Lagrangian and Hamiltonian Mechanics in Under 20 Minutes: Physics Mini Lesson

There's a lot more to physics than F = ma! In this physics mini lesson, I'll introduce you to the Lagrangian and Hamiltonian formulations of mechanics. Get the notes for free here: courses.physicswithelliot.com...
Take the full course on Lagrangian mechanics: courses.physicswithelliot.com...
When you take your first physics class, you learn all about F = ma---i.e. Isaac Newton's approach to classical mechanics. But there's a lot more to mechanics than F = ma, and modern physicists actually more commonly use two other formulations of mechanics: the Lagrangian and Hamiltonian formalisms. In this video, I'll teach you the basics of both. They're not only powerful approaches to classical mechanics, they're also fundamental to the way we think about quantum mechanics!
Get the PROBLEM SHEET, the SOLUTIONS, and the ANIMATIONS: www.physicswithelliot.com/lag...
More to learn:
- All About Pendulums: • Everything You Need To...
- The Principle of Least Action: • Explaining the Princip...
- The Relativistic Action: • The Special Relativist...
- Noether's Theorem: • Symmetries & Conservat...
- Poisson Brackets: • Before You Start On Qu...
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About physics mini lessons:
In these intermediate-level physics lessons, I'll try to give you a self-contained introduction to some fascinating physics topics. If you're just getting started on your physics journey, you might not understand every single detail in every video---that's totally fine! What I'm really hoping is that you'll be inspired to go off and keep learning more on your own.
About me:
I’m Dr. Elliot Schneider. I love physics, and I want to help others learn (and learn to love) physics, too. Whether you’re a beginner just starting out with your physics studies, a more advanced student, or a lifelong learner, I hope you’ll find resources here that enable you to deepen your understanding of the laws of nature. For more cool physics stuff, visit me at www.physicswithelliot.com.

[@iyziejane]

If the Lagrangian and Hamiltonian formulations look pretty similar, to the point of almost being different notations, this is because Hamilton invented the term "Lagrangian" and codified Lagrangian mechanics as we know it, and it was Hamilton's obsession with notation that led him to make the equations look as symmetrical as possible with the P's and Q's, which paid off 100 years later with quantum mechanics

[@architakumar2579]

So basically autism good

[@Simon_Jakle__almost_real_name]

I independently learned or realized that Einsteinian physics is diverted from Newtonian, and Newtonian can be seen in relation to the physics of Gottfried Wilhelm Leibniz (Leibnizian physics?) due to the calculus controversy both men had. Leibniz' material as a variety of Newtons from an "english mind view" whilst germans would have Leibniz as a physics "block" in a "german mind view" or mindset . So am i putting some spot on an alternative genus of physics view, on another branch in some way? Newton-Leibnizian, Lagrangian-Hamiltonian and Einsteinian physics a the three types or groups (so far?)?

[@iyziejane]

@@Simon_Jakle__almost_real_name I'm familiar with Leibniz as a great mathematician and philosopher, though I don't know his involvement in the development of mechanics, I will read to learn more about that. Certainly Hamilton and Lagrange built on the work of Euler and the Bernoulli's as well as Newton, so I agree that the development of mechanics was a fully cross-national effort.

[@Simon_Jakle__almost_real_name]

@@iyziejane I guess i am not a very integer physics mind, because the world of knowledge (and ist effects possibilities) is so vast and the changes beneath humanity happen to swift and kinda-feel absorbing too often, but i went through some rather german based physics history and my recocnition would be: Distinguishing physics i would see a seven level pyramid, beginning in the antique then around the 16th century Kopernikus, Galilei and Kepler to be followed by the Lagrangian-Hamiltonian physics with Kelvin and Maxwell as a next level until physics put foot with Einsteinian-Planck-ian physics (with some Conrad Röntgen) plus some ingenious Material from/by Niels Bohr, Enrico Fermi, John Wheeler and Hawking. And then entanglement with chemistry. Until the mostly too demanding algorithm of Peter Shor. But as often i cant intensivate such a list if i would try to explain the view in my mind, i rather try to spot and count "the genuses of trees" in/near the world of minds (now and then). Furthermore, Carl Friedrich Gauß (Gauss) must have been an astonishing person, not just/only about physics.

