15. Introduction to Lagrange With Examples
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- Опубликовано: 3 окт 2024
- MIT 2.003SC Engineering Dynamics, Fall 2011
View the complete course: ocw.mit.edu/2-0...
Instructor: J. Kim Vandiver
License: Creative Commons BY-NC-SA
More information at ocw.mit.edu/terms
More courses at ocw.mit.edu
Love how he takes his time to write. It is underrated. When profs write, students get a chance to think, write, process what's being introduced.
I noticed that as well.
Indeed! And for students with profs whose first language is not English, it really helps until the student's ear acclimates to the prof's accent if he/she is difficult to understand.
I agree with your 100%
It’s not possible to like this comment enough.
@@markproulx1472 : Thank you for understanding the implication of writing as I've experienced it. Perhaps your experience is similar too? :-)
I'm studying mechanical engineering in Italy, our professor's introduced the Lagrange equations to us making tons of calculations and partial derivatives without even explaining the sense of what he was doing. This, however, is by far the clearest explanation I've found about this fascinating topic, I wish I would have had a professor like him teaching my courses.
exactly. I'm from Pakistan, a country that is much father than even Italy. all we did is become better calculators. never learned when these concepts will come to use or even what the equations mean, how we can use them.
@@AhmadNavidHazara i can understand bro, same is in India.... Best of luck
me too
Posso chiederti dove studi?
well now you do, you just needed to find him here on youtube!
I really appreciate the open courses from MIT. They honestly saved my life. Much more intuition instead of plain math formula are taught in the videos than in the lectures from my college.
Plz solve one paper for me I sent you
This is the most detailed explanation of Lagrange EOM I have found on the net so far! Thank you much!
really? i can't believed, he didn't even explain where "L=T-V" comes from.
Do you have more detailed course on analytic mechanics?
ruclips.net/p/PL69875B9976A7E737
@@lauraesthela6941
I personally think that lectures should be spent more on example problems instead of rigorous proofs and derivations.
The lectures are primarily supposed to be a foundation for your studies anyway, they aren't supposed to cover the material that the course books already talk about at length.
I also think that proofs make much more sense after you have already tried a few example problems - they shouldn't be this mysterious mess of symbols and definitions, they should actually mean something and become obvious to you after you have checked them a couple times, otherwise they are useless.
@@Peter_1986 WELL SAID
This is the most insightful and detailed lecture on Lagrangian Mechanics out there. Thanks J. Kim Vandiver. Your problem solving approach is so stimulating and engaging!
Professor Vandiver, thank you for an incredible lecture on the Introduction to Lagrange Equations with detailed Examples. This lecture really explained Lagrange Equations in full detail.
Lies again? Eat Drink
I had the happy opportunity to have a great math teacher already at the age of 9. And his classes were fundamental to me until the master's degree, already 27 years old. Congratulations to all the great masters of mathematics. BRAZIL BRASÍLIA
Genuine interest of a brilliant engineer for teaching. The best explanation of the Lagrange equation showing the way to solve hard problems. Now I know MIT’s reputation.
As usual institutions like MIT gets the best professors to go with there top line students, so success is almost guaranteed! My professor never came close to explaining the Lagrangian like this professor.
just wow. goes to show how different universities can be. i went to school for applied math with a concentration in stats, so i ended up having to taking a lot of physics classes because those met a lot of my pre-reqs for my degree. and not once in any of my upper division physics classes, did the prof ever mention how you need to test first to see if you can use a lagrangian. apparently they just always gave us problems where it worked, and they just left that entire part out.
Ugh I love this stuff... I should probably go to grad school for me or robotics. Feeing nostalgia big time right now. Thank you mit.
Thanks a lot for this Sir. The way you gradually explain and write makes understanding very easy.
I am a Mechanical Engineer and this is far one of the best class!
I love your site, and intend to study all of your math, chemistry, materials sciences, and physics courses. Thank you. I love the texts and homework sets!
This is great lecture if u wanna visualise and reason... And do things systematically... Greatful to the lecturer and the organisation to make it available to the world...
VAIBHAV DlXIT
Yes!! 🇺🇸MIT!!
his voice is so relaxing
this is the nicest professor ever!
I know nothing about engineering, but I know a good professor when I see one. As a professor myself, I'm always trying to get better at my craft.
We would like to inform you that the video ruclips.net/video/LIzqmOv2lHc/видео.html
has been uploaded on my channel. Please have a look.
Beautiful lecture!....and beautiful the fact that you can play it a 1.5x speed!
