Physics 68 Lagrangian Mechanics (2 of 25) Why Does the Lagrangian Equation Work?
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- Опубликовано: 12 сен 2024
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In this video I will explain why does the Lagrangian equation work in a simple free-fall example.
Next video in this series can be seen at:
• Physics 68 Lagrangian ...
I agree with Brendan. This does not explain why it works. It was only re-doing Lecture 1 and showing that for the case of PE = mgy and no other factors it works out. But I would ask suppose you have a different potential. Well, try the next. Also agree that it is great even if just a re-hash of first lecture. The beauty is still there but still unexplained.
The way I would explain it is in the fact that the equation takes into consideration the Law of Conservation of Energy, and thus can be used to deduce and formulate the Laws of Newton. If you start with a simple conservation equation such as K + P = 0 (The Law of Conservation of Energy), then you can manipulate it to produce a system of equations that all yield the motion of the object to it's net force. With this, all forms of energy conversion can be tied to a system of forces (or a single force in this case), and with this connection the Second Law of Motion can be deduced (F = ma). So to me, the Lagrange Equation was another way to convey the Conservation of Energy Law in that it connected the Mechanical Energy of the system to the net force of the system. I generally (and most people should) saw this the moment I saw the equation was equal to 0. When something is 0, it usually shows something in the universe is conserved and/or harmonic. :)
I believe what Lagrangian equation means ultimately is that the change in L due to change in KE equal to the change in L due to change in PE, but expressed mathematically
The conservation of energy (T+V here) a constant is a consequence of the Beltrami Identity (a simplified case of Euler-Lagrange) being applied to L= T-V.
i.e. L=T-V is equivalent to the conservation of energy by Euler-Lagrange principle of Least Action, so it is the Principle of Least Action that explains all this.
The explanation given by Michel in this video F=ma would work to solve this particular problem, or F= grad(V) more generally, but using L=T-V as our "Functional" in the Euler-Lagrange is a lot easier as it is dealing with scalars, and also allows us to choose any suitable parametric generalized coordinates.
Michel uses x as his generalized coordinate and t as his parameter, x(t) (its a 1 dimensional problem after all).
I'm not sure if Michel has made a follow up video about Euler-Lagrange Principle yet since this video was made some time ago?
@@APresidentsRansom The thing is: the Euler-Lagrange equation doesn't even assume conservation laws to get Newton's Second Law, which is what's puzzling me the most. All that the equation assumes is that there's some quantity E, which is defined as (1/2)mv^2 minus some other quantity V which is not even yet defined, and that we want to minimise this E. Extraordinarily, what we get from this information is both conservation of energy and Newton's Second Law. I am still at a loss as to what all of this means.
Well it showed you how it relates to Newton's second law.
THANK YOU SO SO MUCH! This is all I needed right now. I have struggled a lot to understand the Lagrage method in my physics class. This, simple, plain explanation was all I needed. Don't worry about the negative comments, you helped two struggling physics students in the corona pandemic!
Glad it helped!
After every lecture on the university i go home and look if you have a video about the subject, in 15min you can explain things better than my professor in 2 hours! Thanks Mr. van Biezen
Thank you! I am a big fan of your teaching style. Perfect pace and presentation.
Many thanks!
The first boxed equation is the Euler-Lagrange equation as derived from the calculus of variations.
Aha!
Anthony Shaw
You have shown me that maths is a matter of understanding not speed. I now feel more confident and for that I thank you so much
5 years later, this video is still heroic
I think it's important to notice that the Lagrangian is a function of 3 variables; position, velocity and time. Otherwise the partial derivatives would work out differently!
Thank you. Without your explanation, I could not get into sleep.
Yes! It was a bit of a eureka when I noticed that we are deriving with respect to the variables, rather than the time. It sorted out a lot of confusion for me.
I feel like this didn't explain anything, unless I blanked out on some key point or something (I know it's the same as F=ma). (Also, no offense intended - I really love watching your videos, and they make me excited to learn physics after college)
I agree. This didn't explain why it works only that it works. As I understand it, it works because it is the same equation as F= ma + d/dx (V) but by re-arranging you have it in a form that is supposed to be better but I would want an example to see that.
Brendan F
Since I just read it, I can tell you Paul Nahin's book, When Least is Best, goes into the historical beginnings (brachistochrone problem) and into the derivation of the boxed equation.
agree x100 , any resource to complement the question ?
Not sure if you do it later in the series, but you should show the derivation of this equation in the first few pages of Landau & Lifshitz (using calculus of variations). It's difficult to wrap one's head around the computations, so if you could guide us through them, that would be awesome!
