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: ruclips.net/video/3QCXVKUi7K8/видео.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.
@@udbhav5079 *work. The classical Lagrangian is exactly a multiple of the total work done in a physical process, the action is the work accumulated overa given time. The "Lagrangian equals work times constant plus some arbitry offset" approach is only valid for classical mechanics; in relativistic mechanics you either a) redefine work to be the Lagrangian, for some reason, or b) stick to the good ol' force concept, even though you know that none of this was ever exactly valid in the first palce, because spacetime and stuff, and call it a day (as every scientist on earth does, i'm afraid :/ ).
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 RUclips. 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!
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.
Something really important to keep in mind with regards to Euler-Lagrange equation: partial derivative and derivative are not the same thing! In many places partial derivatives behave as they were plain derivatives but in E-L there is a good chance they do not!
I like the style of the video and the explanations. There's a rather relevant point missing around 5:55 : q and q-dot in L stand for generalized coordinates and their derivatives, and for the srping-mass system we chose q = x. This can also help emphasize the importance of point (3) around 7:40.
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).
@@ParthGChannel i haven't yet studied lagrangian mechanics (by the end of this semester i will) but the first time i understand what it is, was after watching his video
L=T-U can be derived from D'Alembert's principle of virtual displacement or virtual work. Concerning the Euler-Lagrange equations, this is only applicable to systems where FRICTION is NOT involved. If there are systems with FRICTION then you have to add the Rayleigh dissipation function to the E-L equations.
Sometimes RUclips'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.
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.
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.
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.
@@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.
Very well done, a clear and compelling review without a lot of extraneous factoids cluttering the message Like it a lot, you've got a good style and I'm subscribing now. Thank you. 👍
Great job! I am going to share this channel with all the college students. It took me weeks to get started with Lagrangian mechanics (a few decades ago). I wish we had an introduction like this. In a multibody connected dynamic system, e.g. Robots, machines, mechanisms, etc. if one starts with Newtonian formulations, many unknown joint/contact forces appear in the equations and it becomes difficult to solve for the motion. If one uses Euler-Lagrangian equation, it is much easier to solve for the motion.
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.
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).
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.
I would add that the Lagrangian really shines when you're dealing with a problem with constraints. For example, a particle constrained to ride along a curved track (like a rollercoaster). Or the double pendulum (one pendulum hanging from another), in which the coordinate of the bottom pendulum bob depends on the position of the upper one. In these sorts of problems, Newtonian mechanics gets bogged down in dealing with coordinate changes and interdependences, and also dealing with which forces are "constraint forces" like normal forces and tension which hold the particle(s) to travel along the allowed path. But the Lagrangian is much simpler to write down in both cases (since it only depends on the magnitudes of the velocities - directions don't matter! - and whatever functional dependence the potential energy has on position).
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'
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!
Lovely! Lovely!! Very well explained, Parth. I'd studied this long ago and was trying to recall what the Lagrangian was all about, and you explained it so well. Thank you!!
In my view the lagrangian 'kind of' has a physical meaning. Its not anything formal, but T-V represents a desequilibrium, or difference, of the system. If T>V the lagrangian points towards a system that aims at increasing the potential energy and lower the kinetic energy, and viceversa.
Really enjoyed this video, thanks Parth! I'd always heard of Lagrangians and Hamiltonians in the context of QM but never got around to learning what they actually represent. Your explanation and example definitely helped me get a better understanding of the concepts: a nonphysical but useful mathematical tool and the total energy of a system. I was exited to hear Noether's Theorem is based upon Lagrangians, too. I really wish more people knew of the brilliance of Emmy Noether, so I'm glad this may have introduced some to her work and name for the first time. If you've not already seen it, I really enjoy this message Einstein wrote to Hilbert upon receiving her work: Yesterday I received from Miss Noether a very interesting paper on invariants. I'm impressed that such things can be understood in such a general way. The old guard at Göttingen should take some lessons from Miss Noether! She seems to know her stuff.
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.
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.
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 :/
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.
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
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
Nice introduction to LM ... An important point which was overlooked is the way in which LM can incorporate generalized forces (which would appear as extra terms in the E-L equation). Such forces must be taken into account when some physical forces acting on the system are not conservative (and therefore not expressible via potential energy). Such forces also are especially convenient/useful for assessing relevant constraint forces.
