the Lie-brackets (commutator of two operators) commonly used in Quantum mechanics. The point is: You cannot achieve this anti-symmetric closed arrangement with the Lagangian as by the very same calculus \dot L = \dfrac{\partial L}{\partial t}+\dfrac{\partial L}{\partial q}\dot q+\dfrac{\partial L}{\partial \dot q}\ddot q and the acceleration \ddot q does not appear in the general Euler-Lagrange equation (just take any coordinate frame other than carthesian and you will see that the acceleration in a coordinate is not necessarily easily extracted/isolated), so the only meaningful way we can make predictions on the time evolution of the Lagrangian (and therfore its physical meaning) is by using the Legendre Transform again, writing L = H - \dot q p and reasoning \dot L = \dfrac{\partial L}{\partial t} + \{H,\dot q p\}. In general, this is not an easy thing to do, but if 1) time symmetry holds and 2) the momentum is linear in velocity with some constant term p=\dot q/a, then the Lagrangian (plus a constant) is simply int \dot L dt = \int \{H,\dot q p\} dt = \int \{H, a p\} p + \{H, p\}\dot q dt = \int (a*\dot p+\dot p)\dot q dt which is, if you squint you eyes, the total change in momentum, called a force, integrated over a path of motion ds = \dot q dt, which is the classical Newtonian definition of Work. The classical Lagrangian is a multiple of the total work done in a physical process, and the principle of least action states that the total amount of work done within a certain time frame must be extreme (mostly minimized). There you go, classical mechanics is really just "The universe is lazy". And also, most of the facy commutators of quantum operators you learn in QM can be solved by calculating corresponting Poisson brackets; the underlying anti-symmetry of its arguments is transferred from one theory to the other, or as we call it: Algebra remains.
:)
ps sorry for typos :/
Great vid. Hamiltonian mech next?
It was amazing , thanks TH-cam for recommending such an astonishing video 🙃
Wow!! You explained it in the simplest way!! Hats off, man
I enjoyed your video very much. You're concise and clear, and filter out irrelevant mathematical complexity to make an important point. Fantastic.
wow...just derived and used it ,,few days ago in the exam..❤️
how did it go
@@jalajtrivedi6470 first in the class
When can you make a video of Lagrangians in relativity and in quantum mechanics
I really enjoy your content. I'm hoping to study Physics at a higher level and I find your videos useful 🙂
Should I research Hamiltonian mechanics on my own, or wait until I've seen the musical first?
Different Hamilton: Sir William Rowan H, not Alexander H.
Fantastic video, really interesting because as an alevel physics student have never dealt with lagrangian only newtonian mechanical physics. Also, you have incredible head hair sir!
I come to your channel , before starting , any new branch of physics ,
To use different colors in the equations was a good idea.
i like that you explain it like we are 5 yo. Now i understand this hahah
you’re only 5, why are you on youtube
Man your videos are good.. Keep up the good work👍🏻
Really great video!! 👏👏👏
You have the gift of communication.
As an engineering student, I have put this into practice, but I have no clue why xdot and x are assumed to be totally independent of each other when you are doing the partial derivatives.
Because they represent completely independent quantities. x-dot is velocity, which is typically only relevant when finding kinetic energy, whereas x is position, which is typically only important when finding potential energy. The velocity of an object moving at x-dot meters per second is the same whether it's at x = 0 or x = infinity. The only link meaningful link between x and x-dot in this type of problem is that they're happening on the same axis.
@@dwangnoderbora In the case where x (position) is increasing, you can always parameterize the velocity to be a function of position. Then they are literally not independent of each other. Riddle me that.
Deriving the equations of motion for a double pendulum from a Lagrangian and Newtonian perspective is enlightening: it's pretty straight-forward from a Lagrangian perspective, but more challenging from a Newtonian perspective.
Hey buddy your ability of explanation is highly lucid and alluring. Could you tell me from which country you belong and from which university have you pursued your undergrad postgrad and PhD
I was right with you up to, "Now many of you have asked me to discuss............".
Just started learning about Lagrangian mechanics in my Mechanics I class... Really cool stuff! Great video :)
Parth G, Thank you so much for this wonderful video! Please make a series on Calculus of Variations.
Parth, your videos are great! You have gotten so good at this!
I was looking forward to hear an argument "Why Lagrangian Mechanics is BETTER than Newtonian Mechanics F=ma." The video states @7:38 three reasons about complicated system with multiple forces and coordinates. BUT the example is a 1-d case that doesn't need the Langrangian. Maybe a complicated system with cylinder coordinates or other deviation from the simplest cases would have been a better example.
. . . I'm looking forward to your Noether theorem video, but please show also what you tell us.
