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

[@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

[@AmitayushShrivastav]

Thank you

[@winninymeanssweet1920]

@@ronissilva9570 If you put F=ma in the first formulation, then you should put d/dt(dL/dq.)=dL/dt in the second and the two simultaneous differential equation in the third as the law of motion in the three formulations.

[@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!)

[@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.

[@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!!

[@givemeyourfish]

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

[@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!

[@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!

[@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 😆

[@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

[@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.

[@dtcarrick]

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

[@PhysicswithElliot]

Thanks Tom!

[@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!

[@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

[@-danR]

@@trollfacegaming9063 Who's on First.

[@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! 👍

[@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!

[@Ligatmarping]

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

[@pedroafonso8384]

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

[@PhysicswithElliot]

Thanks Pedro!

[@anshumantripathi3977]

This channel will soon reach million subs.

[@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

[@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.

[@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.

[@annakapp7087]

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

[@PhysicswithElliot]

Thank you Anna!

[@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. 🤡

[@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.

[@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.

[@John-mn7op]

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

[@PhysicswithElliot]

Glad it was helpful!

[@rdatta]

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

[@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.

[@stevewhitt9109]

Very best (and simplest) Lagrangian and Hamiltonian explanation

[@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!

[@aqeel6842]

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

[@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.

[@Herr.ahmed.elqafass]

Mr Elliot ,,, IAM happy for writing this massage In the first IAM Mathemation and I already graduated from faculty of Science Mathematics department Eventually Classical mechanic, Fluids mechanic and Quantum mechanics I studied them as a branches of mathematics nooooot physics Thanks alot

[@feynstein1004]

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

[@PhysicswithElliot]

Thank you!

[@feynstein1004]

@@PhysicswithElliot My pleasure. Literally 😃

[@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!

[@CLEOPATRANTOINE]

Both Lagrangian and Hamiltonian formulations were created by Lagrange. Lagrange worked on the Hamiltonian operator in 1811 when Hamillton was only 6 years old and named it with the letter H in honour of Huygens. It is later that the name of this operator was change in Hamiltonian.

[@LilliHerveau]

source?

[@Amoeba_Podre]

@@LilliHerveau the source is that I made it the fuck up

[@xxlvulkann6743]

The statement you provided is not true. While it is correct that Lagrangian and Hamiltonian formulations are named after the mathematicians Lagrange and Hamilton, respectively, the details regarding their contributions and the naming of the Hamiltonian operator are inaccurate. Lagrangian formulation: The Lagrangian formulation of classical mechanics was developed by Joseph-Louis Lagrange, a French-Italian mathematician, in the late 18th century. Lagrange published his work on mechanics in 1788. Hamiltonian formulation: The Hamiltonian formulation of classical mechanics was developed by William Rowan Hamilton, an Irish mathematician, in the 19th century. Hamilton's work on this formulation was published in 1833. The naming of the Hamiltonian operator: The Hamiltonian operator, which plays a central role in the Hamiltonian formulation of classical mechanics, was not named by Lagrange in honor of Huygens. The term "Hamiltonian" itself comes from the name of William Rowan Hamilton, who introduced the concept and notation associated with it. While Lagrange and Hamilton made significant contributions to classical mechanics and the development of the Lagrangian and Hamiltonian formulations, the specific details in the statement you provided are incorrect. - ChatGPT

[@nbnvideo]

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

[@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

[@rajendramisir3530]

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

Superlative video. I have been teaching Science in Patagonia Argentina for half a century and I so appreciate your talents. I shall share with students if you allow me. Cheers from frozen Patagonia.

[@PhysicswithElliot]

Thanks Peter!

@@PhysicswithElliot It is an honour.

[@roberthuber2770]

This is such a helpful introduction! I feel like I understand my mech 1 professor so much better when she says "F is not equal to MA!!! it is equal to P dot!"

[@PhysicswithElliot]

Glad it helped Robert!

[@TheAmcaf]

Applied Math student here with little understanding of physics. Thank you for making me feel slightly less confused

[@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?