Noether's Theorem 8:13 ... based on L, states that there is ... 8:17 ... link in symm and conservation laws 8:24 ... like: (3 mentioned: linear/angular momentum, energy). Does it apply on conservation of charge too?
the way you're explaining things is very good. You're explaining slowly so that even me who is still in school and from Germany can understand everthing. Keep going! You're helping a lot of people and i wanna thank you!
There's no answer to the question in the title. In many situations, the concept of force is way more natural. The friction was already mentioned. The energy may totally conserve, though. How long will it take for a ladder of mass m and length l, which was initially at rest and inclined against the wall at angle \alpha, to hit the floor? The pencil slides across the surface of the table of height h with the given velocity and slides off the edge. How many revolutions it will make before hitting the floor? What is the speed of sound in dry air at sea level at temperature T? What is the signal distortion in a circuit with relay with given characteristics (mass, stiffness, controlling current, all that's needed)? What is the equation of motion of the string with given density and tension with irregularly spaced attached point masses? There's no need to promote the Lagrangian framework as better. It works great for one class of problems, the Newtonian mechanics works great for another class of problems. These classes overlap but not completely.
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!
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
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."
Isn't the kinetic energy of a mass on a spring oscillating under simple harmonic motion? Surely that means that the kinetic energy varies because velocity varies?
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❤️
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.
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.
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
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.
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?
Not true, I use it all the time. In Hamiltonian mechanics you have a greater freedom in choosing transformations. So it is used a lot in Astronomy and Accelerator physics (my field). But it does come from the Lagrangian ultimately. In Lagragian mechanics, the minimization principle makes it clear that you can used all sorts of variables for x,y and z. But in Hamiltonian mechanics, the equivalent of dx/dt becomes a variable of its own. As long as you make transformations that preserves the so-called Poisson bracket, things are still "Hamiltonian". You could go back to the Lagrangian any time...... ALso, first quantization, ie, Schroedinger, is easier with the Hamiltonian. Poisson brackets turn into commutators. In second quantization, ie field theory, then the Lagrangian resurfaces. Clearly these are complementary methods,
@@jceepf what you say about the freedom to choose canonical coordinates and its usefulness is true, but be advised that it is not always true that you can go back and forth from Lagrangian to Hamiltonian mechanics. Constrained systems, like the free relativistic point particle in spacetime formulation, require a more careful analysis (initiated by Dirac, quite unsurprisingly, and finished by Tulczijew).
@Parth G , hi Parth can you tell me why L= T- V ? Why not L= f(T)- f(V)???, in addition I want to know How we came to this standard form of L (Lagrangian) without using the least action principle. Thanks in advance
Ky Lagrangian mechanics se ya Hamiltonian mechanics se heartbeat ya fluid flow ya fir weather,climate ke cases solve kr sakte hain.ku ki double pendulum bhi toh ek chaotic nature show karta hai,toh fir yeh cases solve kar sakte hai Lagrangian mechanics se ya Hamiltonian mechanics se?
Hi @ParthG, You said that we will not depend on Force anyway, instead we are deriving equations of motion in Lagrangian mechanics using the energy functions alone . I am bit confused here.. When you wrote the potential energy expression, I thought you got it by Integrating F•s (work done stored as potential energy). But If we assume we don't know what the force is , How can we get expression for the potential energy in the first place?
The natural way of writing a conservative force is the potential V, not the force, so it's actually the other way around. Since the force is the negative gradient of the potential you can derive the potential knowing the force easily though.
Dealing with energies is much more easier than dealing with forces as force is a vector quantity ( it needs to be added using vector algebra) where as energy is a scalar one ( so we can get rid of the annoying vector algebra). That's why Lagrangian mechanics is more convenient than Newtonian one.
To what points are the forces applied on illustration @ 6:26? The points are different and not in one line with the forces (the vectors), it will make the object to rotate. Learn the basics. With something like that you will repeat a year in a primary school (a normal one). Massless spring, zero friction, constant value of k, zero gravity - so what keeps the object on the surface? It is not physics it is utopia. In physics every spring has a mas, k=f(x) is a function of x, gravity does exists and friction is grater than zero. And what is "spring force"?