Lagrangians made me crazy I started seeing the energy of the curves on buildings and objects around me, and I still have to think about other stuff, because maaaaaaan
Very clear introduction ! Could you do the same for hamiltonians as they look so similar ?
It is a real skill morphing the complicated into the comprehensible.
Please do a video on advantages of Hamilton Jacobi method over Hamiltonian mechanics
What a coincidence! I was just looking for LangragIan just a few minUtes before you uploaded this and there you go ,I am loving this and it's my first time watching your videos . Please keep posting more good mathematical and physics stuffs 👍
I look forward to learning more about lagrangian mechanics with you sir
Nailed it, Langrangian way to go as an investigative math tool, hope to see more how does it unravel more 🤔:)
Hi! I have been your follower time immemorial. I got a question. If Lagrangian mechanics gives the equations of motion of a particle without being so concerned with forces, how is it we still need to get the potential energy using a force (conservative force to be specific, which is derivable from the potential)? In your example F=-kx is a spring force. That's how we come up with the potential energy U = 1/2 kx^2., plugged into the Lagrangian.
I think what the Lagrangian is representing is the quantity of kinetic energy actively being used at any one time, assuming that potential energy is potential kinetic energy
At 7:12, in point-1, as L=T-V, to determine V, shouldn't we know all the forces in advance. Like in spring mass system, we derive it's potential energy as 1/2 kx^2 as we know the form of the force as F=-kx. In that sense isn't Newtonian mechanics superior than Lagrangian Mechanics?
I think so... To determine energy u must know the forces
No, the natural way of writing a conservative force is using it's potential.
@@twakilon OK, for the conservative force, we derive its expression by its potential. Thanks!
Hii parth can you do a video explaining hamiltonian mechanics
Hey parth! I love your videos! Can you make a video on Lagrange multipliers and how to deal with constraints using Lagrange mechanics? I want a video from you because I like your style of teaching
Pure brilliance in your explanation.
To be fair, Euler-Lagrange (EL) is 'better' than Newton-Euler (NE) in certain situation. Indeed, defining constraint force can be tricky for the NE, but there a few situations where NE outshines EL.
If you're dealing with a large complex dynamical systems (i.e., large number of degrees--of-freedom), the EL-equations become very lengthy. The reason for this, is that EL equations require analytical expressions for time-derivates of the generalized coordinates. If the coordinates are highly dynamically coupled, this can becomes exponentially complex quickly...
In these cases, its better to use a numerical approach to compute the dynamics through the NE-equations (this is often done in robotics). I also believe most physics engines use NE-equations over EL-equations since its more suitable for numerical computation.
squiggly L is usually a Lagrangian density (e.g., in field theory).
Oh interesting, we were always taught to use it to represent the Lagrangian itself
Awesome video as always.
Just one question WHY IS EULER IN EVERY THING??
it's good thing - i kinda dont like newton. euler's work is awe inspiring - just beautiful every time i see smth he did.
Because Euler was a genius, most likely.
I appreciate your effort for putting out a video on the solutions of the Schr(accent)odinger Equation. My new request is a video on Kaluza-Klein Theory , I am sure that video will be out soon too.
5:07 okay, The "L" was obvious, but what is q here??
5:10 Euler Lagrange Equation is consistent with Newtonian Mechanics, and mention of resources to study about that
q is just notation, it's just an arbitrary position of a particle, or in the case of q dot, it's speed.
For your second question, this is actually obvious once you differentiate the Lagrangian. x with two dots represents acceleration, so the left side is ma, and by Hooke's law, -kx is the spring force. Therefore F = ma. That's not a complete proof however.
You can also remember that on the left hand side we are differentiation the left hand side by time, since kinetic energy is usually defined in terms of v, this will gives us acceleration. We also can remember that the change in potential energy over space is force. And so the right hand side will be in terms of a force, and the left will be in terms of acceleration and mass.
Once again there's more to it than that, however if you're curious I recommend you search up the Euler-Lagrange Equation, and Calculus of Variations.
@@therealjezzyc6209 hi, thanks for the detailed input. The second point was not a question - i just wrote it to save for myself - like as a reminder that this was mentioned here. but thanks a lot for explanation. I liked it and it saved me time :) otherwise i'll have to search it later.
@@yash1152 oh my bad, glad I could help however.
Then T=(H+L)/2 and V=(H-L)/2: the kinetic energy is the hamiltonian plus the lagrangian divided by 2, and the potential energy is the hamiltonian minus the lagrangian divided by 2. I don't know if it has a meaning or not :-)
Is the potential energy signed? As written x can be positive and negative. With a positive potential when the displacement is positive (stretched spring) and a negative potential when the displacement is negative (compressed spring). That right? Feels counterintuitive since the kinetic energy is unsigned.