Thanks. 1.5x saved a lot of time
i watched it in 2x speed
I'm studying physics in Salamanca, Spain, and this video was really useful
Thank you for them!
I'm stydy in moroco you kan help me or i 'have idee pour continu ma stayding in spain
Thank you FRANCO FERRUCCI. I was becoming bored until I increased the speed. Now he's become a much better lecturer. Perhaps some other lectures would be more interesting with similar treatment !!!!!!!!!!!!!
Thank u so much for sharing this lecture with the world! Recently we reviewed this content in clases but I didn't get it quite good. Now I definitely understand it way better!!!
Thank you MIT . Best technical university in the world
"In 1766, on the recommendation of Swiss Leonhard Euler and French d'Alembert, Lagrange succeeded Euler as the director of mathematics at the Prussian Academy of Sciences in Berlin, Prussia, where he stayed for over twenty years, producing volumes of work and winning several prizes of the French Academy of Sciences. Lagrange's treatise on analytical mechanics (Mécanique analytique, 4. ed., 2 vols. Paris: Gauthier-Villars et fils, 1788-89), written in Berlin and first published in 1788, offered the most comprehensive treatment of classical mechanics since Newton and formed a basis for the development of mathematical physics in the nineteenth century.
In 1787, at age 51, he moved from Berlin to Paris and became a member of the French Academy of Sciences. He remained in France until the end of his life. He was instrumental in the decimalisation in Revolutionary France, became the first professor of analysis at the École Polytechnique upon its opening in 1794, was a founding member of the Bureau des Longitudes, and became Senator in 1799." Wikipedia
I am an engineer and am watching that for fun instead of doing my work... I am a sick man.
Great explanation. Clear and simple. Good job Prof. Vandiver
Holy crap that rod and sleeve problem is insane. I don't think my professor would ever expect my class to solve something like that.
This is MIT though so I don't know what I expected.
Evan Schurr It's lagrangian mechanics, all physics undergrads take it
It was pretty mean. I paused and tried it all. I got stuck on the kinetic energies of the objects actually lol. I was pretty off.
@@xFoKe In the Philippines, Langagian Mechanics and Hamiltonian Mechanics is thought in Master's Degree
Dane Gil Cabrales I’m in my second year of the theoretical physics bachalor and I’m taking classical mechanics with lagrange, hamilton and special relativity. The course is obligatory.
@@yerhing6406 what country are you studying in?
18:12 that is because the rotations are not conmutative. Furthermore, 2 rotations is just a subspace of Lie Algebra, se(3), of Euclidean Group, SE(3), to be holonomic you need the movement be a subalgebra of Lie Algebra, like 3 rotations (spherical) or general planar motion (x and y traslations y z rotation) or planar traslation etc.
what a perfect professor
dat combover tho
I Z dat’s stupid of you
There is two stages in engineer´s life; before and after knowing what a Lagrangian is as well as its practical applications. Vandiver is quite a good choice. Thank you!!!
At 52:00, there should also be a term for gravitational potential energy (PE) of sleeve due to rotation of the rod. The gravitational PE of the sleeve due to stretching is taken into account, but not due to angular motion of the sleeve.
Thank you MIT for sharing.
Much love
Just started this topic on my vibrations class and this is a life saver
شكرا جزيلا على هذا المحتوى الراقي 💜💜🤍
50:30.. I think the refrence of potential energy for the rod is when theta equals 90 degree, max potential energy. and zero potentional enegry when it's straight down, that's why he got confusion about the signs, so the equation is correct.
From what he said, reference position (= Equilibrium position) is when rod is vertically hanging from the pin. Once oscillation starts, the CG of the rod moves up / down. This difference is taken as change in potential energy. This is what I could make out.
His consistently slow paced talking makes using the 1.5x or 2x speed option a great alternative.
1.25 will do
2x speed on anybody is ridiculous
@@user-en5vj6vr2u there are some nptel lectures where you need 2x speed to actually understand what professor is teaching. They speak very very slow.
The conceptual descriptions here are cogent and systematic, but I think the jump from a spring pendulum to that rod/sleeve problem serves to confuse rather than make anything clearer. I understand wanting to give students a challenging problem so that others are easy by comparison, but there seems to me to be too much to keep track of in that problem-rotation, driving force etc.-to solidify the concepts by example. You lose the forest in the trees. I think a more intermediate problem might have been useful to solve to completion.
The best I've come across so far. !!
This lecture seems to be very 'In Depth". It gives a good idea of how to completely analyze a system. The derivations may be a bit too in depth for some people.. Perhaps a brief overview of the various concepts would be useful. I can see how this is a lecture that might be appropriate for students who are studying either Engineering or Science.
