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Thread: Classical Mechanics

  1. #1 Classical Mechanics 
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    OKAY, seeing as how I lack the ability to take advanced physics courses (attending a community college(I've hit the end of both the math and physics series they offer) and rather broke) so my resouces for learning what I want to learn are Here and various other places on the internet that have a tendency to be hard to locate. SO, I would love to talk about some beyond-rudimentary phisics and get some basis points to go look up.

    That said, I've been seeing Salsa get kinda worked up over classical physics, and I'm curious what it entails. I guess the best starting point would be; What's so special about Lagrangian and Hamiltonian mechanics over Newtonian Mechanincs? What's the advantage to using one over the other? How exactly are they different?


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    Don't tempt me--I'll end up writing up the equivalent of a whole course, which unfortunately I don't have time to do. But I'd be happy to get a start discussing this.

    The Newtonian, Lagrangian, and Hamiltonian perspective are three equivalent ways of doing classical mechanics. I emphasize, the underlying physics is exactly the same, but the approaches have different feels to them. Let's set aside the Hamiltonian formulation for now (it's my favorite, but it requires the most mathematical sophistication).

    In mechanics, we are usually concerned with finding the equations of motion x(t) of a particle or group of particles. The equations of motion in Newtonian mechanics are given by Newton's second law:



    where the sum means we add up all the forces acting on this particle. If we can find an expression for all these forces, then we have a 2nd order differential equation, and solving that gives us the equations of motion.

    The disadvantage with this approach is that it's not always easy to figure out what all the forces acting on the particle are. Consider, for example, a bead rolling around on a parabolic dish. To find the equations of motion, you need to know not only the action of gravity, but also the constraint forces that keep the particle attached to the dish. As constraints like this become more and more complicated, it becomes a bigger and bigger hassle to calculate these constraint forces.

    A second disadvantage with the Newtonian approach is that it highly favors using Euclidean coordinates to describe the motion of particles. But sometimes the geometry of the situation makes Euclidean coordinates unsuitable. For example, clearly cylindrical coordinates would be more appropriate for a particle moving on a parabolic dish. So it would be nice to have a theory which worked in any coordinate system.

    The Lagrangian approach has these two features. In Lagrangian mechanics, we make no explicit mention of "forces". The equations of motion are instead derived from the Principle of Least Action. This says the following: Among all possible paths x(t), 0<t<T, a particle will follow the path that is a critical value (e.g. a minimum) of the following integral:



    The above integral of Kinetic-Potential energy is called the action. The Lagrangian formulation holds that the real path a particle will follow is the path which is minimizes the action (technically, it's the path for which the action is stationary, i.e., where the action is invariant, to 1st order, under perturbations of the path).

    Write this as



    where q(t) is the path expressed in some convenient coordinate system (since the choice of coordinates plays no role in the definition of S).

    Minimizing S with respect to choices of q(t) leads to the Euler-Lagrange equations:



    For simplicity I'm assuming 1-dim motion, but this all extends quite easily to any dimensions. Also, I'm assuming for now that the potential U is constant in time.

    It is easy to see that these equations reproduce Newton's laws in Euclidean coordinates. But the advantage is that we can use any coordinates.

    As an example, consider again the bead rolling on a parabolic plate. Good coordinates are the distance r from the center, and the angle about the axis perpendicular to the plate.

    The potential at a distance r away is mgh = mgr^2

    The kinetic energy is



    The Euler-Lagrange equations for the theta coordinate then immediately give us:



    That is, we see immediately that angular momentum is conserved. Meanwhile, the Euler-Lagrange equations for r give



    You can get all this using the tools you already know, but notice how quick this was.

    And, again, it's not hard to cook up problems that are really hard using Newtonian mechanics, but simple using Lagrangian mechanics.

    For example, consider a ball rolling down an inclined plane, where the plane is sitting on a frictionless surface. So we allow the plane to slide to the side as the ball rolls down it. Find the equations of motion. With the Lagrangian formulation, no sweat. But using Newtonian mechanics? Bleh.


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    Quote Originally Posted by salsaonline
    ...And, again, it's not hard to cook up problems that are really hard using Newtonian mechanics, but simple using Lagrangian mechanics.

    For example, consider a ball rolling down an inclined plane, where the plane is sitting on a frictionless surface. So we allow the plane to slide to the side as the ball rolls down it. Find the equations of motion. With the Lagrangian formulation, no sweat. But using Newtonian mechanics? Bleh.
    But to be totally fair you also need to admit that the Lagrangian and Hamiltonian formulations are most useful in theoretical situations, for quantum theories, and for research purposes. For some engineering problems Newton's theory with forces can do things that the variational methods cannot handle. For instance, throw in oridnary Coulomb friction and you need to go back to Newton.

