Block on an Incline: Newtonian, Lagrangain and Hamiltonian Solutions
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- Опубликовано: 4 июл 2024
- Here are three different approaches to the same problem.
Here is the acceleration in polar coordinates
• What is the accelerati...
Introduction to Lagrangian mechanics
• Introduction to Lagran...
00:00 - Intro
00:57 - Newtonian Mechanics
08:12 - Lagrangian Mechanics
13:39 - Hamiltonian Mechanics
18:55 - Other problems and how to solve
How wonderful to see these side-by-side! Nice job!
Thank you! Seeing these side by side and explained with when to use which one and why, finally made my Classical Dynamics learning finally "click"! That point about "Hamiltonian gets tricky if there is energy loss", was something i had heard before, but never really understood why. You clearly have an intuitive understanding of the physics of these systems, as well has their mathematical treatment, and you are able to communicate them. Thanks!!
Uhhhhuuuuuui
Thank you for the great video. I learned Newtonian mechanics early in life, and now I’m trying to learn Lagrangian and Hamiltonian mechanics late in life so this is very helpful.
I'm currently making a series of videos for my classical mechanics course - I will get to Lagrangian mechanics this semester (Hamiltonian is second semester).
Here is my playlist of video so far (still making the videos)
ruclips.net/p/PLWFlMBumSLSbZvcPMA0nH60x1ebX4XSqB
17:43: This is exactly the reason why I never use s as a variable. I prefer r, x, q. I will use eta, rho, tau and omega before I use s. Another letter I never use is d, unless it’s a problem which requires no derivatives. I have friends who fell for the “d in d/dt cancel out” already.
Not so much a problem for me because I write sin in cursive and would write a free standing variable name s in print. I don't use d though. When you're dealing with Laplace transforms you're dealing with the _s-domain_
DId those who cancelled the d's in d/dt flunk out? Hard to believe they could have gotten a degree of any sort.
@@walshrd It was in high school 😅. Only now, 2 years later, I am doing a degree in physics.
@@shaiavraham2910 Physics is a noble science, but where are the jobs? Get a degree in engineering with a minor in physics. Your upcoming career will be much better and just as challenging.
That - 2 points was best thing I saw today
You sir are a huge asset to students of science and engineering
great job! There is a very straight forward way to understand.
Excellent explanation of the problem in three diferent ways ! vow !!
You r a wonderful teacher..nice presentation of comparision, thanks
Brilliantly explained! Thank you so much.
Thanks!
Nice explanation. Hope to see more examples and case studies in details from you. Thank you.
Thanks. Clearly explained 👍
You’re amazing! Thank you!!
Wonderful video. I wish I knew this in high school.
Excellent explanation and problems.
Man, I love you. Greetings from Argentina
Yeahp. This is the Physics "Click" that I need. Thank you!
Nicely explained. Thanks a lot
Great explanation.
That was awesome!
omg, you are the best physics teacher i have ever seen, thank you so much
thank you so much for the kind comment. I'm glad you enjoyed the video.
It is just beautiful !
From France (the country of Lagrange) :congratulations for exposing the same problem with three différents ways of solving it ;at the same time we can associate the names of Newton Lagrange Hamilton. MERCI MONSIEUR.
Sorry to disappoint you, but Lagrange was italian, born and raised in Turin from italian parents (although one of his great-grand fathers was french). He became a French citizen and changed his name from Giuseppe Luigi Lagrangia to Joseph Louis Lagrange at 60 years of age, when Napoleon annexed Piedmont to the French Republic (following a suggestion of Lagrange himself). He already moved to France a dozen years earlier.
Lagrange was italian, from Turin, although he worked a lot in Germany and in France.
The most trivial of comments:
Back in the last millennium, when I studied [the ;-] calculus, we used a prime (') notation vs the dot.