Step by step video solutions for civil engineering questions
So much better and more efficient than my lecturer, thanks :)
waw.....after leaving university for almost 10 years. This is so cool!!
51:15 Lo+L2/2 also needs to be multiplied by cos if we want height
thanks to your explanation the greatest insights of the greatest mathematicians are easy to understand
My physics professor at my local community college is just as good as this guy. My professor received his Ph.D. from the University of Chicago and I think is just as smart as any physics professor at MIT.
You are lucky to have a community college lecturer who seemed to be as good as one from MIT. And it is possible to have a lecturer from MIT to be worse. A place like that hires people who can obtain research funding from which a fraction of the salary comes. And the hiring process involves ONE public talk, based on which the candidate’s EQ (ability to understand the question) and IQ (cleverness of the answer to the question or the way in which the answer is delivered) are judged. Mercifully (for the students) there is a 6-12 year period of observation before a candidate receives tenure and becomes a part of the permanent staff.
Wonderful lecture and incredible instructor!
holonomic means path independent
nonholonomic path dependent
at 19:28 he said this system is NOT holonomic, but he chose 2 coordinates to describe the white mark on a ball. this has nothing to do with SYSTEM, choose 3 coordinates to describe the white mark on a ball, then the SYSTEM becomes holonomic (basically describable), when you say system i am talking about something intrinsic, only can be controlled by the system and NOT BY YOU. simply saying work done on the system by you OR work done by the system.
This professor is GREAT. I am impressed. Thanks!
Thankyou. Professor.
I don't know why I watched this at 1AM but I actually understood the lecture and didn't watch the previous ones :D
I have a retake of the exam tommorow YOU SAVED ME ❤️
step 3 @ 22:18 could be confusing as he wants you to find T AND V, not T PLUS V - a very different thing. Seems a trivial gripe I know, but this could be a real barrier to progress if reviewing notes later
At 50:10 he clearly says that 'it is the change in height' that the rod goes through - so the expression gives 'change' in potential energy.
However, that is not what we are aiming to write; as per my understanding, we are writing the 'potential energy' of the system, not the change in it.
If any of you has an explanation, please share with me.
Anurag Anad: Potential energy can be referred relative to any chosen origin; it is not an absolute quantity as you seem to think. Only changes in potential energy have any physical significance.
This explanation is wonderful.
profesor sorry for words, you are awesome
29:15 when the camera gives up
😂
According to J. L Meriam (Dynamics), which stated the Lagrange Equation based on L =T-V is valid for conservative forces only. For non conservative forces the Lagranger Equation shall modified for T (kinetic energy) not L. Please advise.
Spend three days to study lagrangian mechanics, this is really amazing
Very clear introduction. Thank you.
If you don't have time just watch the algorithm (20:22) and the example right after.
Thank you so much for this video! i´m an student for the upm (universidad politécnica de madrid), in spain. If there are any videos about hit transfer i would apreciate it.
Try going through "Elementary Applied Partial Differential Equations With Fourier Series And Boundary Value Problems" by Richard Haberman for any help in heat transfer theory. I found this book extremely helpful with explanations :)
Cool!! I will read it, thanks.
For heat transfer lectures you can use www(dot)nptel(dol)iitm(dot)ac(dot)in
Very clear explanation thank you, but instead of practical problems I am more interested on HOW Lagrange got into this
dr leonard susskind from standford also has a series of videos on this very topic in his classical mechanics course. if you go to the part in the course where he first introduces hamilton, he briefly comments on what lagrange was doing and why he did it, and then jokes about how he has no clue what hamilton was doing or why he was doing it, or what he was smoking at the time.
I really like this lecturer
can a 16 year old enjoy this class for the love of physics.........YES
I'll try too
You know, it'll be a very big deal if you understand all of this. This requires stuff like calculus of variations!
The lecture is very useful ... but I need the previous lectures on this subject. please , provide me with possible linkage..... also explanation is wonderful
The playlist is (ruclips.net/p/PLUl4u3cNGP62esZEwffjMAsEMW_YArxYC). For more info and course materials (assignments with solutions, exams, lecture notes) see the course on MIT OpenCourseWare: ocw.mit.edu/2-003SCF11
Thank you very much
***** you actually responded? wow
MIT OpenCourseWare what book he suggest?
