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Primacy of conservation laws

  1. Nov 9, 2012 #1

    I am hoping for some input concerning shall we say the philosophy of mechanics, particularly with relation to the concept of force.

    Most text books treat force as being something of a primary concept. A fundamental aspect of nature much like energy and mass and therefore something which cannot be readily defined, merely related to other concepts by rote application of Newton's laws.

    To me it's not really. To me the conservation laws, and the quantities to which they refer, are the more sensible choice of foundation upon which to lay a conceptualisation of reality.

    The explanation I have sketched out of force, then, is roughly as follows

    - certain quantities are found by experiment to be conserved. For now, mass-energy, momentum and charge will do
    - conserved quantities are really useful because we can think of them as being transferable, and transferable with quantifiable amounts and rates
    - force is just a name for the rate of transfer of momentum
    - "a" force is an influence upon the rate of transfer of momentum

    To me this

    - is simple
    - makes Newton's laws seem like stating the obvious (2nd by definition, 1st=special case of 2nd, 3rd by definition as every transfer has a source and a sink)
    - explains jet force in a way that is much more direct than appeal to Newton's 3rd, and reduces jet "force" to just a way of talking about straightforward momentum transfer
    - is consistent with the standard model of particle physics and more broadly (by conservation of 4-momentum) with general relativity

    To me the primacy of force is a long-obsolete hangover from the days when "contact force" was still considered a viable proposition. Recasting contact forces in terms of the Colomb interaction to me kind of forces the issue towards the above conceptualisation.

    My only quibble is I'm kind of making this up as I go, wilfully ignorant of accepted views of the matter. I have not heard of such a conceptualisation clearly stated from any other source, although to me it seems implicit in the Standard Model -- or perhaps explicit, if you read the right books? But then again I've just picked up Feynman Lectures on Physics Vol 1, which includes a chapter on Characteristics of Force. Feynman -- who, of all people, you might expect to view force as nothing more than a transfer of momentum by gauge bosons or otherwise -- says nothing of the sort and just really seems to throw up his hands and say it's all too complicated(!)

    So I guess I would like to know whether a) this view makes any sense and b) where are some good places to start reading about this approach as used by smarter people than me?
  2. jcsd
  3. Nov 9, 2012 #2


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    Your reasoning is essentially correct, and that kind of thinking is part of what got us from Newtonian Mechanics to more advanced physics. If you look at Lagrangian Mechanics, Classical Field Mechanics, or Quantum Field Theory, you deal primarily with energy and momentum - the conserved quantities. Forces can be extracted from solution if you need them, but the concept of a "force" is not directly used in solving a problem.

    If you look deeper, into things like Quantum Electrodynamics and Quantum Chromodynamics, you see that even conservation laws aren't fundamental. What's fundamental are certain symmetries. From each fundamental symmetry arises a conserved current and gauge fields that carry forces. So you end up with both conservation laws and forces as a consequence of symmetries. The mathematical formulation of this comes from Noether's Theorem.

    In classical physics, invariance of the laws of physics under translation gives you conservation of momentum. Under time - energy. Under rotation - angular momentum. Since only global symmetries are considered, you don't really get a gauge field.

    In Quantum Electrodynamics, the U(1) local symmetry gives you both the conservation of electrical charge and the electromagnetic forces that couple to this charge.
  4. Nov 9, 2012 #3
    If you conceptualize force as rate of transfer of momentum then you run into difficulty in scenarios where force is present but no transfer of momentum is taking place. An example would be leaning against a wall. There is a force between me and the wall, and an equal and opposite force between my shoes and the floor. The net force on me, as a whole, is zero but the forces are not applied in the same place which creates strain on both me and the floor/wall.
  5. Nov 9, 2012 #4


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    And there is a corresponding momentum flow as a result. You->wall->floor->you.
  6. Nov 9, 2012 #5
    Would you like to put up some maths to back up this proposal?
  7. Nov 9, 2012 #6
    Lovely -- I started to draft a reply but got bogged down in detail. You've said it in a nutshell.

    Balanced forces are balanced momentum flows. As with flows of mass or charge, you can have flows occurring without any net accumulation or depletion in any body.

    As for the strain - it's essentially Hooke's Law. The (directed) rate of momentum flow by the Coulomb interaction between molecules depends on their separation. Their equilibrium separation (strain) therefore depends on the (directed) rate of momentum flow they are required to sustain (tension or compression).