This has the advantage of working in ASCII text. y = x; y' = dy/dx
I wouldn’t say it’s that trivial cause the prime notation is general notation for the derivative but the dot can only be used for the time derivative (because time derivatives come up so often in physics). I still use prime notation occasionally (Newton’s) but I prefer Leibniz notation.
the prime is still used in some cases, the dot is specifically for derivatives over time (at least everywhere I've seen it being used). I used it a lot on my thermodynamics courses
Sadly my mechanics education
was Newtonian virtually completely
(the others weren't mentioned at all)
and then when introduced to Quantum mechanics
Hamiltonians were used virtually without reference to the classical systems
How much more helpful introducing all three would have been.
I had to backfill my understanding of classical mechanics
during my masters to understand the quantum problems I was looking at.
😁😁😁😁😁😁😁😁😁😁😁😁😁😁😁😁😁😁
You are lucky. Lagrangian Mechanics is nothing but a mathematical trick of Newtonian Mechanics. The teacher here has been using Newton's law to derive the Lagrangian equation.
@@chuckstarwar7890
As they are two ways of describing a mechanical system
they are distinct and I think
if you look at Leibniz's approach to mechanics
where he used energy as the basis
to understand the situation.
As they are two ways of looking at the same thing
they are naturally interchangeable
The question then becomes which is more useful
to help describing and solving problems in the real world
Newtonian style mechanics works well for simple systems
and that is what gets taught first in schools
especially with the tricks to make it easier in Newtonian approach.
But some simple systems like a mass on a string oscillating at large angles
the Newtonian approach becomes much more problematic
whereas the Lagrangian approach is easier.
I think you are assuming that Newtonian mechanics is the default
(because virtually everyone meets mechanics with the Newtonian approach)
and that equations for energy and momentum etc are inherently Newtonian
I don't view them as such.
Why don't you just study on your own? I knew about Lagrangian mechanics and Hamiltonian mechanics whilst taking my classical physics undergrad (lower divisionals), and I could solve problems using such methods. I also knew multivariate calculus, linear algebra, and differential equations before even taking physics, yet many in those physics classes were literally learning these at the same time as taking their physics course (via the physics course), which is just absurd; learning the mathematics via a pure math approach is always superior, which is why I aced every exam without studying, both in math and physics courses.
I never understood why people complained they were sufficiently programmed by others, when it is 100% your responsibility to learn stuff. I didn't major in physics and go to college to be taught; I went to enhance my learning, of which I was already doing on my own; isn't this the entire point of higher education? That you are the impetus behind your learning?
Then I hear all these stories, such as yours, of people in literal graduate programs that don't know this stuff; HOW? I literally knew this stuff BEFORE I EVEN WAS AN UNDERGRAD. The benefit to this is that I massively outperform my peers (which is why I am in the 3rd best university for physics; go google it to find which one, I don't reveal such details directly).
@@chuckstarwar7890 It is not a trick at all, and the fact you think this just exposes that you don't understand what is going on on a fundamental level. You can also use Lagrangian 'equations' to derive Newton's laws, so you clearly don't understand what all this really means. Do you not know what axioms are? Also, if Lagrangian mechanics wasn't equivalent to Newton's mechanics, that would be a massive problem (and it wouldn't work for solving real physics problems). The fact you think this is a 'mathematical trick' is honestly baffling to me.
Some people are born to teach, you are one of them
thanks!
Very good! Or you can use the principle of conservation of energy KE+PE+Wk = constant which is very useful for rolling cylinders etc. This is how most high school students start with these problems.
Thank You
Wish I could like it more than once!
thanks!
Omg you are the best.
thanks!
Thank you so so so so so so so so so so much
"Getting the differential equation is the physics". I used to tell my tutoring students this all the time. Once we had translated the word problem into equations, I'd say "The physics is _over_. We can now hand this to any mathematician, without telling them _anything_ about where the equations came from, and they can solve for x, t, phi, whatever. It's not until they get it down to a number, _then_ the physics starts back up by translating the number into something we can _say_ about the original problem (like where the ball landed, how long fish tank took to drain, etc)".
Nice one
Kane's method is another candidate to compare as it is supposed to generate the most computationally efficient EQM. The method is most famous for solving spacecraft dynamics relations.
The tangent of the coefficient of friction gives the angle of the slope when the box begins to slide.