[@primingdotdev]

I always wanted to learn more advanced physics but kept taking more math classes instead. This video scratched an itch I had for a long time!

[@PhysicswithElliot]

Glad to hear it!

[@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.

[@nathangodefroy3738]

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

[@PhysicswithElliot]

Thanks Nathan! Glad it helped

[@swayamjoshi7667]

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

[@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.

[@ytpah9823]

🎯 Key Takeaways for quick navigation: 00:00 🎙️ Elliott introduces the three formulations of classical mechanics: Newtonian, Lagrangian, and Hamiltonian mechanics. 00:14 📚 Newtonian mechanics, described by \(F = ma\), is the one most people initially learn, but Lagrangian and Hamiltonian formalisms are more widely used by modern physicists. 00:27 🔬 Lagrangian and Hamiltonian mechanics are essential for understanding quantum mechanics, particularly the physics of very small objects like elementary particles. 00:40 🕰️ Elliott uses a simple pendulum as an example to illustrate each approach, starting with Newtonian mechanics. 01:20 📏 Elliott explains that you can describe the position of the pendulum using either the arc length \(s\) or the angle \(\theta\), and both are equivalent. 02:03 ⚖️ Only two forces act on the pendulum: gravity \(mg\) and tension \(T\). Newton's approach focuses on summing these forces to get \(F = ma\). 03:02 📈 The equation of motion for \(\theta\) is derived as \(\theta'' = -\frac{g}{l} \sin \theta\), using Newton's second law. 03:45 🌐 A simple solution to the equation of motion only exists for small angles \(\theta\), where the motion approximates a sine or cosine function. 04:53 📜 Lagrangian and Hamiltonian mechanics were developed years after Newton and offer new practical and theoretical insights into the structure of mechanics. 05:22 🧠 Lagrangian and Hamiltonian approaches may seem more mathematically complex than Newton's but offer deeper insights and new problem-solving strategies. 06:03 🎚️ For Lagrangian mechanics, Elliott starts by defining the Lagrangian \(L\) as the difference between the kinetic energy \(K\) and potential energy \(U\). 07:14 📝 The Euler-Lagrange equation, which is derived from the Lagrangian, provides another way to understand the motion of systems, based on the principle of least action. 08:58 🔄 The Euler-Lagrange equation allows you to transform theta dot into theta double dot, and it's used to get equations of motion using Lagrangian mechanics. 09:15 📜 The Euler-Lagrange equation is reminiscent of Newton's Second Law; it describes how force is the rate of change of momentum. 09:44 🛠️ Lagrangian mechanics is a useful strategy for obtaining equations of motion, often more straightforward than using F=ma. 10:12 💡 The Lagrangian formalism simplifies handling constraints and understanding symmetries in a system. 11:10 ⚡ Hamiltonian mechanics begins with the total energy (K+U) and leads to Hamilton's equations. 11:58 📊 Hamilton's equations yield a pair of first-order differential equations for theta and p, unlike Euler-Lagrange's second-order equation. 12:29 ⚠️ While Hamiltonian is the total energy in simple cases, it's not always so; the general definition involves derivatives of the Lagrangian. 14:59 🌐 Phase space in Hamiltonian mechanics allows for a geometrical understanding of system dynamics. 15:56 🔄 In phase space, energy conservation leads to motion along lines of constant energy. 17:23 🌠 Lagrangian and Hamiltonian mechanics are not just important in classical physics but also form the foundation for quantum mechanics.

[@chaizixuan6531]

An amazing mini-lecture!

[@treborg777]

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

[@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

[@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

[@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!

[@ammyvl1]

Woah. Admittedly, the Euler-Lagrange equation, and Hamilton's equations came out of left field (as someone who has never encountered either, but has a sufficient bearing in calculus to understand), but even so, it was really cool to see them work like this, and I'd definitely want to check up their derivations and read more after watching this.

