Why Lagrangian Mechanics is BETTER than Newtonian Mechanics F=ma | Euler-Lagrange Equation | Parth G
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- čas přidán 2. 06. 2024
- Newtonian Mechanics is the basis of all classical physics... but is there a mathematical formulation that is better?
In many cases, yes indeed there is! Lagrangian mechanics, named after Joseph Louis Lagrange, is a formulation of classical physics that is often more convenient to use than Newtonian Mechanics.
The first concept worth knowing about is that a quantity called the "Lagrangian" is defined as L = T - V where T is the kinetic energy of the system we happen to be studying, and V is the potential energy. In this video, we see how to find the Lagrangian for a simple mass-spring system by considering the mass of the block, spring constant of the spring, and motion of the entire setup. We see how to write the kinetic energy (or more specifically the speed of the setup) in terms of the distance moved by the system.
Once we have found the Lagrangian for the system, we then see that the "big boi" equation of Lagrangian mechanics is the Euler-Lagrange Equation. This is a complicated (looking) equation that allows us to substitute in the Lagrangian, and churn out an equation of motion for the system. Now the Euler Lagrange Equation (or EL Equation) is consistent with classical Newtonian mechanics - something I'd like to show in a future video. But the important thing is that there is some calculus (normal derivatives and partial derivatives) to be done, after which we will find an equation of motion.
Now for the mass-spring system, the equation of motion could have easily been found by considering the forces acting on the system (in this case the force exerted by the spring), and saying that the sum of all the forces was equal to the net force, ma. This is essentially the same as applying Newton's Second Law to our system. And in this case, using forces is MUCH simpler. So why did we go the long way and use Lagrangian mechanics?
Well, in many other scenarios, working with forces can be very complicated and fiddly. Working with energies, which is what Lagrangian mechanics does, can be much easier. Additionally, the EL Equation is very well suited to working with multiple coordinates. For example, if an object displays motion in multiple directions, such as x, y, and z, and there are different forces acting in each direction, then with Newtonian mechanics we would have to resolve all the forces in each direction. With Lagrangian mechanics, we simply find the Lagrangian and find an equation for each coordinate - x, y, and z.
That just about covers the very basics of Lagrangian mechanics, but it's worth mentioning that it goes much deeper. For example, Noether's Theorem, which talks about the fundamental link between symmetry and conservation laws (conservation of energy, conservation of momentum, etc) is based on the EL equation. It is a very interesting look at some deep universal concepts, and it's based on Lagrangian mechanics!
For those of you interested in finding out more about the Euler-Lagrange Equation, please check out this wiki page: en.wikipedia.org/wiki/Euler%E...
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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: czcams.com/video/3QCXVKUi7K8/video.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!
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.
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!
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
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
The depth of content is so well-balanced for such a short video, really enjoyed it!
"...an ideal system"
me: wait that's not a spherical cow?
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
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...
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).
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!
You have explained this very well, I understood it without having had very advanced calculus, only integration and derivatives. So good job!
Wow being an msc student this is easily the best introductory explanation i have heard . Keep going forward u r a great teacher 👍
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
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!!
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!
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.
Great Explanation. The point is you kept everything simple while still useful and let us see its potential, definitely subcribed
I came here from Walter Lewins playlist of classical mechanics . Your video was added in that playlist
Walter lewin✨
Same
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.
Sometimes CZcams'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.
Oh yes! this channel is a great find. Can't wait to see the video on Noether's theorem!
Waited so long for this one! Can you do some problems from Lagrangian Mechanics?
Where have u been for these many days, bro ur videos are a nerd's dream come true.
Parth, your videos are great! You have gotten so good at this!
Congratulations on the clarity of your presentation! You have natural teaching skills.
This was the best physics video I've watched in a while. Great video Parth
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.
Surely this is one of the best explanations of the Lagrangian on CZcams. 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!
Glorious explanation. I can only dream of having professors this effective at my uni...
currently in a robotics major and lagrangian mechanics is probably the coolest thing i have learned
Ayyyy! Thank for your video, man! Watched few videos about Langranian Mechanics every each of them gives different view of it. Thank you
Thanks for your well explained videos that always helps me picture and understand my physics courses better.
Just started learning about Lagrangian mechanics in my Mechanics I class... Really cool stuff! Great video :)
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❤️
Interesting and very well explained. Glad YT recommended it.
Great Video. Very well explained. Really liked the key points at the end, find myself finishing maths videos and not coming away with anything. Thanks
Parth G, Thank you so much for this wonderful video! Please make a series on Calculus of Variations.
Excellent video. Very interesting, informative and worthwhile video. Parth is a brilliant explainer.
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.
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.
I enjoyed your video very much. You're concise and clear, and filter out irrelevant mathematical complexity to make an important point. Fantastic.
I really love your videos. Great if you can make video on practical problems based on the theory in each case.
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.
It really depends on the scenario. They're certain times when thinking of stuff vectorally allows you to make quick approximations
Excellent explanation - very clear and easy to follow!
Great presentation, everything is clear and elegant and surprising!
Popped up in my recommendation and changed my life..thank you yt!
I remember how amazed I was at how usefull Lagrangian mechanics are dealing with complicated mechanics problems, when I learnt about them.
It was amazing , thanks CZcams for recommending such an astonishing video 🙃
I really enjoy your content. I'm hoping to study Physics at a higher level and I find your videos useful 🙂
I look forward to learning more about lagrangian mechanics with you sir
Not sure why this was in my recommended but subscribed! Great job on the video. Wish I had this in Physics 2 last year.
Wow!! You explained it in the simplest way!! Hats off, man
Man your videos are good.. Keep up the good work👍🏻
Very well researched and presented, thank you.
LOVE U PARTH, THANK YOU FOR TAKING THIS TOPIC .
Squiggly L and H are usually used for Lagrangian and Hamiltonian densities which are slightly different from Lagrangians and Hamiltonians.
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 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.
Your video are so informative and helpful... They keep my eagerness high to lear more
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.
Pure brilliance in your explanation.
Another gem found in youtube.
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
Great stuff -- and a relief to have the hope of getting Hamiltonian and Lagrangian under control, if not today maybe in due course!
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.
Feeling blessed, this awesome video came on my recommendation. Lovely Explanation
this is the first video of you I saw, And your channel just got a new subscriber
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
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.
Thank you for this video. Bringing in the Hamiltonian explanation helps forming the picture in my "trying to catch up" head.
great content! simple and knowledgable! :)
all your videos are very well explained. thanks for sharing your knowledge!
Subscribed! Amazing video, man.
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!
Really good😍 I'd surely watch a lot of your very clear and satisfying videos
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'
Parth, you're the best!
Hey Parth! Would love if you made a video on string theory.
By the way these concepts helped me a lot!!!!
Great video bro. Keep the good work going.
Many thanks. Very good and clear explanation
thanks a lot, I really liked the way you explained it in this video.
Nice clear explanation. Good job.
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.
well that was really clever thank you parth bhaiya for telling me this fun physics
Love your videos Parth
Really great video!! 👏👏👏
You have the gift of communication.
8:03 The blue and orange lamps in the back are a vibe
Very clear! I love this video.
As always...hats off Parth !
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!
Great intuitive explanation!
Thanks for that refresher!
Marvellous! Now I want more details!
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
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!
Good explanation I loved it ! New sub 👊
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