    It is not a simple matter of what is the best approach. It is a matter of picking the right tool for the job.
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    I understood a fair bit of this, none of the math seems to be over my head.

    Coulomb friction? How does this gum up the works of the non-Newton methods of doing mechanics?
    Wise men speak because they have something to say; Fools, because they have to say something.
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    Quote Originally Posted by Arcane_Mathematician
    Coulomb friction? How does this gum up the works of the non-Newton methods of doing mechanics?
    It provides a means of dissipation of energy so that potential energy is not simply a function of position; i.e. it is not described adequately by a scalar potential function.

    Newton's methods handle the problem with an empirical model that describes the force due to friction. Lagrangian and Hamiltonian methods do not directly address "force". At the most fundamental level that is one of the beauties of those approaches, but it is also a limitation for certain types of problems. But at the deepest levels variational mechanics offers insights that don't come easily or at all when you need to be able to describe all the forces in order to handle a mechanics problem as with the standard Newtonian approach.

    It is worth emphasizing again that Lagrangian, Hamiltonian, and Newtonian mechanics are all just different formulations of Newtonian mechanics at the most fundamental level. To handle friction with the variational methods would, I think, require that the model include atomic interactions, and no such workable model exists. The limitation of which I spoke occurs with macroscopic models and is not a limitation at the most fundamental theoretical levels, but it is a restriction when one is interested in macroscopic "engineering" problems of some types.

    While there are reasonable empirical models for handling friction, it is real problem in terms of theory and having a model based on true fundamentals. Feynman tried and failed to develop such a theory, and theoretical physicists don't come any better than Feynman.

    There are two points here. The simplest one is that you need to recognize the limitations of physical approaches, and use the approach that is appropriate to the problem at hand. The second point, more subtle is to note that with the current emphasis on fundamental laws of physics, that there would still be a great deal of physics to be done even if a "theory of everything" -- a set of physical principles that is consistent and which is known to govern the behavior of all elementary particles and all forces (including gravity) -- is formulated and verified. There are properties of systems and materials that become evident only when there are large numbers of particles involved. Such properties are sometimes called "emergent". Thermodynamics is one example. Another is fluid dynamics and in particular turbulence. There is no really good fundamental theory of turbulence either. The fundamental laws of physics are important, but there is important work and large gaps in knowledge beyond those fundamental laws. [/i]
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    okay, so friction is a general problem when you look at a problem without looking at forces. I know what friction is, and since you had originally said 'coulomb friction', I wonder, are there different types? or is it just another way of referring to friction?
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    Im very aware of the usefullness of lagrangian. It can be used to easily detect standing waves
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  9. #8  
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    Quote Originally Posted by Arcane_Mathematician
    okay, so friction is a general problem when you look at a problem without looking at forces. I know what friction is, and since you had originally said 'coulomb friction', I wonder, are there different types? or is it just another way of referring to friction?
    There are different models for friction, depending on the circumstances. There is is Coulomb friction, viscous friction, and other models as well. Coulomb friction is the usual model seen in elementary mechanics classes (I generally lump the usual kinetic and static models described by a coefficient of friction under the banner of Coulomb friction). There are good empirical models useful in engineering applications, but that is not representative of deep physical understandidng.

    Since there is no fundamental understanding of friction I don't know how to answer the question from a pure physics perspective as to whether these are different physical processes at the most basic level. It is some sort of dissipative interaction among atoms and molecules involving the electromagnetic force, the result of which is an increase in random motion (heat) but beyond that simple statement there is not much in the way of deep understanding.

    So, in fact, I doubt that you really know what friction is. Nobody else does either.

    http://en.wikipedia.org/wiki/Friction
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    I would think the best way to explain classical mechanical friction (static/kinetic) would be, using the names from high school physics, essentially 'normal force' interactions on the atomic scale, no? Since there is no truly smooth surface in reality, we always have some bumps and waves, ditches, divots, imperfections in general. Atomically speaking, you have particles that, while attached to others on the surface, collide with other divots and ditches and what have you. I hope I'm not off base, but that's what I see that kind of friction as being (no opinion on electrical friction(resistance)).

    Heat comes from those collisions. When the atoms hit, they undergo energy transfer that adds heat to the system, since there is constant force on the system that inputs more energy, right? I don't know, I think I can visualize the exchange but I don't know how to explain whats running through my mind.
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    Quote Originally Posted by Arcane_Mathematician
    I would think the best way to explain classical mechanical friction (static/kinetic) would be, using the names from high school physics, essentially 'normal force' interactions on the atomic scale, no? Since there is no truly smooth surface in reality, we always have some bumps and waves, ditches, divots, imperfections in general. Atomically speaking, you have particles that, while attached to others on the surface, collide with other divots and ditches and what have you. I hope I'm not off base, but that's what I see that kind of friction as being (no opinion on electrical friction(resistance)).
    You're not wrong. What the bear is saying is that there's no fundamental law that can tell you, for example, exactly what the coefficient of kinetic friction is for two specific types of surfaces rubbing against each other. We know that ultimately friction is caused by electrostatic forces between the atoms in the two surfaces, but we don't know, at a deep level, why the numbers in the laboratory come out exactly the way they do.