    Thanks K^2 for your earlier post, too -- yet more impetus for me to get around to learning about symmetry in physics. Nice to know I'm intuitively on the right track (if only in a tiny way) Any suggestions for introductory books or materials, suited to a time-poor person, would be gratefully received!
    Last edited: Nov 9, 2012
  8. Nov 9, 2012 #7
    True, but modeling strain in terms of momentum flow is awkward. At least it is to me. Modeling inelastic collisions in terms of force would be equally awkward. It's best not to get stuck on one idea. Be familiar with a wide variety of conceptual tools so you can pick the best one for any given job.
  9. Nov 9, 2012 #8


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    So dp/dt = F isn't enough? I understand that you might be tempted to point out that dp/dt = ƩF, and ƩF=0. But do keep in mind that d/dt is a linear operator, and we need to decompose it if we want to look at individual flows rather than the net change.

    Or do you want me to go deeper and write out some Feynman diagrams for a virtual photon exchange, showing that any application of force results in momentum flow?

    Or do you need me to go a step deeper and derive relation between stress tensor and momentum flow? If you want me to be creative, I can do that using the fact that stress tensor is a conserved quantity under coordinate transformation, while angular momentum is a conserved quantity under rotation specifically.

    When you are asking about something so fundamental as momentum flow due to an interaction, I really don't know what sort of level of math you are looking for.
  10. Nov 10, 2012 #9
    Well, I don't think you need go as far as a person leaning.

    Just a 1Kg mass sitting on a table in a gravitational field will do, although the analysis presumably also applies to any body force.

    And granted that the table initially deforms minutely when first placed, but considering only the subsequent steady state.

    And please look again at the title of the subforum and ask yourself if at least some of your responses would be more appropriate if this were one of the more advanced physics subforums.
    Last edited: Nov 10, 2012
  11. Nov 10, 2012 #10


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    Valid point on section. I'll stick to classical.

    For a mass supported by a normal force in a gravitational field, again, you are looking at two flows. Body receives upwards momentum to normal force and looses it to gravitational force. That lost momentum is transferred to Earth via gravity and is returned back via forces of deformation into the support surface. There is still a net flow, even if you consider this classically.
  12. Nov 10, 2012 #11
    Are you implying that there can only be continuous deformation without limit?

    I fail to see what this extra layer of complication adds.
  13. Nov 10, 2012 #12
    It simplifies, not complicates.

    Define momentum, posit that it is conserved, and that's all you need. Newton's laws become 1) stating the obvious, 2) a redundant definition and 3) stating the obvious.

    Where it doesn't seem to help, just substitute the word "force" where you see "flow of momentum". They mean the same thing. The word "force", while redundant, is still a useful shorthand for sources of momentum flow, since many problems can be analysed in terms of forces (discrete sources of momentum flow) where an expression can be found relating the size and direction of force (i.e. rate and direction of flow) to other quantities.

    It's also more than just a simplifying framework. It says that momentum transfer is primary, and force is just a name for it. Compare to the usual Newtonian view that force is primary and that force causes the transfer of momentum. You all know a lot more about modern physics than I do, but I figure this is closer to the current understanding ... isn't that why we now speak of "fundamental interactions" and not "fundamental forces"?
  14. Nov 10, 2012 #13


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    It resolves this "difficulty". If there is an interaction force, there is a transfer of momentum taking place. Just like net force can be zero, net transfer can be zero.

    Naturally, if the net transfer is zero, we don't need to actually worry about each individual interaction. And that's precisely how it would be formulated in, say, Hamiltonian mechanics. [itex]\dot{p}=\{p,H\}[/itex]. No need for concept of force as such. And yeah, no need for complication of tracking momentum transfer due to each interaction individually.
  15. Nov 10, 2012 #14
    Thank you both for telling me that at some underlying and fundamental level that the use of a self consistent system of mechanics, developed over several centuries, is either somehow wrong or can be simplified.

    There are other subforums here for such discussion.

    This subforum is all about the classical system I just described.

    I do not know of any text on structural mechanics or bridge design that follows this modern route to success.

    In fact I cannot recall any bridge design text using the word momentum at all.

    Many fluid mechanics texts use the phrases "destruction of horizontal momentum" and "appearance of vertical momentum" to descibe what happens when a flowing fluid is directed at a wall.