"Getting the differential equation is the physics.
Solving the differention is the maths."
- Physics Explained
Love that quote lol
In the Hamiltonian you have m.g.h and m.g.sin(t) added. That doesn't pass dimensional analysis.
cool video
Nice example to show the three methods. However Lagrange and Hamiltonian use generalized coordinates (the minimum number of coordinates to describe the motion. (in other words the degrees of freedom). In this case its a one degree of freedom system. It is more illustrative to place a coordinate system on the top of the ramp, use a rotation to be parallel with s and it follows you derivations to the "T". other wise it can be confusing. Newton might be harder in some cases but if you are designing and need the force then you have to use Lagrange multipliers and it might be difficult to determine the result. Also nonholonomic systems are more difficult. you can also use the extended Hamilton's principle to get equation or Kane's equations of motion. which handles nonholonomic better than Lagrange in my opinion. Great job which I had the valor to make a video. keep up the good work
The boss has spoken!
There were no nonholonomic constraints in any of the problems that were presented. Nonholonomic constraints cannot be handled directly by Lagrange's equations unless the equations of constraint are linear in the derivatives of the generalized coordinates. If that is true, then applying Lagrange multipliers with the notion of generalized forces allows Lagrange's eqs to be applied. When the constraint forces are truly nonholonomic (which there are very few of these problems...the constraint equations for a tethered ball wrapping itself around a vertical pole is one), generalizations of Lagrange's equations are used and historically attributed (by some) to Gauss, but Gauss never really pursued the ramifications of those extensions of Lagrange's eqs. Kane took up the task and used an orthogonality projection mathematics that eliminates the need to calculate any nonhonomic constraint forces.
Just like a bead that is constrained to lie on the path of a string is a one-dimensional problem (one degree of freedom) and that the distance travelled along this string from an initial starting point is the generalized coordinate for position. There is also a generalized momentum coordinate.
More about Lagrangian &Hamilton mechanics
Dimension analysis shows your equations are right
whats the banging in the back ground?
Great but I think on Lagrangian solving, u=mg(h-s.sinQ) should be which on yours s is missing.
Kinda late but here is a thing that needs to be aware of (big error): The block is moving downward (i.e., towards to the negative y-direction) so the potential energy must be NEGATIVE, which is -mg(h-s·sinθ). This will give the correct result that a=-g·sinθ, not +g·sinθ. It would be better if you could explain where to place your reference point (u=0) to avoid the misunderstanding regarding the sign of the potential energy. So many people make the same mistake on RUclips and I am really surprised that no-one actually notices it. If you use Newtoniam formalism you will still get the same result: x-axis := -mg sinθ = ma.
Is there a way to set up the hamiltonian directly without using the lagrangian?
There was an error but this is how if you are aware you can spot📍
How simply he taught Hamiltonian!
Amazing!!!
Im aware. Tell me about this error.
An s was missing on lagrangian
So, after watching this video I understand that:
Use Lagrangian rather than Newtonian for mechanical problems.
Use Lagrangian IF there are some strange constraints on the motion.
This is why I find physics interesting
All this is not really physics. The only physics here is Newton's guess as to how local forces determine motion thru a simple mathematical relationship: F = MA. This is called a constitutive equation and cannot be proven. Newton said that a force produces a scaled acceleration in the same direction as the applied force. Lagrange's and Hamilton's equations are mathematical rewrites of Newton's fundamental equation. These equations give us insight into the motion of a dynamical system and its fundamental characteristics, but no new physics.
In the position equation for Newtonian mechanics, where did the 1/2 come from.
it’s the reverse power rule for integration. integral of x * dx is going to be 1/2 * x^2 + c, or x(initial) in the case of the problem’s context
From integration.
gsinθ is a constant here so ∫gsinθ.t.dt can be written as gsinθ ∫ t.dt and ∫ t.dt is equal to t²/2 + constant.
Why is the force in the x direction equal to mgsintheta and not mgcostheta if we use cos to find the the x component of a vector?
it depends on how you define the angle theta. In this case, theta was the angle the ramp is inclined - so if you do the geometry, then for mg theta is the angle between the force and y-axis (not x-axis). The opposite side of this right triangle is in the x-direction so you use sine.