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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: ruclips.net/video/3QCXVKUi7K8/видео.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.
@@udbhav5079 *work. The classical Lagrangian is exactly a multiple of the total work done in a physical process, the action is the work accumulated overa given time. The "Lagrangian equals work times constant plus some arbitry offset" approach is only valid for classical mechanics; in relativistic mechanics you either a) redefine work to be the Lagrangian, for some reason, or b) stick to the good ol' force concept, even though you know that none of this was ever exactly valid in the first palce, because spacetime and stuff, and call it a day (as every scientist on earth does, i'm afraid :/ ).
Having studied this intimately in grad school, and applied the principles in my M.Sc. thesis, I find your explanation clear and concise. Well done!
Sure when you familiar with what will be "in a separate video" & "that's in for another video".
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 RUclips. 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!
I remember how amazed I was at how usefull Lagrangian mechanics are dealing with complicated mechanics problems, when I learnt about them.
The depth of content is so well-balanced for such a short video, really enjoyed it!
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.
Something really important to keep in mind with regards to Euler-Lagrange equation: partial derivative and derivative are not the same thing! In many places partial derivatives behave as they were plain derivatives but in E-L there is a good chance they do not!
I like the style of the video and the explanations. There's a rather relevant point missing around 5:55 : q and q-dot in L stand for generalized coordinates and their derivatives, and for the srping-mass system we chose q = x. This can also help emphasize the importance of point (3) around 7:40.
i was also expecting that
A good point.
"...an ideal system"
me: wait that's not a spherical cow?
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).
And there is our Andrew Dotson who solves Projectile motion with Lagrangian formalism.
Yes
Overkilling a simple problem
Absolutely fair and valid lol, love Andrew's work
@@ParthGChannel i haven't yet studied lagrangian mechanics (by the end of this semester i will) but the first time i understand what it is, was after watching his video
L=T-U can be derived from D'Alembert's principle of virtual displacement or virtual work.
Concerning the Euler-Lagrange equations, this is only applicable to systems where FRICTION is NOT involved.
If there are systems with FRICTION then you have to add the Rayleigh dissipation function to the E-L equations.
Where have u been for these many days, bro ur videos are a nerd's dream come true.
I came here from Walter Lewins playlist of classical mechanics . Your video was added in that playlist
Walter lewin✨
Same
Sometimes RUclips'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.
Ayyyy! Thank for your video, man! Watched few videos about Langranian Mechanics every each of them gives different view of it. Thank you
You have explained this very well, I understood it without having had very advanced calculus, only integration and derivatives. So good job!
Waited so long for this one! Can you do some problems from Lagrangian Mechanics?
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.
Wow being an msc student this is easily the best introductory explanation i have heard . Keep going forward u r a great teacher 👍
currently in a robotics major and lagrangian mechanics is probably the coolest thing i have learned
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.
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 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.
Very well done, a clear and compelling review without a lot of extraneous factoids cluttering the message Like it a lot, you've got a good style and I'm subscribing now. Thank you. 👍
Great job! I am going to share this channel with all the college students. It took me weeks to get started with Lagrangian mechanics (a few decades ago). I wish we had an introduction like this.
In a multibody connected dynamic system, e.g. Robots, machines, mechanisms, etc. if one starts with Newtonian formulations, many unknown joint/contact forces appear in the equations and it becomes difficult to solve for the motion. If one uses Euler-Lagrangian equation, it is much easier to solve for the motion.
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.
Finally someone who states clearly the pure mathematical (and not physical) nature of this theoretical item!
Popped up in my recommendation and changed my life..thank you yt!
Humble request need a video on symmetry of space and time and how it leads to conservation laws.
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
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.
I would add that the Lagrangian really shines when you're dealing with a problem with constraints. For example, a particle constrained to ride along a curved track (like a rollercoaster). Or the double pendulum (one pendulum hanging from another), in which the coordinate of the bottom pendulum bob depends on the position of the upper one.
In these sorts of problems, Newtonian mechanics gets bogged down in dealing with coordinate changes and interdependences, and also dealing with which forces are "constraint forces" like normal forces and tension which hold the particle(s) to travel along the allowed path.