My plan was to do that later.
Sir, your explanation is extraordinary! I'm relaxed with it, and no worries about the physics' courses I am taking. Thank you for these efforts.
Glad you found our videos. All the best on your studies of physics. 🙂
This man just saved my life
I am sure that is an exageration, but we are glad our videos were helpful. 🙂
SUPPORT MICHEL VAN BIEZEN ON PATREON!: www.patreon.com/user?u=3236071
This is probably a fake DONT DONATE.
I believe what Lagrangian equation means ultimately is that the change in L due to change in KE equal to the change in L due to change in PE, but expressed mathematically
That was a great interpretation Muhammad Nada. I am 62 years old and from North of Cochin, India, Jose Francis
@@josefrancis7126 nobody cares
You may check a structural dynamic book in which it shows how to derive Lagrange equation about generalized coordinates.
Hello. Thank you for saving my grades. Ly my dude
hqaving my ynamics exam in three hours. gonna see how many videos i can watch in that time. thank you so much for doing this!!
You are welcome. All the best on your exam.
Way clearer than I ever learned in college. Thanks for the great review!
Thank you. Glad you found our videos! 🙂
Hi Michel, what level of differential equations would one need to read John Taylor’s classical mechanics? I am nearly finished with first year intro Newtonian mechanics and I’m very much looking forward to Lagrangian mechanics.
I am only a few chapters in to a differential equations book, but I finish the course in late September. It covers all the same things you cover in your differential equations playlist. Would this be enough?
Thank you in advance.
Yes, you would need some rudimentary understanding of differential equations. The hardest part is to determine the KE and PE equations for more complicated systems. We have shown a number of examples and will add more as we have time.
dear Sir, pls accept my wholehearted salutation to ur excellent job of diffusing most worthy knowledge, and moreover with such extreme accuracy and clarity, ( and in such a soft humble manner ) , leaves me speechless, u deserve the best, and if i was God, i would have made u sit in the best place in the heaven..all my best regards with u..
Rumman,Thank you for the comment. I feel very fortunate and blessed to be able to reach students like yourself around the world.
You haven't really explained why the Euler- Lagrange equation works
high class basic touch..really sir you r boss..if I ever go to America I will be lucky to meet with you..thanks for the videos..
+MOHAMMAD ALI Tanvir You are welcome. Glad to be of help.
Nice. Just refreshing my memory after 30 years -- the only thing is maybe leave the potential as an arbitrary function and then show that the gradient of it is a force.
What you have just stated is correct: force IS the gradient of the potential - this is a definition in Physics, a consequence of which is Newtons Law F=ma.
More generally a field is gradient of a potential if the field is a conserved one of course.
Thank you Michel! That was awesome.
Should have provided a simple, intuitive derivation of the Euler-Lagrange equation. Merely showing that it reduces to F=ma makes it no more transparent.
The videos are meant to be watched in sequence, which should offer a better understanding.
Thanks sir.....its very great way to understand the Lagrangian mechanis Thank you a lots
Good explanation of the Lagrangian Equation, it made it better to understand :). Thank you for the video!. Also I can imagine how it must have been in the past to go through all that mathematics to discover that the Lagrangian is also equal to F=ma, all that work lol.
I never thought I will understand it that easily. Thank you very much sir.
Happy to help
Thank you for the easy explanation with a really good example 😀
It helps to start with simple examples. 🙂
Sir plz guide how to prepare for ipho and can you plz start a series on it ?
F=ma!! this is awesome! I'm delighted rediscovering physics in such great way! it's been 25 years ago since I saw it last! I feel I'll go through all the vids with plenty of likes :)
great explanation!
very nice! professor explained very easy about lagrangian mechanics that I could understand although I am high school student
Your are teaching like magic real scientist thank you sir
Michel van Biezen. Is there anywhere in these excellent lectures where the Euler / Lagrange equation is derived? Also why at 1.58 is 1/2 m first derivative of x times x held to be constant when differentiating partially? Thanks in advance for any advice you may care to offer.
We still have to make those videos.......
Thank you! I understand much better now. Thank you for dedicating your time on making these videos for us!
Thank You Very Much Sir
You explain it so simple. There is really no excuse for complicating it like my college professors do. I wish you were my professor!
This "explannation" might be easy, but it is nowhere near general. Didn't explain anything.
Wow, you made it look so simple so quickly...