15:42 - how is mg sin(theta) an energy term ?? It's a force term
Great video! I'm interested in simulating n-body gravitational systems, and that led me to learning about Hamiltonians and Lagrangians. There is one part I'm confused about. As much as I thought I understood partial derivatives, I found it disturbing to just assume that ∂(ds/dt)/∂s = 0. In other words, treating s-dot as a constant, not affected by changes in s. That is weird, because s-dot definitely changes when s changes. Can anyone explain this?
Haha, I just thought the same
Third person worry about that as well. ds/dt can it be a function of s?
I know this is not gonna answer your question directly but i think you will understand this issue if you learn the chain rule.
@@tantumpropter5576 ok
I had the same concern when I learned this. But it works out. The formal calculation is done "as if" one could vary s and s-dot independently. This computed variation is valid if they were two independent variables. Later in the work, you indicate that s-dot is actually constrained to always be ds/dt which imposes the dependency constraint (and restricts you to the specific variation in the (s, ds/dt) direction.
But why we should have these three models? What these do one over another? Which method do solve complex problems easily?
The tricky part is that they all three describe the energy and/or forces in a system, just in different ways. Each is a different way to describe how the system can behave. He did not solve the pendulum problem in the different ways, but going through it one could see that’s it’s much easier to get the values for the energies in the Lagrangian then to describe the forces in the Newtonian approach.
My first Lagrangian problem was to determine the shape of a cable suspended between two points. The force vectors needed for a Newtonian approach are hard to describe because there are two dimensions that depend on the shape the cable ends up in. It is much easier to describe the potential energy as a function of the height of the cable at any point. Since there’s no motion, there’s no kinetic energy and lots of Lagrangian terms drop out, making it much easier to figure out.
why is dL/dS/dt is equal to dL/dS ?
Deriving the equations is somewhat informative. But I really need to see concrete examples with actual numbers to understand the differences between them. If I'm trying to model a system and I want to know where an object will be at time T, how can the different methods tell me that? I've watched a dozen videos on lagrangian mechanics in the past day or so and I still have zero idea what I might be able to do with it.
This playlist by Prof. Michel van Biezen may be what you're looking for. He's got a similar ones on Hamiltonian and Newtonian mechanics with step by step derivations and plenty of examples. Enjoy.
ruclips.net/video/4uJaKJASKnY/видео.html
ftPVXWK0GOFDi7FcmIMMhY_7fU9&ab_channel=MichelvanBiezen
Please explain triangle in details
The triangle is just trigonometry
What are the dots for?
The dot represents a derivative with respect to time. So, a single dot over a variable means d/dt (first derivative). Double dot means a second derivative.
@@DotPhysics thank you
Very instructive but from the half of the video I started praying that the missing "s" in the potential term _mg sin theta_ wouldn't have led to troubles...
God physics and mathematics are so beautiful
What about friction (dissipative force/energy)?
Friction isn't too easy with Lagrangian or Hamiltonian (but it is possible). The best option is to use Newtonian mechanics
@@DotPhysics Yes. that is the problem. Lagrangian or Hamiltonian are used in Qm and GR. Yet dissipative terms are lacking
wait whats the q again in hamiltonian? it wasnt defined :(
q is any generalized position value. It can be the distance of the block down the incline.
thanks. whats the intuition behind Lagrangian? why do we wanna take KE - PE?
Phy Xpl = PCP
Once you put a box around it, THAT'S IT!
Hello, very nice video. But there is problem in the Newtonian explanation. It is not true, that you don't know what N is! And no cheating is needed. You draw that right away - N = mg cos(theta).
Yes, you can FIND N - but only by applying a constraint. It's not the same as knowing the gravitational force.
Good job, coming from a dynamicist with Ph.D. in Aerospace Engineering
Of the three: Hamiltonian mechanics translates better to quantum mechanics.
"The Hamiltonian isn't always the energy" - What? Why? I thought that was true by definition... Can somebody tell me an example in which H is not E?