[@ammyvl1]

rewatched and it's still awesome I love this video

[@sujitsadhnani750]

just what i had been searching all day

[@richardneifeld7797]

I enjoyed your video. I suggest a video comparison the axioms of classical mechanics, quantum mechanics, special relativity, and general relativity, and perhaps quantum field theory.

[@davidwright8432]

That takes a tad more math than this video is pitched at.

[@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.

[@niloymondal]

The wavy lines are the ones in which the particle keeps spinning in one direction instead of oscillating. Thanks for the video, simulation tool and the insight!

[@PhysicswithElliot]

Yep!

[@mosquitobight]

And the waviness of the lines is caused by the fact that the pendulum slows down on the upswing and speeds up on the downswing. If there were no gravity, the horizontal lines in the phase space would be straight (and there would be no elliptical lines).

[@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

[@ericphantri96734]

The Lagrangian was simply the simplifying version of action and reaction principle of Newton third law because if action over size then you can not handle the reaction that simply said choke on large bite or never bite more than you can chew same thing in military or economic But on the Hamilton is simpler if the matrix of vector of square matrix of all current vectors of all the body in the space reduced at position of couple instead of using the actual position of the body we use the point of couple like grid points pair then find the solution but it must be square vector and the solution of entire matrix is the magnitude of the final vector and the direction is from the center and the angle and range extrapolation from any member at the furthest rim so instead of vector analysis of multi body problem in mechanic you try the range and field flux and the cross of the two is the energy vector of position like wind, or heat , or pressure result at any time no matter how many bodies in the entire system

[@stevennowo]

Wow, it's so amazing. I tried to understand it, I'm from Colombia, I don't have a really good english, but u explain so clean. I'm going to suscribe!

[@PhysicswithElliot]

Thanks Steven!

[@SyedShah-os7ck]

Really amazing and simplified explanations

[@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

[@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"

[@michaelwalker3935]

Halfway through the first video ans and subscribed. Excellent job. I took P-Chem II and i so wish id had this as a resource. Math abd Science faculty are arrogant and obnoxious these days. Its as if they don't really understand what they're teaching but make you want to believe theyre experts.

[@ronaldmarcks1842]

How I wish this guy had taught me high school physics. He's awesome.

[@PhysicswithElliot]

Glad you liked it Ronald!

[@robertschlesinger1342]

A very worthwhile refresher video.

[@RomanoPRODUCTION]

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

[@shoubhitwandile4565]

I have a small doubt At 8:35 why did you take theta and theta dot as independent variables. They should be treated as dependent variables the same way as when we have time dependent position of particle and we treat acceleration and velocity as dependent variables.

[@TheOrganicartist]

I wish youtube's algorithm had directed me to this channel sooner. The video is great. Thanks for the inspiration.

[@PhysicswithElliot]

Thank you!

[@AppliedMathematician]

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

[@wayneyadams]

Those little backwards 6 symbols (stylized lower case d) are called partial differentials (or derivatives), they tell you the rate of change of the coordinate in the numerator as the coordinate in the denominator changes. Let's look at a real world example. You are standing on an uneven stretch of ground with a hill in front of you. Let's call the east-west direction X, and the north-south direction, Y. You want to calculate the change in elevation (height) when you walk from one point to another on the hill. Let's say the point is some distance in the X direction and some distance in the Y direction away from you. We'll use Z for the elevation. So, we are asking for the change in elevation, dZ. Here's the plan. You will walk in the X direction first and calculate the change in elevation, then turn and walk in the Y direction finding that change in elevation. Adding them together gives you the total change in elevation, dZ. To keep this simple. let's assume the changes are smooth continuous upward changes, in other words, you are always walking uphill. Let's say the hill has a slope so that the elevation changes at a rate of 20 cm per meter as you walk in the X direction. That is what the partial derivative gives you. It is the change in elevation in the X direction ignoring changes in the Y direction. Let's say you walk 10 meters. Your change will be the rate of change, the partial differential, times the distance you walked, dX. 20 cm/meter x 10 meters = 200 cm. Now you turn and walk in the Y direction. Let's say the elevation changes at the rate of 5 cm per meter in the Y direction. Let's say you have to walk 20 meters in the Y direction to reach your final destination. Just like before, you multiply the rate of change of the elevation, the partial derivative, times the distance you walked, dY. 5 cm/meter x 20 meters = 100 cm. Remember that you are already 200 cm higher because of the first part of the walk. Your total change in elevation for the walk is the 200 cm change from the walk in the X direction plus the 100 cm change from the walk in the Y direction. dZ = 200 cm + 100 cm = 300 cm. That is all partial differentials do, they break down paths into small independent pieces that are then added together to get the total. Now to keep everything honest. in real world applications all those changes would be very small, and dZ would be the change of your elevation as you walk from one point to the next. I used large numbers to help clarify the process with understandable quantities that we can all relate to. When we break a vector (a path in some direction) into pieces like this, the pieces are called components. Of course, this can be extended to any number of coordinates. Wayne Y. Adams B.S. Chemistry (ACS Certified) M.S. Physics R&D Chemist (9 yrs.) Physics Instructor (33 yrs., retired)