[@n0nenone]

@@Simon_Jakle__almost_real_name sophisticated Englishmen be like

[@sanori-cs]

1:49 Newtonian formulation 5:44 Lagrangian formulation (L = K - U) 10:59 Hamiltonian formulation (H = K + U)

[@ronissilva9570]

I would say: Newtonian formulation (Σf=mä) Lagrangian formulation (L = K - U) Hamiltonian formulation (H = K + U)

[@aug3842]

@@ronissilva9570ä would snap! i think ur thinking of ẍ lol

[@ShanBojack]

​@@aug3842exactly

[@user-cr7qf3ik9g]

Thank you

[@davidgustavsson4000]

I wish I had learned this before quantum mechanics. We essentially had a half semester course racing from "what is an operator" through "what's a Hilbert space" to "this is the Schrödinger equation, good luck!". It hasn't even occurred to me to try using Hamiltonian mechanics in classical physics.

[@reckerlang2163]

QM be like: Wave functions live in Hilbert space. What is a wave function? IDK This is Schrödinger eq., solve it More TISE in 1D square well and SHO There exist some operators...collapse of wave function "Bra" and "ket", I can't "c" Some random n, l, and m stuff Here is spin, which is a kind of angular momentum, except it has nothing to do with movement Every word professor said makes sense, but after a lecture everyone is more confused than ever Prof: think QM is bad? Get ready for E&M! Me: deliberately looking for a way to switch major despite being almost done with undergrad Also me: dead inside😭

[@celsogoncalves7348]

@@reckerlang2163 ​ @Recker Lang These concepts aren't really as tough as it seems. If you're familiar with classical physics, specially electromagnetism, you can assimilate them very easily with good texts. Quantum Mechanics by mcintyre made QM concepts natural to me, showing the cradle experiments and how they led to the current understanding of those phenomena.

[@reckerlang2163]

@@celsogoncalves7348 Haha thanks for the advice. I found Griffith’s “Intro to QM” kinda good too tbh. I am definitely not quitting now cuz I really like physics. Cheers my fellow physicists 🥂

[@thomasrhodes2178]

We were taught Hamiltonian Mechanics in Classical Mechanics to lead us into QM and Schrodingers Equation more than its use in CM. Schrodingers Equation seemed natural this way.

[@reckerlang2163]

@@themongoman Very valid point! Even with Griffith, we see a lotta stuff where we have to skip due to “lack of knowledge of mathematical methods”, thus no actual “solving” the problem. Seeing QM in undergrad is both exciting and terrifying b/c like you said we are not ready to see this kinda stuff even after taking modern physics, ordinary Diff. Eq, and linear algebra. Math is everything in QM, and I remember there was once a friend of mine who is working toward his master degree in theoretical physics tried to explain me outer product and spin using group theory, yet I know nothing about it :( (my math major roommate just learned group theory this year lol) I suppose this is also why there is only 1-2 “real” QM course in undergrad. Thank you for your advices! Physics 4 Life! (OMG I have never received reply this long on KZbin, thank you so so much for typing all this up to help a physics newbie out, much appreciated!)

[@mintakan003]

Future topic suggestion. Noether's theorem. Symmetry. Why is this so important for physics and math?

[@PhysicswithElliot]

Thinking of doing Noether next!

[@thesuperkat943]

@@PhysicswithElliot you could say it’s the topic for, a-Noether video

[@PhysicswithElliot]

@@thesuperkat943 *clap clap clap*

[@jay-5061]

@@thesuperkat943 bruh

[@daniellockhart8594]

Yes!!

[@douglasstrother6584]

As a Physics Freshman, I recall reading the terms "Lagrangian and Hamiltonian Mechanics" in the course description for the Upper Division Classical Mechanics couse and thinking "What does that even mean?". I figured that I'll learn that when I get there. I got there about 40 years ago!