Michael you are a boss! Thank you
I dont speak english very well I am Mexican 70 years old..Michel is skillful teacher
thank you this is really helpful
This is true only for free-fall motion, correct? Otherwise, we wouldn't be able to say that the total force F is equal to -mg
+kilgour22 Correct. We'll show different examples with different scenarios to show how we work with the Lagrangian in those cases.
Aren't x• and x related throughout the journey of the particle? Why is the derivative of x• wrt x taken as 0?
Because we take the partial derivative and the way the theory of Lagrangian mechanics is devised it requires us to look at those as two separate variables.
@@MichelvanBiezen Now although X and X dot depend on each other, and for every X there is a corresponding X dot throughout the journey of a body, the laws of physics don't require the X dot as a function of X (or X as a function of X dot) to be same for all the journies. Depending on the initial conditions, the mapping could carry from situation to situation. Is this what those lagrangian equations are reflecting?
can't understand why the acceleration is negative????!!!!Acceleration would be downwards doesn't it means its in the same direction as the "g" acceleration due to gravity.....then why is "g" +ve and a -ve????!!!please explain!!!please really need these equations to solve questions with more ease!!!
As indicated in the video a = - 9.8 m/sec^2 because the object is acted upon by gravity. The constant g is a vector quantity and is directed downward as a vector, but the magnitude of g is just a positive 9.8 m/sec^2 (since the magnitude of a vector cannot be negative). Thus it depends how you write the equation. Define you variables and stay consistent throughout the problem.
when the object go down can we say acceleration positive!))
Thank you Sir!
You are welcome.
U r simply awesome sir 🖤
Thank you so much 😀
Using this strategy, don't you loose the vectorial nature of the equation of motion? Since your starting with energy, the equation of motion you end up with (expressed by force) is a scalar function is it not?
What are you talking about?
Oops. Sorry, this comment was supposed to be a response to another video question. (I can see why you were so surprised).
That said, let's get back to your question. Yes, indeed, when you work with the Lagrangian, you work in terms of scalar quantities. However, since potential energy can be related to a direction, as is the case with PE due to gravity, you can relate the sign of the differential equation to the F = ma equation and derive the direction from that.
Moreover, you apply the Euler-Lagrange equation to the system's Lagrangian in generalized coordinates in order to obtain the equation of motion for each object in the system. You write these generalized coordinates in terms of a chosen coordinate representation. Once you've conveniently solved the problem using this process, you can express the solutions to the equations of motion in terms of this representation using a coordinate transformation. That's what gives you direction: you wind up with vector components at the end of the day. You could in theory do that transformation before solving the EoMs, but then you have to solve one in each direction and you wind up with way more equations to solve. Using generalized coordinates allows you to solve the least number of EoMs possible to fully determine the motion of a given system.
This is perhaps more clear In more elaborate examples. Here, there is only one degree of freedom (movement in the vertical direction), and he doesn't write out the transformation of that single generalized coordinate to the cartesian coordinates he's using, since it happens to be trivially along a single axis (so "x" here equals y). On that note, I also much prefer using q's to represent generalized coordinates, since then they aren't confused with the x dimension in cartesian coordinates.
d(∂T/∂v)/dt - ∂V/∂r=0
d[(∇v)T]/dt - ∇V =0
I changed the partial derivative of T with respect to v, for the velocity gradient of the T, and the partial derivative of V with respect to x for the gradient of the V. If you notice, by definition the gradient have the vectorial nature, so if you solve this last step you have the second law of motion in vectorial form.
(∇v)T = ∂[m(Vx)²/2]/∂Vx*î + ∂[m(Vy)²/2]/∂Vy*j + ∂[m(Vz)²/2]/∂Vz*k = m(Vx)*i + m(Vy)*j + m(Vz)*k
So
d(∂T/∂v)/dt = m(Ax)*i + m(Ay)*j + m(Az)*k
And, the definition of concervative force is F = -∇V
Fx=[∂V/∂x]*i ; Fy= [∂V/∂y]*j; Fz=[∂V/∂z]*k
We have:
Fx=mAx ; Fy=mAy ; Fz=mAz
how the mass times gravitational acelaration directly equal to yhe positive force since the multipy is negative
Since the Lagrangian operates in terms of the KE and PE, it loses track of the direction (which is part of the Newtonian method). KE and PE are scalars, while F is a vector. If we only want the magnitude, direction can be ignored. (magnitudes of vectors cannot be negative)
I am from Iraq, I love you so much
Thank you. Glad you found our videos! Welcome to the channel! 🙂
Thank you very much
Thanks a lot sir 👍👌👌👌👌
Most welcome
what's the link to the series?