[@wayneyadams]

My favorite (being facetious, it was a bear to solve) Mechanics problem in graduate school was this. You have a solid disc that rolls along a surface without slipping. A spring is attached to the axle at one end and a wall at the other. We extend the spring by rolling the wheel forward some distance and release it so the wheel begins to roll back and forth under the influence of the spring. A small peg is attached to the wheel at the edge and a simple pendulum consisting of a nearly massless rod with a mass attached to the lower end. The pin is placed so the pendulum hangs straight down when the wheel is in its starting position (no tension in the spring). The wheel is driven by the spring causing it to roll back and forth while the pendulum swings back and forth as a result of the rolling wheel. The problem is to write an equation for the motion of the weight at the end of the pendulum using the set of variables listed in the problem. These are not numerical quantities; the solution is the equation written with variables into which numerical values may be substituted. Honestly, I have not kept up with advanced physics and could not solve this problem today without spending hours reviewing and relearning advanced Physics, so if you decide to tackle it, don't expect me to provide you with the answer. 😄

[@PhysicswithElliot]

Oof sounds tough!

[@FirstLast-gm9nu]

Did the problem come with an accompanying diagram? It'd take me awhile to figure out what exactly the constraints of the problem are, let alone even attempt to solve it

[@wayneyadams]

@@FirstLast-gm9nu I wish I could post a diagram, but this crude comment system does not allow for that. Let me try to clarify the problem. You have a spring with force constant, k, attached to the axle of a solid disk with mass M, and radius, r. The disk rolls without slipping so that it will move with harmonic motion along the x axis with x=0 at the equilibrium point of the spring (neither stretched not compressed). A small peg protrudes from the side of the disk at distance r from the axle, in other words right at the edge. Attached to the peg is a very thin rod of length L which has negligible mass. A bob of mass m (as distinguished from M of the disk) is attached to the end forming a simple pendulum. At time, t = 0 the disk sits at x = 0. The peg is directly over the axle so the pendulum hand straight down directly over the axle at x = 0. The disk is rolled forward some distance x and released. The spring causes the disk to move with harmonic motion around x = 0, while the pendulum swings back and forth. The angle of the pendulum with the vertical is q. Write a time dependent equation for the position of the pendulum in terms of the angle, q with the vertical, using the variables listed below along with any constants you may need. The equation should look like: dq/dt = equation in terms of the given variables k = force constant of spring M = mass of solid disk r = radius of disk moment of inertia of a solid disk: I=1/2Mr^2 L = length of pendulum m = mass of pendulum bob q = angle pendulum makes with the vertical We assume it is a simple pendulum so the mass of the rod is assumed to be zero, or small enough to be ignored. As I said, I can no longer solve this monster without spending hour to relearn a whole lot of physics and math I have not used for decades. This problem does not require the Lagrangian to solve. I'll post a classic Lagrangian problem below.