[@silverspin]

Inspires me as an undergrad

[@douglasstrother6584]

@@silverspin Stick with it! Learn how to draw pictures and visualize all of the crazy Physics Stuff; it's essential for building intuition. Be open-minded about finding your knack: you may find that you have an affinity and talent for something you haven't even tried yet.

[@Bayners123]

What a clear summary, with well thought out supporting materials. You cut to the essence but leave pointers for people to find the details. Great work!

[@PhysicswithElliot]

Thank you!

[@lelomambueliane4915]

Can it be denied that this guy solves the most difficult problems? kzbin.info/www/bejne/ppzaamWVf9WpZ6c

[@NovaWarrior77]

Absolutely awesome. I finally found somewhere that got past the H=KE+PE of Hamiltonian mechanics AND actually explained the point. Thank you.

[@PhysicswithElliot]

Glad it helped!

[@PunmasterSTP]

This really blew my mind, and once again I'm so glad that educational material exists on KZbin. Thank you for spreading your knowledge; it was mechanawesome! 👍

[@S1nwar]

the Lagrangian was the most beautiful thing when meeting it in the early courses of studying physics. the way you can just throw away all the complicated geometric/vektor assesments you have in newtons method and just use the energies is so efficient

[@givemeyourfish]

I went to graduate school for engineering and that was the best explanation of the Lagrangian/Hamiltonian I have ever listened to.

[@dtcarrick]

As a physics teacher I can safely say this is amazing! Succinct and encouraging for a student. Well Done.

[@PhysicswithElliot]

Thanks Tom!

[@rdatta]

Very well done! Brilliantly conceived and the use of a consistent scenario makes for a really instructive study.

[@JeffSchneiderMusic]

Me thinks this is going to be a great KZbin channel!

[@alvarol.martinez5230]

damn didn't expect you of all youtubers to comment on a video like this!

[@chriscopeman8820]

@@alvarol.martinez5230 Amanda Chaudary from Cat Synth Tv did a video on "square root of pi" for pie day. Multi-dimensional people have multi-dimensional interests.

[@fernandosantosviana7971]

lagrangian and hamilton are just talking about energy wich comes from newton phisics, no big deal

[@LSC69]

It's "methinks," one word.

[@JeffSchneiderMusic]

​@@LSC69 Noted. And that's why I'm not the Schneider with the PhD 😆

[@tedsheridan8725]

Very clear and well presented. I briefly learned Lagrangian and Hamiltonian formulations 20 years ago in Dynamics and promptly forgot them. Now I'm teaching myself more physics and they keep popping up. Thank you!

[@PhysicswithElliot]

Thanks Ted! Glad it was helpful!

[@chaizixuan6531]

An amazing mini-lecture!

[@obetancourtra]

Thanks for this. I've worked with a considerable amount of lagrangians and hamiltonians in my macroeconomics class to determine optimal paths of investment or consumption. It's always interesting to see where our mathematical tools come from.

[@PhysicswithElliot]

Glad you liked it Orlando!

[@DrDeuteron]

And it’s great to see where our 401ks go.

[@oak3785]

these videos hit different and get more appreciation post graduation, forgot what got me into physics in the first place but your videos bring me back in

[@SyedShah-os7ck]

Really amazing and simplified explanations

[@endormaster2315]

I cannot describe how wonderful this video is. You have encouraged me to learn in my own about a topic I didn't know I liked

[@annakapp7087]

Please keep making more physics videos. This was so helpful.

[@PhysicswithElliot]

Thank you Anna!

[@nbnvideo]

Wow, really really wish that this had been available before I studied quantum physics! Thanks for making the vid!

[@John-mn7op]

Most helpful 20 minutes that I’ve ever spent on this topic!

[@PhysicswithElliot]

Glad it was helpful!

[@stevewhitt9109]

Very best (and simplest) Lagrangian and Hamiltonian explanation

[@Ligatmarping]

Nice work! Im a math guy who started studying a little physics after many years; I like it a lot.Greetings from Argentina.

[@rajendramisir3530]

Interesting and fascinating. I like the Hamiltonian Flow. Path of least action vs Path of least resistance(electron flow). Just beautiful stuff!