But the Lagrangian is much simpler to write down in both cases (since it only depends on the magnitudes of the velocities - directions don't matter! - and whatever functional dependence the potential energy has on position).
Congratulations on the clarity of your presentation! You have natural teaching skills.
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'
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!
L = difference between Kinetic and Potential energy. I assume this means L is related to the potential for change.
Lovely! Lovely!! Very well explained, Parth. I'd studied this long ago and was trying to recall what the Lagrangian was all about, and you explained it so well. Thank you!!
In my view the lagrangian 'kind of' has a physical meaning. Its not anything formal, but T-V represents a desequilibrium, or difference, of the system. If T>V the lagrangian points towards a system that aims at increasing the potential energy and lower the kinetic energy, and viceversa.
Great Explanation. The point is you kept everything simple while still useful and let us see its potential, definitely subcribed
Really enjoyed this video, thanks Parth! I'd always heard of Lagrangians and Hamiltonians in the context of QM but never got around to learning what they actually represent. Your explanation and example definitely helped me get a better understanding of the concepts: a nonphysical but useful mathematical tool and the total energy of a system.
I was exited to hear Noether's Theorem is based upon Lagrangians, too. I really wish more people knew of the brilliance of Emmy Noether, so I'm glad this may have introduced some to her work and name for the first time. If you've not already seen it, I really enjoy this message Einstein wrote to Hilbert upon receiving her work:
Yesterday I received from Miss Noether a very interesting paper on invariants. I'm impressed that such things can be understood in such a general way. The old guard at Göttingen should take some lessons from Miss Noether! She seems to know her stuff.
Squiggly L and H are usually used for Lagrangian and Hamiltonian densities which are slightly different from Lagrangians and Hamiltonians.
Another gem found in youtube.
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.
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.
As a mathematician and a image processing specialist, Euler Lagrange equation is very important in minimazing energy functionals
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 :/
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.
Excellent video. Very interesting, informative and worthwhile video. Parth is a brilliant explainer.
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.
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
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
8:03 The blue and orange lamps in the back are a vibe
This is more complex but much more efficient than the simple thing we've learned!
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
This was the best physics video I've watched in a while. Great video Parth
Thank you for this video. Bringing in the Hamiltonian explanation helps forming the picture in my "trying to catch up" head.
Glorious explanation. I can only dream of having professors this effective at my uni...
Man, you look like a very empathic, spiritual and warm person, and the info you share are so well structured. It is good i found your chanel.
Nice introduction to LM ... An important point which was overlooked is the way in which LM can incorporate generalized forces (which would appear as extra terms in the E-L equation). Such forces must be taken into account when some physical forces acting on the system are not conservative (and therefore not expressible via potential energy). Such forces also are especially convenient/useful for assessing relevant constraint forces.
Noether's Theorem
8:13 ... based on L, states that there is ...
8:17 ... link in symm and conservation laws
8:24 ... like: (3 mentioned: linear/angular momentum, energy). Does it apply on conservation of charge too?
the way you're explaining things is very good. You're explaining slowly so that even me who is still in school and from Germany can understand everthing. Keep going! You're helping a lot of people and i wanna thank you!
There's no answer to the question in the title. In many situations, the concept of force is way more natural. The friction was already mentioned. The energy may totally conserve, though. How long will it take for a ladder of mass m and length l, which was initially at rest and inclined against the wall at angle \alpha, to hit the floor? The pencil slides across the surface of the table of height h with the given velocity and slides off the edge. How many revolutions it will make before hitting the floor? What is the speed of sound in dry air at sea level at temperature T? What is the signal distortion in a circuit with relay with given characteristics (mass, stiffness, controlling current, all that's needed)? What is the equation of motion of the string with given density and tension with irregularly spaced attached point masses?
There's no need to promote the Lagrangian framework as better. It works great for one class of problems, the Newtonian mechanics works great for another class of problems. These classes overlap but not completely.
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!
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
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."
Isn't the kinetic energy of a mass on a spring oscillating under simple harmonic motion? Surely that means that the kinetic energy varies because velocity varies?
There is a small imprecision at 8:36. From those symmetries a conservation law follows but not the other way around.