[@wayneyadams]

Here is a classic graduate level Mechanics Lagrangian problem. Write an equation for the shape of a frictionless ramp that minimizes the travel time of a point mass sliding down from height, h. Hint: It is called a brachistochrone curve. The name comes from the Greek, brachistos meaning shortest, and chronos meaning time. Further hint: There are several methods for solving this problem, but try to do it using the Lagrangian since that is what my professor required.

[@wayneyadams]

@@PhysicswithElliot It took hours and about a third of a legal pad, due to the dead ends. I guess I could have cheated like the Communist Chinese student did, but then what does one learn if one cheats, after all I was paying money to be educated. That same test had the trajectory of a relativistic bullet, and the ramp problem I posted to another comment.

[@InferiorPotassium93]

this is incredibly good content. thanks for making it

[@dennycote6339]

Stunning, it makes sense and ive never seen Hamiltonians before. What a great video. Im deeply curious now.

[@dennycote6339]

I just figured out why this is a more intuitiv eway to see physics. WE actually use hamiltonian mechanics to do things,. I dont consider the force i throw a ball at, i consider its endpoint and the momentum i have to impart to it to connect that endpoint to the endpoint of my hand as it releases the ball. in doing so i follow the agreement with the universe to use the least Action to do so, Im LAZY! i might need that Action later.

[@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.

[@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🙏

[@carlosmadriaga1409]

I'll take bs physics even I'm too bad at math, and not doing well at my hs causes Dyslexia. love these kinds of videos dude, thanks.

[@whatever6874]

Did you do it?

[@Kiky_MedPhysicist]

Thank you sir for your dedication! 🙏

[@blaxpy]

Should be studying for my physics 1 exam but instead watching a video about 3rd year topics lol

[@tobechukwublessed4274]

My first experience to Lagrangian and Hamiltonian Mechanics, it's really fascinating. Please recommend a text book for this

[@yadav-r]

As a junior game developer, looking forward to learn how certain things were achieved in the AAA video-games, it seems they use lot of Physics & maths. You have explained things very clearly. Thank you very much sir, for sharing this in such an easy way to understand.

[@PhysicswithElliot]

Glad it helped!

[@joegreenthal2931]

Great video. When you're serious of classical mechanics is completed I'd like to see the solution for the 1st 2nd order differential equations using numerical analysis.. This analysis of course would would be another set of videos but you could tie it back into these classical mechanics videos.

[@PhysicswithElliot]

Thanks Joe! You might like playing with the animations I wrote, which work by solving the differential equations numerically. I'll think about doing a video about numerics at some point in the future! The animations are linked here: www.physicswithelliot.com/lagrangian-hamiltonian-mini

[@WechselWissen]

Question: At 2:38 you draw the presumably tangential component of gravity. Tangential to what? Wouldn’t the line have to be tangential to the circle, that the ball moves along?

[@eriknystrom5839]

Very nice…. I’m now waiting at L2 , soon meeting with the JWT.

[@nicepajuju3900]

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

[@brightibezim1486]

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

[@ckcmedia43]

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

[@drewpass5697]

nice video, clear and coherent. It's amazing how much the way one speaks makes a difference for understanding content. You sir are as clear as can be and your voice is soothing lol no homo

[@PhysicswithElliot]

Thanks Drew!

[@dbracale]

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

[@pacotaco1246]

Routhian Mechanics would be a fun follow up video

[@ezrachen8976]

9:13 you considered them force and what is assumed to be linear momentum but it's actually torque and angular momentum which is easily identifiable by doing a quick units check

[@AwfulnewsFM]

That's why they are called generalized force and momentum, they are independent of coordinate systems

[@johnpayne7873]

New viewer, old physicist. Channel takes me back to reading Feynman’s lectures. From your comparison between Lagrangian and Hamiltonian formalists, I have developed a deeper intuition of the Principle of Least Action. Maybe a topic already covered?