[@InferiorPotassium93]

this is incredibly good content. thanks for making it

[@abdullahkarolia3418]

I've been confused for a whole semester on Lagrangian mechanics and this actually made it very clear, I might actually pass now, thanks!

[@DivyanshMMMUT]

5:36 that point that you mentioned is such a key to start loving physics if I have to put it I would say love for physics is not a love at first sight it Starts from zero and grows more and more and you can now never hate it.

[@Mayank-mf7xr]

My favourite of these is the Hamiltonian formalism because of its use in Statistical Mechanics and Quantum Mechanics. It really gives a new and very powerful perpective to ask and answer difficult questions about systems we cannot hope to deal with using bare Newtonian Mechanics.

[@anshumantripathi3977]

This channel will soon reach million subs.

[@justchecking905]

Finally you have enabled me to understand these three formulations of mechanics that I first learned in graduate school in 1968. I have no need of them now as a retired scientist but thank you!

[@maalikserebryakov]

There are more formulations

[@trollfacegaming9063]

@@maalikserebryakov who asked

[@dbracale]

Wow. The best video I have seen in the last year! Great explanation. I learned a lot!

[@aqeel6842]

More math in Lagrangian and Hamiltonian Mechanics? Wonderful, I look forward to learning it

[@vtrandal]

You, my friend, deserve millions of subscribers. Such wonderful content you are delivering here! Thank you! I wish you the best in all you do.

[@PhysicswithElliot]

Thank you Vincent!

[@AppliedMathematician]

Yes, do a few calculations using Lagrange mechanics! That really helps to appreciate it, especially for constrained systems.

[@nathangodefroy3738]

Thank you for making a hard subject more approachable. Great channel!

[@PhysicswithElliot]

Thanks Nathan! Glad it helped

[@nicepajuju3900]

Wow this mini lessons are very good!! very clear and straightforward presentation

[@paulwolf3302]

Well done. I've heard of these but don't think I learned this in undergrad engineering. I was able to listen to your explanations and understand them, taking your word for it that you were doing the math right. That takes too much patience, and I would never use it in my profession, but if I can just grasp the basic ideas that is really educational and you never know how you might use them.

[@robertschlesinger1342]

A very worthwhile refresher video.

[@milind11]

Good stuff. Keep pouring the knowledge.

[@pedroafonso8384]

This so underrated.. please dont stop doing content like this!

[@PhysicswithElliot]

Thanks Pedro!

[@brightibezim1486]

Thank you. Enjoyed your Physics videos. Been a long time since I jumped from Physics to Programming.

[@sujitsadhnani750]

just what i had been searching all day

[@treborg777]

I wish this had been presented in my grad school classical mechanics course.

[@liemlucas]

Great video and explanations. We essentially exclusively use Lagrangian mechanics in microeconomics bc of the simplification of all the moving parts involved

[@MuhammadRamzan-tf6br]

Amazing, thank you so much. It was music to the ears listening to you!

[@jimsal78]

Was just thinking the -p²/2m was very reminiscent of shrödinger. Then I watched the end and you were speaking of quantum mechanics using the Hamiltonian. I've been out of physics now for a few years and had forgotten how much I enjoyed doing it. Thank you.

[@ioanab7076]

Yes very clear video, you make these concepts very enjoyable to watch and listen to.

[@gideonk123]

Fantastic explanation! Regarding the 2 different types of curves in phase-space after 17:00, I presume the internal ones, which touch the horizontal axis (dp/dt = 0) are where the pendulum swings back and forth (momentarily zero velocity when changing directions). The 2 external curves are where the pendulum swings/rotates around the pivot point: one is clockwise rotation and the other is counter-clockwise rotation.

[@PhysicswithElliot]

Yep!

[@saatviknarain2413]

Great explanation. Thanks!

[@urieldaboamorte]

I'm an Econ undergrad and it's nice to see how similar these approaches are to what I saw in an intro to Dynamic Optimization.

[@kingstonstreet3726]

I’m just here to support you and I don’t know anything about physics but I will watch to support and learn about it

[@sujitsadhnani750]

phenomenal work

[@feynstein1004]

This was easily one of the best videos I've ever watched. Subbed

[@PhysicswithElliot]

Thank you!