@@sepehrjafari7847 he suggests yo read some Stellar on firt 40 second off lecture, do You hace more detail on ir? Thanks
Brilliant explanation
This is irrational due to mathematical points, which have no mathematical foundation for observation. No one has ever seen a point; hence no one ever will. Only "pictures of points" can be seen, which is due to the first false assumption of Euclid. Gause spent a lifetime looking for inconsistency which is a point that, as defined, is not observable. A point is pointless:?)
50:37 L0 is not the "unstretched spring length". It's the _equilibrium_ spring length (up to half L2) when M2 is hanging from it.
1:11:00 And he reaches the same conclusion in his static analysis at the end (without ever correcting his definition.)
+tokamak you are just another miserable kid that feels better when ''correcting'' academics...
George Anagnwstopoulos You're an even more miserable kid who tries to put down others correcting others (and fails at it because rigor and sharing are both at the heart of modern science as well as personal growth) and cannot even feel better afterwards!.. Instead of wasting your time on the internet by watching these lectures when you clearly lack the necessary drive, passion, and conscientiousness to make any use of them in your personal life and trying to bring down others to your own lowly, cynical level, go learn how to cook pizza or clean toilets and make some money, Greek boy.
+tokamak
Reading your comment made me slightly vomit.. you seem so full of yourself.
Philandros Good! Now you're less full of yourself then.
Unstretched spring length is synonymous with Equilibrium spring length, it's just less formal. A spring with a mass on it, in equilibrium, is unstretched, any peturbations or imbalances and it stretches and contracts.
Listen for the smoke monster from Lost at 25:21
metal4bld lol
metal4bld c
Nice.
Fantastic teacher!
Awesome..."I am an engineer and I like practical"... lol exactly what i needed
Sheldon Cooper won't agree :p
와우! 칠판도 9개다 전동으로 움직이고 필요하면 내려서 보고, 따라 적고...... 이 시스템 좋다. 강의를 듣는 학생이 많은지 글자도 크게 쓰고,,,,,, MIT 가보지 않았으니.......... 지금은 바뀌었겠지
Motion equation of spring mass system sounds not right because the term mg doesn’t have to be there . @26
I could be mistaken, but I believe that except for the potential energy of the spring, the potential energy terms in the second problem are incorrect. First, the dependence on theta for the rod should be 1+sin not 1-cos. 1-cos would give 0 potential energy at theta=0, which is not correct (should be simply MgL1/2 - the center of mass is L1/2 above ground (the ground being the distal end of the rod when theta is 0, i.e., L1 from the top of the system - one would imagine - though this was never specified), and mgL/2 at pi/2, when the rod is horizontal, which is also wrong, and should be simply MgL (the entire rod is L1 above ground). using 1+sin gives the correct result (MgL/2 at 0 and MgL at pi/2). The only way to imagine 1-cos as correct would be to assume ground is at L/2, and perhaps that is what is being done here, but I can't see why that would be. Maybe it would work out to the same thing in the end, since my way and the way presented here only differ then in a translation of the zero point for gravity, but I doubt it.
For the sleeve, the problem is more convoluted, but also incorrect as presented. If theta is zero, then according to the presentation V for the sleeve = Mg*(Lo + L2/2 - x1), and if the spring is unstretched (x1 = Lo + L2/2), V=Mg(Lo+L2/2-Lo-L2/2) = 0. This would mean that the ground is defined as being at -(Lo+L2/2) from the top of the system, which is not consistent with either a logical placement of ground (L1 from top or the top of the system itself), or the position to which it appears to have been set if the potential presented for the rod was correct (L1/2 from top). Also, if theta is pi/2, the expression presented reduces to Mg(Lo + L2/2), implying that Lo+L2/2 = L1, which is not likely and certainly not specified. For a correct expression, assuming the ground is at L1 from the top, then when theta is 0, V for the sleeve is 0 when x1 = L1, and increases as x1 becomes smaller than L1, so there's an L1-x1 in there, and if theta is 0, then V=Mg(L1-x1), and this component of the potential for the sleeve drops to 0 at theta=pi/2, but the height of its center of mass increases to L1, so, at pi/2 V=Mg(L1), suggesting an L1sin term. long story short, what I get is Mg(L1-x1)cost+MgL1sint, which at least is easy to see for theta = 0 and pi/2.
Hi Glen:
the movement and the potential energy of the sleeve has nothing to do with the length L1 of the rod. The most easy places to define the ground are the places where x1 = Lo + (L2/2) (no spring strength) and theta = 0 (no gravitational "strength"). Then, because gravitational potential energies are all related with heights from the ground, and theta is measured from the vertical, all the heights in this problem must be expressed in terms of cosine functions of that theta.
