@@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.
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
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?)?
@@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.
@@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.
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.
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😭
@@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.
@@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 🥂
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.
@@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 RUclips, thank you so so much for typing all this up to help a physics newbie out, much appreciated!)
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 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.
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!
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!
🎯 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.
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.
@@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.
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.
This really blew my mind, and once again I'm so glad that educational material exists on RUclips. Thank you for spreading your knowledge; it was mechanawesome! 👍
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
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.
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
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.
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.
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!
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
Y is it that we understand RUclips tutorials so much better than our classes? Are RUclips 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.
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.
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
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!"
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.
2 года назад+2
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.
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?
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.
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.
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.
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!! 🥰🥰
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
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!
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.
You might like my video about the action in relativity (ruclips.net/video/KVk1QNTWBxQ/видео.html), where the physical meaning becomes much clearer: it's the length of the curve that the particle traces out as it moves through spacetime.
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.
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!
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).
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.
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
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.
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.
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)
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.
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.
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
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
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🙏
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.
So a pendulum gives you an eye picture in the phase space. Interesting :) Thanks for the visualisations! On the blackboard it was more like "ok so it's the phase space" but I never even could imagine how the graph looks like lol.
11:41 what’s the more general way to determine the momentum , if not already calculated from Lagrangian method? Just manually crank it out from “mass times velocity “?
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 RUclips… although it would be cooler if I could see a ‘phase’ space for the Lagrangian… ( would it be the same?).
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?
This is the best video I've ever watched on this topic. Thank you so much for making it. Now i just have to learn about operators... and how to do differential equations 💀
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
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"
1:49 Newtonian formulation
5:44 Lagrangian formulation (L = K - U)
10:59 Hamiltonian formulation (H = K + U)
I would say:
Newtonian formulation (Σf=mä)
Lagrangian formulation (L = K - U)
Hamiltonian formulation (H = K + U)
@@ronissilva9570ä would snap! i think ur thinking of ẍ lol
@@aug3842exactly
Thank you
@@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.
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
So basically autism good
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?)?
@@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.
@@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.
@@Simon_Jakle__almost_real_name sophisticated Englishmen be like
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.
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😭
@@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.
@@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 🥂
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.
@@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 RUclips, thank you so so much for typing all this up to help a physics newbie out, much appreciated!)
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!
Inspires me as an undergrad
@@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.
Future topic suggestion. Noether's theorem. Symmetry. Why is this so important for physics and math?
Thinking of doing Noether next!
@@PhysicswithElliot you could say it’s the topic for, a-Noether video
@@thesuperkat943 *clap clap clap*
@@thesuperkat943 bruh
Yes!!
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!
Thank you!
Can it be denied that this guy solves the most difficult problems? ruclips.net/video/pkw92_Jpv1E/видео.html
I went to graduate school for engineering and that was the best explanation of the Lagrangian/Hamiltonian I have ever listened to.
Absolutely awesome. I finally found somewhere that got past the H=KE+PE of Hamiltonian mechanics AND actually explained the point. Thank you.
Glad it helped!
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!
Thanks Ted! Glad it was helpful!
🎯 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.
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.
Glad you liked it Orlando!
And it’s great to see where our 401ks go.
Me thinks this is going to be a great RUclips channel!
damn didn't expect you of all youtubers to comment on a video like this!
@@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.
lagrangian and hamilton are just talking about energy wich comes from newton phisics, no big deal
It's "methinks," one word.
@@LSC69 Noted. And that's why I'm not the Schneider with the PhD 😆
As a physics teacher I can safely say this is amazing! Succinct and encouraging for a student. Well Done.
Thanks Tom!
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.
Yep!
This really blew my mind, and once again I'm so glad that educational material exists on RUclips. Thank you for spreading your knowledge; it was mechanawesome! 👍
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
Very well done! Brilliantly conceived and the use of a consistent scenario makes for a really instructive study.
Please keep making more physics videos. This was so helpful.
Thank you Anna!
I've been confused for a whole semester on Lagrangian mechanics and this actually made it very clear, I might actually pass now, thanks!
Nice work! Im a math guy who started studying a little physics after many years; I like it a lot.Greetings from Argentina.
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.
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
This so underrated.. please dont stop doing content like this!
Thanks Pedro!
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.
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
Thanks!
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.
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!
There are more formulations
@@maalikserebryakov who asked
@@trollfacegaming9063 Who's on First.
Very best (and simplest) Lagrangian and Hamiltonian explanation
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.
Thank you Vincent!
This channel will soon reach million subs.