[@feynstein1004]

@@PhysicswithElliot My pleasure. Literally 😃

[@snowman8241]

I would like to give you some serious credit for your teaching abilities and methods. This is movie is excellent material to study for a teacher, and has great pedagogic value. I'm not trying to shit on teachers. I have the education to be a high school teacher myself, and I find your movie very inspiring and that it shows me new way to view physics. Bravo!

[@namitkamani4732]

Love from India Mr.Elliot❤I am really enjoying your videos...they are very conceptual...you explain so nicely everything..Please make whole playlist of quantum field theory from basics....God bless you🙏

[@beatriztamayogutierrez7646]

Best video on youtube by far! So helpful

[@eliasjazz]

If only there had been this channel during my university times , I would have been one of the best in my class, excellent explanation , thank you

[@PhysicswithElliot]

Thanks!

[@BlastinRope]

Yeah but when this video exists so do algorithms whose purpose is to feed you a functionally infinite amount of content that it predicts you will waste your time on, so it balances out.

[@olivierbarbe9269]

Great video thanks, very clear and engaging

[@Sorvah]

this is the physics content I've been searching youtube for

[@PhysicswithElliot]

Thanks Jenssy! Let me know about the things you'd be interested in learning about!

[@heheheh7191]

What an amazing channel! You’ll blow up some day.

[@martinfierz]

Thx for the nice video! Tip: when you introduce something new (like Lagrangian and Hamiltonian mechanics), then produce a SIMPLE problem for viewers to try to solve on their own, and only after that a more complex problem

[@DeepakKumar-yo7xd]

Appreciating your effort. Well done!

[@PhysicswithElliot]

Thanks Deepak!

[@youxizhang1171]

Thank you very much ! This video explain in a most clear way .

[@jflopezfernandez]

Awesome explanation, thank you!

[@ClaudioPedrazzi]

Thanks a lot for sharing, very didactic indeed. Exactly what I was looking for, to get a quick introduction in the two different approaches!

[@user-zo1wv8cx5s]

Thanks for the explanation, it really helped me understand

[@GM_Neo]

All of a sudden I'm glad I kept this video in my watch later for over a year because coincedentally I took calculus and understand some of it

[@amahlendlovu9992]

Thank you so much for saving my semester. I'm doing a second year classical mechanics course and I haven't been understanding most of lagrangian and hamiltonian. But now I do. Excellent tutorials

[@PhysicswithElliot]

Glad it helped Amahle!

[@vibbruh]

Y is it that we understand KZbin tutorials so much better than our classes? Are KZbin teachers just much much better or is our focus not on our classes or the methodology of teaching in our institutions is bad? And very nice video btw.

[@eduardodejesuslararuiz7311]

This is amazing!

[@hernandezdiazjuanpablo9817]

What a great video, i really liked it !

[@spiderjerusalem]

Instantly subscribed. Great channel!

[@nataliiaproshunina-lc5gm]

Thank you sir for wonderful channel to learn physics!

[@amandeepsaha]

The best channel I've seen

[@jeongdaeseo384]

how wonderful explanation! thanks very much.

[@frankreashore]

Another amazing video. Thanks.

[@paulsutton5896]

The trouble (for me) is that until Lagrange draws attention to it, "action" is an entirely meaningless quantity. Unlike "total energy", "action" has no physicality. We might as readily have called upon Lagrange's inside leg measurement.

[@PhysicswithElliot]

You might like my video about the action in relativity (kzbin.info/www/bejne/gYfOYoSEibx1rrM), where the physical meaning becomes much clearer: it's the length of the curve that the particle traces out as it moves through spacetime.

[@paulinnanjing]

I did my physics degree in the 1980s and either nobody bothered to explain this to me or I wasn't paying attention. Even the maths units I covered didn't go there. Thank you for bringing some belated clarity.

[@peterpoperzius3776]

great video, elliot. thank you

[@sdsa007]

Amazing stuff! I’m on my way to towards understanding Schrodinger’s famous equation! This is the best compare/contrast between Lagrangian and the Hamiltonian on KZbin… although it would be cooler if I could see a ‘phase’ space for the Lagrangian… ( would it be the same?).