    At least, that is my understand of what he's saying.
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    okay, I get your point now.
    Wise men speak because they have something to say; Fools, because they have to say something.
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    As I have it, one aspect of the difference between static and kinetic friction is the amount of work that has to be done to lift the "bumps" out of the "pits". When movement gets going, momentum lessens friction resulting in the climb restarting before the bump has had a chance to fall all the way down into the "pit". The smaller the bumps and pits (smoother), the less friction there is as the amount of work needed to force the surfaces apart is less. Lubricants might both provide smaller molecules to fill the pits and/or take care of some of the Coulomb force by having less between its own molecules than between the two surfaces. Anyway, that is how I always thought about it.
    Disclaimer: I do not declare myself to be an expert on ANY subject. If I state something as fact that is obviously wrong, please don't hesitate to correct me. I welcome such corrections in an attempt to be as truthful and accurate as possible.

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    Quote Originally Posted by KALSTER
    As I have it, one aspect of the difference between static and kinetic friction is the amount of work that has to be done to lift the "bumps" out of the "pits". When movement gets going, momentum lessens friction resulting in the climb restarting before the bump has had a chance to fall all the way down into the "pit". The smaller the bumps and pits (smoother), the less friction there is as the amount of work needed to force the surfaces apart is less. Lubricants might both provide smaller molecules to fill the pits and/or take care of some of the Coulomb force by having less between its own molecules than between the two surfaces. Anyway, that is how I always thought about it.
    No one is claiming that we lack a heuristic understanding of friction. I think the only claim being made is that we can't make precise mathematical claims about friction based on a simple model, the way we can for, say, the electrodynamics of point charges.
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    Yeah, sorry, there were some other posts made while I was writing mine. I understand. Sorry for continuing off-topic Arcane.
    Disclaimer: I do not declare myself to be an expert on ANY subject. If I state something as fact that is obviously wrong, please don't hesitate to correct me. I welcome such corrections in an attempt to be as truthful and accurate as possible.

    "Gullibility kills" - Carl Sagan
    "All people know the same truth. Our lives consist of how we chose to distort it." - Harry Block
    "It is the mark of an educated mind to be able to entertain a thought without accepting it." - Aristotle
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    No worries.
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    Back on topic... It's the lack of understanding of the intricacies of friction that makes it outside the scope of Lagrangian Mechanics, right?
    Wise men speak because they have something to say; Fools, because they have to say something.
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    Quote Originally Posted by Arcane_Mathematician
    Back on topic... It's the lack of understanding of the intricacies of friction that makes it outside the scope of Lagrangian Mechanics, right?
    It is the inability, in the presence of friction to describe the potential energy succinctly. It is not simply a function of position. You are now dealing with a system that is not conservative, due to the dissipation of energy as heat through the mechanism of friction.

    Don't get sidetracked by this fact. It is simply meant to illustrate that you should choose a tool appropriate to the problem, and Lagrangian mechanics is not the right tool for all problems. But it is an excellent tool for many problems. Understand it and understand the limitations in the macroscopic world.

    As salsaonline pointed out Lagrangian and Hamiltonian mechanics is fundamental for much, if not all, of modern physics. It is important stuff. Just not a panacea.

    The other point of acknowledging the issue of friction is to point out that there is a lot of physics that would remain to be done even if a "theory of everything" unifying the four fundamental forces were to be formulated. Friction is just one topic that would still be available for a fundamental, quantitative, accurate explanation. The "everything" in "theory of everything" omits a few things.
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    Let's be honest though: Lagrangians, Hamiltonians, and Quantum Field Theory is sexy stuff. Yes, okay, fluid dynamics needs to be studied more. But can't blame people for getting distracted by the hot chick in the room.
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    So, the application of Lagrangian and Hamiltonian mechanics is more useful in the "perfect world" scenario. Or, I suppose, to a reasonable degree of accuracy in models where friction can be ignored. And, where forces are either grossly calculated or unknown, when you know the energy of the system Lagrange comes in quite handy.
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    There's nothing in Newtonian mechanics that can't be handled by Lagrangian mechanics. Sometimes one or the other perspective is more convenient, that's all.