You can find all the physics videos easily from the home page of this channel. (Type in "Michel van Biezen RUclips" and you will find it). It is chapter 68 of the physics playlists.
Nicely explain
Thanks and welcome
Please can I get the derivation of the Lagrange equation of motion
The closest we have to that at the moment is in chapter 70: PHYSICS 70 PRINCIPLE OF LEAST ACTION When we have time we would like to add the Lagrangian derivation.
Should be written as steps; to be consecutively completed.
See the rest of the videos in the playlist.
This is a gold mine.
Can we extend the same to N body case where we'll land into say N equations?
Why is d/dt of partial derivative of L with respect to xdot equal to partial derivative of L with respect to x????
How do we get this result?????
Note that it is another representation of F = ma. If you work through the math as shown in the video the equation reduces to F = ma so we know it is correct.
Michel van Biezen Sir,
In the video u start off by saying that the differential equation is general and can be applied to any system. Then u apply it on free falling body and the equation reduces to F=ma (Newton's laws)...
My question was how do we get that differential equation for any general case???
There are additional videos showing examples in the playlist of how it can be applied to other cases. We still have to make videos with the proofs of how these are derived in the future.
Abhijit R : bro. you tell that why we take L=K.E - P.E. ?????
Wait F=ma and W=F and W=gm
However the equation shows us that -gm=ma
-mg=-W
-W=F
Can someone explain to me?
Weight is also known as the gravitational force.
So the equation should be W=F not -W=F
I actually cant believe this is Physics, my Economics Prof is really overdoing it
What does your economics professor do that is different?
thanks
How could you easily skip the concepts of phase space and configuration space, not mentioning minimizing the functional, and try to explain how Lagrangian works... This is just a fancier way to do high-school mechanics, it's not Lagrangian. For Lagrangian, you have to talk about the minimizing of action, by Euler-Lagrangian thm.
If not this can you please suggest some other lectures which are more relevant
@@harshitaharshloomba7701classical mechanics by Edward disloge
Very clear explanation - thanks
I am have a naive question : Is L = KE-PE a general equation or will the definition of lagrangian change with respect to the system we are analysing?
That definition should not change. However there are a number of other ways in which the Lagrangian can be defined. When I have some more time, I'll continue with the analysis of the Lagrangian and Hamiltonian.
Thank you for the reply.
@@MichelvanBiezen Sir, in the context of the question raised I would like to ask if the equation for Lagrangian will remain same in case of quantum mechanics and nuclear physics, more specifically the matter-energy interchange like in nuclear reactions or will it be needed to be modified as per condition?
now this is making sense
@Michel van Biezen
everything is ok ..but I have one doubt, which is trivial for the experts but the person who starts learning this it will create a big problem which has already happened with me. In the example, you are denoting the height as "Y-AXIS" but you are solving the equation as "X" coordinate. But all the motion is happening along the y-axis only.
The x is used as a general coordinate.
Thank you sir
You're welcome!
why you take y=x? i don't understand...
You can use any variable. (It is called a general variable and it is often done with Lagrangian Mechanics)
Professor you are showing up all those other profs! It's like they actually want to make people feel dumb.
oddly enough, this raises questions about the sign...
No it doesn't because v=dx/dt is +ve up if x is +ve up. The fact that he showed v pointing downwards on his diagram is irrelevant to the analysis, i.e. a = -g.
Beautiful. :-)
This man is a Godsend, and that's coming from an avowed atheist
thank u for recognizing my comment from ur such a vast place in experimental academics..Sir, i have created an algorithm that can simplify and assert control on anything, of any range..i havent yetexposed it to anyone ( by the way, all my findings r experimentally backed) , and would consider a great achievement if u personally back me up..we can create the greatest monopoly ever..pls bless me with ur Guidance..
shut up crackpot
who the wrecked are those seven peoples ,i wonder!!
Totally lost time to watch that because the question from the title is answered at 0%.
why a=-g teacher?
Since we defined g as + 9.8 m/sec^2 and we a is in a downward direction we must set a = -g
The worst video ever. The title is a big lie!!!
I still don't know "why" though....
It sounds like we'll havev to make another video, trying a different approach.
@@MichelvanBiezen please do.
@@MichelvanBiezen i think when the video says “how” people are expecting a video that shows where the Langrangian comes from and how the Euler-lagrangian equation was derived.
You haven't really explained why the Euler- Lagrange equation works
Sreehari E which book is best for classical mechanic and for EMT
Thank you so much sir