It looks like energy, but it's possible you could use some generalized coordinates (not cartesian) and you get stuff that is not actually energy
@@DotPhysics: Oh, thank you for the prompt reply.
What's the (mathematical) definition of energy then?
[Mathy translation: let's say we have a Hamiltonian system
(M, ω, H) where (Μ,ω) is a symplectic manifold and H a function on M (the Hamiltonian). The "energy" E should then be a function on M. What function in terms of ω and H? ]
Okay now add friction force and see.
The math alone is simple. The real deal is how to put physic problem into math equations, all this tricks, placing adfitional variables.
Dunno Newton or Langrang 😉 , but I know Hamilton (and Abu Dhabi) 😂
SPANISH.....PLEACE....!
👷🇨🇱...............👉 Santiago OF CHILE
But... N = Mg cos(theta) ??? There is no cheating: We can calculate the Normal force.
You can do that only, if you assume that the block ist not moving in that direction. And this was his point: We should get this as a result of the calculations rather than using this as a fact we know by looking at the problem before solving it.
exactly
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Why isn’t the teaching of physics delayed until a
student has had calculus “one” Newton didn’t know about energy or that it is conserved. So why use his approach initially/solely in an introductory course. Biology etc aren’t taught based upon the knowledge basis of the 17th century.
Cause otherwise it's way too hard for students to understand properly the principles behind what they learn and why it's true.
Learning the physics without calculus is not ideal, I agree. But it is possible to build physical intuition in this way. Personally, when I learned calculus it just made a bunch of things click together better; I already had a decent grasp of the concepts.
You might want to read Newton's Principia to find out what Newton really did know. You will change your tune then.
The Newtonian solution has an error in the 2nd step as the integral of sin(theta) is -cos(theta). It all works out in the end OK but that is very sloppy and degrades from the overall "high level" quality of the video.
Here, 'theta' refers to the slope of the inclined plane and hence it is a constant. So, we won't be considering its integral. In other words, you can simply take it outside the integral, and then integrate it. Therefore the step is correct.
Theta is a constant. The integration is with respect to time, not with respect to the constant theta.
@@walshrd Yes I see that now. Thanks for pointing that out.
So why didnt you set up Lagrange's eqs accounting for sliding friction in your block problem? Is it because you really do not understand the underlying principles of Lagrangian mechanics and the notion of generalized forces? Hamilton's equations are NEVER used when energy is added to or subtracted from a dynamical system. Why? Because, in general, the Hamiltonian can be the total energy and an integral (constant) of motion if the system is autonomous (explicitly time independent): dH/dt = partial H/partial t = 0 if t is not explicit in the Hamiltonian. This gives H = constant for all motions of an autonomous dynamical system.
Lagrange's equations are projections of the Newtonian equations parallel to the surfaces of constraint within the system. For this, you will have to have some knowledge of how orthogonal coordinate systems would work in your favor: Since constraint forces are normal to the surfaces of constraint, they have zero projections parallel to these constraint surfaces (orthogonality) and therefore cannot contribute to the calculation of the equations of motion and can be ignored. That's why Lagrangian mechanics is such a powerful tool in mechanics.
Why didnt you show us your derivation of the eqs of the gravitational 2-body problem using any approach? Each of your 3 ways is equally easy. In all 4 of your examples (counting the double pendulum) energy is conserved. The Hamiltonian is the total energy and H = constant for all motions in every case. Could you write the equations of motion for the double pendulum using a coordinate system whose axes are fixed to the first pendulum and whose origin is fixed at the attachment point of the 2nd pendulum? That's the "best" coordinate frame that can be used to write the equations of motion. You would use Lagrangian mechanics to compose the equations of motion, but how would you figure out what the tension in the supporting rods was at any given time? What if that knowledge was required by the designers of this system? Lagrangian mechanics will do it for you in a straight forward manner, whereas a Newtonian mechanics approach will be an enormous headache. Can you make the tensile force calculations using Lagrangian Mechanics? That involves extracting information from generalized forces. All this is basic to the underlying principles of Lagrangian mechanics that should be thoroughly understood.
not good
you say “OK” too much as well as “So.” Why do academic/lectures have this handicap?