    Friction is easily handled in the Lagrangian formulation by introducing an explicit time dependence into the potential energy part.
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    For arcane, and anyone else interested in Lagrangian mechanics: Here is an elementary discussion given by David Tong:

    http://www.damtp.cam.ac.uk/user/tong/concepts.html

    I've downloaded his lecture notes on other things and have found them to be pretty good. This discussion is aimed at beginning college students, so should be pretty accessible.
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    Thank you for the link, very helpful and appears (haven't started in on it yet, I have a nice slate of time to myself tomorrow) to be very coherent for someone at my understanding level (more rudimentary than I thought it was...).
    Wise men speak because they have something to say; Fools, because they have to say something.
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  24. #23  
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    Quote Originally Posted by salsaonline
    There's nothing in Newtonian mechanics that can't be handled by Lagrangian mechanics. Sometimes one or the other perspective is more convenient, that's all.

    Friction is easily handled in the Lagrangian formulation by introducing an explicit time dependence into the potential energy part.
    Ok, I'll bite. I'm not saying that you are wrong, but how would you do this problem using Lagrangian mechanics ?

    Given a mass on a plane inclined at an angle with the horizontal, with a coefficient of static friction , find the maximum value of for which the mass does not slide down the plane.

    It is a pretty simple Newtonian calculation to show that .

    For simplicity, I would be happy with the case .
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    I haven't worked this out, but I think this would work:

    Solve the Euler-Lagrange equations subject to the constraint that the particle does not move at all. The associated Lagrange multipliers will give you the forces of constraint. One of these will be the normal force, the other will be the force of friction. You should get that the ratio F/N = tangent theta.
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    Quote Originally Posted by salsaonline
    I haven't worked this out, but I think this would work:

    Solve the Euler-Lagrange equations subject to the constraint that the particle does not move at all. The associated Lagrange multipliers will give you the forces of constraint. One of these will be the normal force, the other will be the force of friction. You should get that the ratio F/N = tangent theta.
    That works out just as you projected. But it is not entirely satisfying.

    It requires the problem to be set up as clearly 2-dimensional when the natural approach would be to work with an "x" coordinate alone along the plane. The x constraint gives the frictional force. The normal "y" coordinate gives t he normal force, and nothing else. pertinent to the problem, so you need non-trivial insight to tell you that something is important there and to go look. You do indeed find that the ratio of the forces corresponding to the Lagrange multipliers is . Beyond that one would need some insight to relate that to a "coefficient of friction", which is probably not too surprising since the concept of a coefficient of friction is ad hoc in the first place. In fact the coefficient of friction does not enter into the problem formulation in any way. I do not find these "problems" particularly troubling.

    However, with regard to the general issue of handling friction within the Lagrangian framework, I am still a bit skeptical. I went back and reviewed the development in Goldtsein's Classical Mechanics. He develops Lagranges's equations using D'Alembert's principle of virtual work, starting from Newtoian mechanics. It is fairly easty to show that Newton's laws follow from Lagrangian mechanics, so it is quite clear that the two are equivalent (no surpise). But he is rather coy about the question of friction.

    Here is what I have concluded from that small bit of review.

    The normal formulation of Lagrangian mechanics is dependent on potential energy arising from a "generalized potential". This is not quite as restrictive as a true potential function. It does allow one to, for instance, formulate electrodynamics in a Lagrangian setting. It also permits certain models of friction -- in particular Rayleigh dissipation functions. But it does not appear that it would be tolerant of ordinary Coulomb kinetic friction models -- the opposing force being independent of velocity and proportional to normal force.

    There does appear to be a means for formulating mechanics via a variational principle even in the face of dissipative processes and non-holonimic constraints. When that is done the Lagrangian (T-U) is replaced by a new function, T+W where W is the work done by the system one the environment. However, since W does not arise from a generalized potential it is not at all clear how one can determine W analytically without resorting to a Newtonian force model. In complicated cases, say a bead sliding on a curved wire, it is not dlear how one could determine W even with a Newtonian force model, short of a numerical simulation. I don't see an elegant solution with either approach.

    Lanczos (The Variational Principles of Mechanics is a bit more direct -- "Forces of a frictional nature, which have no work function, are outside the realm of variational principles, while the Newtonian scheme has no difficulty in including them." He is probably a bit too dogmatic in this statement and too optimistic regarding Newtonian mechanics, but I think he is close, and close enough for macroscopic problems of the type typically encountered in engineering.

    Friction is not a problem for most of theoretical physics (outside of attempts to formulate a basic theoretical understanding of friction itself), since it is not a basic physical process or force. Frictional models are ad hoc and completely empirical. Hence they do not impede understanding of the foundations of physics via variational methods. I am not in the least bit surprised at the difficulty in handling ad hoc ideas within the context of a theory that is really directed at the foundations.

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    I'll have to think about it more. Anyway, thanks for the added perspective.
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