[@PhysicswithElliot]

Covered here!: kzbin.info/www/bejne/qYbOaqxoaKuDfs0

[@Kirillissimus]

As far as I see it: Newtons mechanics: objects move where combined forces push them to; Lagrangian mechanics: an object moves where it gains the most motion using the least "potential energy"; Hamiltonian mechanics: objects can move wherever they want as long as their "total energy" is conserved. And although all of the ideas are kinda intuitive and they obviously should be true, even the Lagrangian one is way too generalized and abstract for my liking and the Hamiltonian goes even further than that. But you have to be able to become deeply and intimately connected with the machines you design and test to truely understand what is going on with them and the whole idea of "energy" is really no good for that. I would even go as far as introducing the "inertial" force in order to simplify the newtonian sum(F)=ma to sum(F)=0. This way it is even easier to feel it.

[@glabrouswashere8078]

It’s essential that it’s understood that these are not alternatives that give different results. They are different ways of writing down the same laws of physics, each derivable from the other. Solutions to Newtons law conserve energy and minimise action. Paths that minimise action are solutions to Newton’s laws and conserve energy. And you’ll never understand e.g. planetary (i.e. near frictionless) motion if you rely on ‘feeling’. Unless you can feel the Force, of course.

[@Kirillissimus]

@@glabrouswashere8078 Yes, of course they are all mathematically equivalent but conceptually they start from different ideas and they provide different approaches to the same task. I just say that one of the approaches is much more intuitive than the others and it is much easier to use mostly just by drawing stuff as lines, circles and arrows on a sheet of paper and therefore start to go by feel and intuition after a while while others force you into abstract thinking and mathematics right away.

[@RizkyMaulanaNugraha]

There is actually a next step of revelation, which is called Feynmann Path Integral of (Quantum) mechanics: objects *will* move to any possible trajectories available, but the one we observe is the one found in the highest probability region paths. This reconciles some mystery on why Lagrangian and Hamiltonian is true for big objects, despite small quantum objects behaves probabilistically. It turns out the least action path for big objects is the only highest probability region that still obey Hamiltonian equation.

[@yaoooy]

@@RizkyMaulanaNugraha if you observe only that objects take the least action path, how can you prove with an experiment that the object actually takes any possibile path and it is not just a probability calcolation ?

[@RizkyMaulanaNugraha]

​@@yaoooy Not sure if I got your question right. For big object, the probability amplitude is highly concentrated in the least action path, hence the classical trajectories. In the quantum scale, the experiment didn't agree that particles take a least action path. It is rather governed by the wave function. So if the wave function is scattered (for example in the slit experiment), the probability of finding the particle is consistent with the path integral (if we take into account that a wave function can be split and interfere with itself to produce the path). Another way to see it: energy quanta is discrete, so there is no way it can be split in half (it is always observed as a whole). However the location in which it is detected only makes sense if we thought that the wave function somehow split before being observed and interfere with itself to produce a new wave function, in which if we take the probability amplitude, the location of the observed particles are consistent with the interference pattern. The particle somehow knew how to add up all these possible paths together at the same time, so that if the same experiments repeated many times over, the statistical results follows the probability amplitudes.

[@jacobflores8666]

The second-order differential equation for the mass on a pendulum is a nonlinear dynamical system, correct? It's a fairly straightforward problem to show the behavior of that system but it does require some tools from nonlinear dynamics (which are tools from multivariate calculus, ODE, and linear algebra by proxy) to accomplish. The way you would go about it is to first break the second-order differential equation down into a system of first-order differential equations. Then the trace and determinant of the 2x2 Jacobian matrix (found by doing a Taylor expansion around the fixed points of the system) of the first-order system determine local behavior in some epsilon-neighborhood around the equilibrium points in the system. You can then get an idea of the total behavior of the system depending on the type of equilibrium points you have (i.e., a saddle point or a center).

[@waynelast1685]

Good video. Could you elaborate why we want to use H and L , and in which situations?

[@alexdukec.7551]

Easy sub. Don’t need to think twice. Great hob!

[@PhysicswithElliot]

Thanks Alex!