Is force even a real thing? Im thinking its not; its merely a convenient abstraction, a mathematical artifact, not a real physical phenomenon.
All so-called "forces" - at least the ones we experience and observe daily - can be reduced, simplified, or reinterpreted in terms of gravity and electromagnetism. Never minding strong and weak nuclear, but we cant talk about that. From Einsteins general relativity we know gravity is merely space-time curvature. Everything else is electromagnetic. From Einsteins special relativity we know that magnetism is just a relativistic reinterpretation of the electric, from high velocity electrons. From quantum electrodynamics we know that the electric force is merely the result of a transfer of momentum between charged particles via the virtual photon. Thus, electromagnetic forces are understood as exchanges of momentum between fundamental particles. Revisiting gravity, we can indeed understand a falling body as unchanged momentum in a curved space. It seems to me that the only true phenomenon from which all others derive is momentum.
Now follow me on this logic. The translational symmetry of space is why we have the conservation of momentum. Momentum, by way of quantum electrodynamics, is where we get electric forces and thus all other everyday forces between bodies. Forces applied over distance is our concept of energy. Thus the conservation of momentum from space translational symmetry is where we get energy and its conservation. From which we get time translation. Could it be that space then gives rise to time? Or that space and time gives rise to matter?
Im just wondering if Im just taking a brain sht, or if the reasoning is loosely rational.
I would be ok with saying all the physics stuff isn't real. We are just trying to model reality - and we have come up with some nice methods (force and energy), but they aren't inherently real.
@@DotPhysics I'd have to say force is at least real in a certain context. If we define a force as "Something that causes a change in motion" (kinda vague but eh) then we definitely can observe that as our reality in the classical sense. As for the rest yes model != reality, but hopefully model ~= reality in some sense or another.
in spite of chalkboards full of hundreds of equations, all of them roughly about the same thing, physics seems to be a dead science stuck in an awful rut .... the last new thing in physics was the higgs particle that came along in the 1960's i think.
we have these terrifically elaborate symbolic methods of dealing with waves and particles and forces and energies but no matter how cleverly they are manipulated we still can't exceed the speed of light or broach any other fundamental limit and so all discovery has come to an end
all that remains are open ended fantastical conjectures that defy proof or experimentation and in their place the youngsters fill up their time with exercises in lagrangians and hamiltonians....thereby endlessly substituting the process for the results and calling it victory.
Galileo, d=1/2at^2 : no energy, force or mass involved. Newton gummed up Physics for over 300 years. Kepler ‘laws’ equally mass and force free. Galilean relative motion has the earth approaching the released object, not vice versa; which Newton erroneously gave us. Euler, LaPlace, Legrange more abstract math confusion instead of physical ( physics) answers. “The Final Theory: Rethinking Our Scientific Legacy “, Mark McCutcheon.
No. This is not the time for uninformed speculations. Try reading about classical mechanics written by someone who knows the subject and is not a crank. Your 1/2at^2 is the one dimensional constant acceleration a point mass will receive when placed in a one-dimensional, constant field of force. Now go back to school.
If you insist on using paper and pen to demonstrate your point (or is this a lesson ) shouldn't your demonstration be written clearly to be understood ? Otherwise, its wasted effort
trails the explanation -- it might as well be useful to use Publish or Perish.
I disagree that this was good presentation. For professionals in physics and math it was nothing new, for hobby folks the language was too mathematically "cool" to capture the physical meaning of each approach.
So what? It was informative and for engineers, this is useful info.
I don’t understand how this was a bad presentation for you.
Blud couldn’t understand the maths 💀
I'm a physical chemist, with a math background, but not a lot of classical physics, so I've never really done much lagrangian or hamiltonian mechanics - so this was super helpful.
I don’t know what a “professional in physics” would be doing watching an introductory video to using the Lagrangian and Hamiltonian to solve a basic problem. Of course it’s nothing new, *you aren’t the audience*; the audience is people who want to learn how to apply the concept for the first time