[@rudyd8403]

A vivid memory is when my lecturer switched from fixed ("newtonian" Elliot calls it though everything he talked about is actually newtonian) to generalized coordinates like Lagrangian. I later went back to earlier chapters in my trextbook and found it much easier to solve some of the problems there with the new lagrangians. I'm an EE but won't forget the excitement that that revelation brought.

[@michaeljeynes6495]

Can you recall which mechanical problem would be the easiest or most basic problem which the Lagrange methods solve better than the usual?

[@maalikserebryakov]

The Lagrangians and hamiltonian formulations were made after newton died and hence are not newtonian. Read the names. 🤡

[@stijncousin4891]

The Lagrangian formalism can also be derived from the principle of virtual work, which in itself is already a very strong formalism for classical mechanics. I prefer this approach since it more naturally accounts for non-conservative forces too. Maybe an idea for a future video?

[@makespace8483]

Good job! Thanks.

[@biswajitbhattacharjee5553]

These views of classical mechanics has huge huge success and benifit for "physics, engineering" like calculus. But real problem is now appearing in the name of String theory , Quantum field theory , standard model no doubt quantum theory. Very good class, thank you.

[@omaralhafez5014]

This is really good video 😢 I was so confused why Lagrangian and Hamiltonian have been ever studied, now I'm convinced and satisfied 😊😊

[@ckcmedia43]

Thank you very much dear eliot.3 in 1 shot.

[@nihilisticgacha]

thank you soooooo much for this simplified yet extremely informative introduction!!!!!! I'm not studying physics but somehow the course uses a lot the terms you mentioned in this video without giving us proper explanation! and i'm too dumb and short on time to start a whole course on physics just to understand these concept. you are such a lifesaver!! 🥰🥰

[@PhysicswithElliot]

Glad it was helpful!

[@lelomambueliane4915]

Can it be denied that this guy solves the most difficult problems? kzbin.info/www/bejne/ppzaamWVf9WpZ6c

[@setyokabul]

Thanks...🙏 This helpful for my study..

[@RomanoPRODUCTION]

I don't have a headache yet but yes at speed 1.5 it's mind blowing. Thank you for your educational skills

[@swayamjoshi7667]

im a 12th grade student this was quite fascinating to me as physics has always been fascinating

[@MichaelBrueckner]

Remember encountering this almost 50 years ago (TU Berlin) - Theoretical Physics I (I think, you'd use 101). We used to talk of Eulerian observer and Hamiltonian observer. One sitting at the river bank and the other swimming, sort of fun thought experiment using the respective equations.

[@aymanmahrous2957]

Awesome 👌, many thanks.

[@frh_astroboy8215]

Brooo u explain a lot... Thanks I love ur teaching method. I learn more in less time❤❤❤❤❤❤❤

[@mritunjayjha2872]

Awesome!!!! Spark needed for continuing from Newtonian to Lagrangian and Hamiltonaion.....

[@johnpapadopoulos9057]

Marvelous. Enlightening.

[@sanjayrajpoot7277]

Nice Explanation

[@mariaclaratapascogiraldo3802]

Thank u so so much, your videos helped me a lot!

[@Roxyhehe532]

I'm really enjoy this video 🎉

[@TheAndresOpeth]

Wow... this video was extremely awesome. I really appreciate this kind of content. I surcribed to the channel already. Looking forward to seeing more content like this. Also is outstanding that you take the time to upload problems. Congratulation this is a quality content.

[@PhysicswithElliot]

Thanks Andrés!

[@suzimurphy1904]

MS Physics here and this is a great throwback to those days when I was learning this stuff; but I have to say that even today I am frustrated by the same thing that I was "back in the day" .... the choice of sign for the potential energy, which Im sure cannot be arbitrary ... choosing a "+" sign completely changes the way the math works. It would have been nice "in situ" to cover what that decision was based on and why it matters. In fact, IIRC, most of the students at the time that were getting this stuff wrong in tests / homework, were making that particular sign error "mistake"