    I ask for a mathematical description, equivalent to the classical analysis, using the modern theory to compare and see if it is actually simpler and easier, as claimed.
  16. Nov 10, 2012 #15
    I don't understand your "not readily defined". In classical mechanics these things except for energy are (were?) rather well defined, with balances, scales and rulers - thus there were not so long ago the kg (for mass) and the kgf (the force by one kg on Earth). Perhaps that was the reason for the primacy of those in classical mechanics. "Energy" is a hard to measure concept, it started out as a bookkeeping value.
    :bugeye: Momentum ("quantity of motion"): p=mv
    "The Quantity of Motion is the measure of the same, arising from the velocity and quantity of matter conjuctly."
    In classical mechanics, stationary v=0 => p=0. Or if in inertial motion, p=constant.
    No "momentum flow" in classical mechanics: it is a conserved quantity, not a fluid.
    Not force in general, but what Newton called "impressed motive force" was defined just as you propose to do:

    F=dp/dt "The alteration of motion is ever proportional to the motive force impressed".

    Of course, the conservation laws that were developed from classical mechanics played an increasingly important role in history; indeed one can turn things around in a convenient way, based on modern knowledge.
    Please elaborate: how does the Coulomb interaction affect Hooke's law?
    I don't have that book at hand, it will be interesting to look up. :smile:

    And I did not look into that myself more than standard textbooks, but perhaps this is a good starter ("what else"!):
    Last edited: Nov 10, 2012
  17. Nov 10, 2012 #16


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    Lagrangian and Hamiltonian Mechanics are topics in Classical Mechanics. I'm not sure what your complaint is.

    You are trying to artificially limit discussion to a static case. First of all, yes, any structural mechanics problem can be solved using Lagrange Multipliers without talking about forces. Of course, what you are actually analyzing is stress, so you have no choice but to involve forces at some point, and you might as well start balancing forces from the beginning.

    Dynamics problems, however, are greatly simplified by use of Lagrangian and Hamiltonian Mechanics in generalized coordinates. That's kind of why you usually learn them in a Classical Mechanics course.

    But hey, if you want Lagrangian analysis of a mass supported by the floor, here it is.

    Lagrangian and constraint.
    [tex]L = \frac{1}{2}m\dot{y}^2 - mgy + \lambda f(y)[/tex]
    [tex]f(y) = y-H = 0[/tex]

    Equations of motion.
    [tex]\frac{\partial L}{\partial y} - \frac{d}{dt}\frac{\partial L}{\partial \dot{y}} = \lambda - mg - m\ddot{y} = 0[/tex]
    [tex]\ddot{y} = 0[/tex]

    [tex]\lambda = mg[/tex]

    Why is this simpler? Because I did not have to stop and ask what forces are acting on the body and what I need to do to have them add up to zero. Sure, with one body it's easier to just balance forces. What if you have a dozen different bodies with different constraints between them? Still feel like setting up all of these equations? In contrast, I can just write down the Lagrangian, write down the list of constraints, and just feed the whole mess into Mathematica to be solved. This method completely eliminates the need to go through each degree of freedom by hand.

    And by the way, this is how you solve a problem with constraints. That is the standard approach used for the past 200 years. I don't know if that qualifies as "modern theory" in your books. Maybe you're still going by Principia as the only text on classical mechanics.
  18. Nov 10, 2012 #17
    :rofl: (sorry couldn't help it!)
  19. Nov 10, 2012 #18
    Is this a physics forum or an engineering forum? As stated in the OP, this is about the philosophy of mechanics. The change is of entirely no consequence when it comes to calculation. It's just a way of thinking about what you actually mean when you use the word "force". Do you say it "just is" or do you explain it as something that emerges from something else that "just is"?

    As we say where I come from ... "no need to get ya knickers in a knot!"
  20. Nov 10, 2012 #19
    Alteration of motion is momentum flow. If you object to the word "flow", that's why the third law is there. Whatever momentum is lost by one object is necessarily gained by another - it flows from one to the other any time a force is acting. I doubt Newton would dispute that. The issue is simply whether the force causes the flow, or the force is just a name for the flow.
  21. Nov 10, 2012 #20
    Is this the Science Advisor's reply to the following?

    @ russell2pi

    I clearly have a different definition of the principle of conservation of linear momentum than whatever you are thinking of.
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