More math in Lagrangian and Hamiltonian Mechanics? Wonderful, I look forward to learning it
Wow, really really wish that this had been available before I studied quantum physics! Thanks for making the vid!
This was easily one of the best videos I've ever watched. Subbed
Thank you!
@@PhysicswithElliot My pleasure. Literally 😃
this is the physics content I've been searching youtube for
Thanks Jenssy! Let me know about the things you'd be interested in learning about!
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
Glad it helped Amahle!
Y is it that we understand RUclips tutorials so much better than our classes? Are RUclips 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.
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.
Can you recall which mechanical problem would be the easiest or most basic problem which the Lagrange methods solve better than the usual?
The Lagrangians and hamiltonian formulations were made after newton died and hence are not newtonian. Read the names. 🤡
An amazing mini-lecture!
Thank you for making a hard subject more approachable. Great channel!
Thanks Nathan! Glad it helped
Most helpful 20 minutes that I’ve ever spent on this topic!
Glad it was helpful!
Really amazing and simplified explanations
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
Interesting and fascinating. I like the Hamiltonian Flow. Path of least action vs Path of least resistance(electron flow). Just beautiful stuff!
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!"
Glad it helped Robert!
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.
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.
Thanks Peter!
@@PhysicswithElliot It is an honour.
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?
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.
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!
Glad to hear it!
just what i had been searching all day
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.
A very worthwhile refresher video.
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!
Thanks Steven!
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
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.
That takes a tad more math than this video is pitched at.
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!! 🥰🥰
Glad it was helpful!
Can it be denied that this guy solves the most difficult problems? ruclips.net/video/pkw92_Jpv1E/видео.html
Wow this mini lessons are very good!! very clear and straightforward presentation
im a 12th grade student this was quite fascinating to me as physics has always been fascinating
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
I wish this had been presented in my grad school classical mechanics course.
this is incredibly good content. thanks for making it
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!
I don't have a headache yet but yes at speed 1.5 it's mind blowing. Thank you for your educational skills
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.
You might like my video about the action in relativity (ruclips.net/video/KVk1QNTWBxQ/видео.html), where the physical meaning becomes much clearer: it's the length of the curve that the particle traces out as it moves through spacetime.
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
Applied Math student here with little understanding of physics. Thank you for making me feel slightly less confused
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.
rewatched and it's still awesome I love this video
Wow. The best video I have seen in the last year! Great explanation. I learned a lot!
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!
Yep!
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).
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.
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
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.
Stunning, it makes sense and ive never seen Hamiltonians before. What a great video. Im deeply curious now.
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.
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)
Yes, do a few calculations using Lagrange mechanics! That really helps to appreciate it, especially for constrained systems.
I wish youtube's algorithm had directed me to this channel sooner. The video is great. Thanks for the inspiration.
Thank you!
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.
Thank you. Enjoyed your Physics videos. Been a long time since I jumped from Physics to Programming.
Good stuff. Keep pouring the knowledge.
Appreciating your effort. Well done!
Thanks Deepak!
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.
source?
@@LilliHerveau the source is that I made it the fuck up
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
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
Thanks Drew!
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
Amazing, thank you so much. It was music to the ears listening to you!
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.
Did you do it?
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🙏
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.
Glad it helped!
Can someone tell me what is the board that the author uses in his video? Looks very convenient for online teaching
Procreate!
Thank you sir for your dedication! 🙏
So a pendulum gives you an eye picture in the phase space. Interesting :)
Thanks for the visualisations! On the blackboard it was more like "ok so it's the phase space" but I never even could imagine how the graph looks like lol.
That's what I was hoping for when I made the animation! Glad it helped
11:41 what’s the more general way to determine the momentum , if not already calculated from Lagrangian method? Just manually crank it out from “mass times velocity “?
How I wish this guy had taught me high school physics. He's awesome.
Glad you liked it Ronald!
Thank you very much dear eliot.3 in 1 shot.
Yes very clear video, you make these concepts very enjoyable to watch and listen to.
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 RUclips… although it would be cooler if I could see a ‘phase’ space for the Lagrangian… ( would it be the same?).
What an amazing channel! You’ll blow up some day.
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.
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?
Covered here!: ruclips.net/video/sUk9y23FPHk/видео.html
This is the best video I've ever watched on this topic. Thank you so much for making it. Now i just have to learn about operators... and how to do differential equations 💀
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
Love this video. Just to ask, if the system was damped would the curves on the phase space all slowly converge to the origin?
👍
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"
Thanks a lot for sharing, very didactic indeed. Exactly what I was looking for, to get a quick introduction in the two different approaches!