Great Sir. Adding subtitles would have made it easier to understand.🙏
This video does have subtitles. Double check your settings.
He suggest to read on Stellar, could yo tuve more detail which book is?
¿A esto le llama introducción?
Comenzó con ejemplos no muy triviales para alguien que no conoce del tema, menos mal existen otros canales que lo explican mejor; ahora recién podré continuar con el siguiente vídeo, pero sin duda esto no puede llamarse introducción 77
51:56 Why don't have to take account of the length change due to the mass of sleeve?
KX=M2g
X=M2g/K
M2g(L0+L2/2+M2g/K) - M2gX1cos (theta)
I thought the reference should be the final possible state? equilibrium state?
Couldn't stop thinking about the song from ZZ Top
The best lecture....
I find this really well explained. However, it bothers me to death that he doesn't show where these equations come from and why they're mathematically and physically correct. Students just take it as correct and apply a formula to solve the problems
Hope one day I can be a part of the actual class
excellent teaching
Feels a little like a late Thursday afternoon lecture...
At the beginning he refers to better notes "in the cellar" it sounds like. Does anybody know what an where this is? I've got the OCW course materials, but he seemed to me to say there was something better on the net someplace...
Thanks,
-dlj.
He mentions Stellar which is MIT's course management system. It is a platform for learning, course management and collaboration, serving the MIT community. Most of the Stellar site is not accessible to the public.
Where does the professor takes this force [F(t)=F_0 cos (omega t)] from? (video: 1:06:17) Who does exert (apply) it? Is it a new datum of the problem? Thank you, nice video.
When phi 1 is fixed phi 2 can still move ....same way when x1 is fixed x2 can still move.... so why not independent..???
I mean to say... phi 1 and phi 2 are fixed one by one but x1 and x2 were fixed simultaneously... why so??
it would have been independent if you fixed x1,y1,x2 and after that y2 would still be free.Here he just fixes x1 and x2 and the system is not free anymore so it isnt independent
Excellent lecture.However, I'm in trouble understanding Hamilton's principle and Kane/Jourdain 's Principle. Is there any lecture including these content?
Great course
thank you professor
At the beginning he mentioned to "visit STELLAR website " for short notes ,,! Please provide the link here !
At 57:00 I think he forgets to write the Kinetic Energy related from the bar, wich would be M1 * L² * Theta-dot²
how do you do that ???
@56.00 why isn't the calculation for kinetic energy for sleeve done wrt A since it is rotating about A and not rotating about G
There are two components to it's kinetic energy. The first is due to its translational energy, and the second is due to the fact that it's rotating about it's center of mass (while simultaneously translating). So you are correct that A is the axis of rotation for it's entire motion, but to simplify the calculations, the translational motion is treated separately from the rotational motion. This is because Izz wrt A of the sleeve would be time dependent in the inertial frame, but Izz wrt G is not. It's essentially using superposition to isolate a fixed point of rotation.
What about the dry friction?
I done so many lagrange problems, but i can’t solve lagrange question of my final exam, my professor creativity is so great that the whole class cannot even solve the question, jesus even rotor dynamics or 3D rigid body are much easier than lagrange problems
강의 속도가 느린 것도 아주 좋아요. 외국인을 배려한 강의인듯 하네요.
super lecture.
I followed this through and worked the examples ... in the final collection of d(δΤ/δx)/dt terms there is a Coriolis component 2*m2*x*(dx/dt)*(dθ/dt) to which he refers (at 1:14:46). I cannot see where this was derived in the KE term at 58:00 nor thereafter. Can someone help me by telling me what I am missing?
01:30 "So I defined what's called the Lagrangian last time." - which is WHEN? (what video?) I watched a couple of videos preceding this one and I coulndn't find anything about this.
This is a great video. Great lecture on Lagrangian Mechanics.
A fine supplement to a study of LandauLifschitz's course.
We would like to inform you that the
video ruclips.net/video/LIzqmOv2lHc/видео.html
has been uploaded on my channel.
Please have a look.
thank you for the explanation. i was following lectures from the start. I found that there may be lecture missing about how to calculate KE and PE for translating and rotating frames. you keep on referring that lecture like " as we calculated in previous lecture etc etc". will you please add that lecture or give me some reference which is more alike to your previous lecture to successfully calculate KE and PE for any system.. please help.
Great Work!!
Where does the coreolis term (2M2*X1..) comes from , in the therm 1 of lagrange equation for the theta component?
I just don't see it in the derivatives...