When can you make a video of Lagrangians in relativity and in quantum mechanics
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❤️
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.
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.
Parth, your videos are great! You have gotten so good at this!
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.
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
Thanks for your well explained videos that always helps me picture and understand my physics courses better.
Oh yes! this channel is a great find. Can't wait to see the video on Noether's theorem!
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.
Man your videos are good.. Keep up the good work👍🏻
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!
Hamiltonian mechanics : why doesn’t anyone love me :(
Normal people: Because no one wants to solve two differential equations when they could just solve one.
Me, an intellectual: I like ZZ Top
"He got his own musical! Ain't that enuff?"
Not true, I use it all the time. In Hamiltonian mechanics you have a greater freedom in choosing transformations. So it is used a lot in Astronomy and Accelerator physics (my field). But it does come from the Lagrangian ultimately.
In Lagragian mechanics, the minimization principle makes it clear that you can used all sorts of variables for x,y and z. But in Hamiltonian mechanics, the equivalent of dx/dt becomes a variable of its own. As long as you make transformations that preserves the so-called Poisson bracket, things are still "Hamiltonian". You could go back to the Lagrangian any time......
ALso, first quantization, ie, Schroedinger, is easier with the Hamiltonian. Poisson brackets turn into commutators. In second quantization, ie field theory, then the Lagrangian resurfaces.
Clearly these are complementary methods,
@@jceepf what you say about the freedom to choose canonical coordinates and its usefulness is true, but be advised that it is not always true that you can go back and forth from Lagrangian to Hamiltonian mechanics. Constrained systems, like the free relativistic point particle in spacetime formulation, require a more careful analysis (initiated by Dirac, quite unsurprisingly, and finished by Tulczijew).
@@ilrufy7315 true. I was wrong to say that it is always possible.
this is the first video of you I saw, And your channel just got a new subscriber
Very well explained sir. I would like to from you on which topic your are researching ?
@Parth G , hi Parth can you tell me why L= T- V ? Why not L= f(T)- f(V)???, in addition I want to know How we came to this standard form of L (Lagrangian) without using the least action principle. Thanks in advance
I'm watching this instead of paying attention to physics class lmao
me too
Parth G, Thank you so much for this wonderful video! Please make a series on Calculus of Variations.
Ky Lagrangian mechanics se ya Hamiltonian mechanics se heartbeat ya fluid flow ya fir weather,climate ke cases solve kr sakte hain.ku ki double pendulum bhi toh ek chaotic nature show karta hai,toh fir yeh cases solve kar sakte hai Lagrangian mechanics se ya Hamiltonian mechanics se?
Really great video!! 👏👏👏
You have the gift of communication.
Hi @ParthG, You said that we will not depend on Force anyway, instead we are deriving equations of motion in Lagrangian mechanics using the energy functions alone . I am bit confused here.. When you wrote the potential energy expression, I thought you got it by Integrating F•s (work done stored as potential energy). But If we assume we don't know what the force is , How can we get expression for the potential energy in the first place?
The natural way of writing a conservative force is the potential V, not the force, so it's actually the other way around. Since the force is the negative gradient of the potential you can derive the potential knowing the force easily though.
@@twakilon then, how we know the potential? it cannot come from nothing, right?
It really depends on the scenario. They're certain times when thinking of stuff vectorally allows you to make quick approximations
Hamiltonian
1:38 Hamiltonian
1:54 Hamiltonian Mechanics and
1:59 mention of Hamiltonion Operator (remember: from Schrodinger Equation)
Dealing with energies is much more easier than dealing with forces as force is a vector quantity ( it needs to be added using vector algebra) where as energy is a scalar one ( so we can get rid of the annoying vector algebra). That's why Lagrangian mechanics is more convenient than Newtonian one.
To what points are the forces applied on illustration @ 6:26? The points are different and not in one line with the forces (the vectors), it will make the object to rotate. Learn the basics. With something like that you will repeat a year in a primary school (a normal one).
Massless spring, zero friction, constant value of k, zero gravity - so what keeps the object on the surface? It is not physics it is utopia. In physics every spring has a mas, k=f(x) is a function of x, gravity does exists and friction is grater than zero. And what is "spring force"?