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Formalism of Newtonian Mechanics

  1. Sep 22, 2006 #1
    Hi,

    I was wondering, how would one formulate Newtonian Mechanics as a rigorous mathematical model? Would one take force to be an external quantity defined by various force laws (Coulomb, UG) and accept Newton's Second Law of Motion as an axiom (and first law as a definition of inertial frames)? Or would one take the first and second law to be definitons of force (in whitch case it seems to me the concept of inertial frame becomes impossible to define)? Also, would one take mass to be externally defined?

    It seems to me both of the approaches, which are mathematically incompatiable, are used in textbooks. What's worse, they are intermixed according to convenience, confusing me about inertial reference frames.

    While it seems that later formulations of classical mechanics (Lagrangian, Hamiltonian, Hamilton-Jacobi) are introduced in a rigorous manner, the structure of Newtonian Mechanics is never presented in any detail. I am a fanatic for rigour, proofs and highly abstract and formal statements (even though I prefer physics over mathematics any day). For me this state of ignorance is rather disturbing. Can you help me out?

    Thanks.

    Molu
     
  2. jcsd
  3. Sep 22, 2006 #2

    radou

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    You don't have to be alarmed about that, since Newtonian mechanics in its 'most innocent' form is taught for a good reason before something more, as stated - rigorous, such as Lagrangian, Hamiltonian etc.
     
  4. Sep 23, 2006 #3
    Do you mean it is not possible to give a rigorous foundation to newtonian mechanics?
     
  5. Sep 26, 2006 #4
    Well? Should Newton's laws be treated as definitions or laws?
     
  6. Sep 26, 2006 #5

    radou

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    There are definitions in mathematics, not in physics. At least not in this context. So, Newton's laws are laws. :smile:

    You are probably refering to classical mechanics in general - which means non-relativistic or non-quantum mechanics. Maybe it's best to say that Newtonian mechanics means classical mechanics at an earlier stage of development. Classical mechanics, including the Lagrangian and Hamiltonian approach, does not include a 'force' in such an explicit way as Newtonian mechanics does; it's more, let's say, based on an energetic approach, as far as I got it.
     
    Last edited: Sep 26, 2006
  7. Sep 26, 2006 #6

    ZapperZ

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    Why are we so concerned about giving labels to these things? This is as nonsensical as calling Pluto as planet or not. It make NO difference! Call it a monkey if it pleases you. It changes nothing.

    Zz.
     
  8. Sep 27, 2006 #7
    Whether or not we call Pluto a planet is entirely a linguistical issue, but whether or not N2L is a definiton is not at all so. In any theory, one starts with certain definitions, certain axioms and certain allowed logical manipulations and then proceeds to derive theorems by applying these logical transformations on the axioms.

    If N2L is a definition of what we call force, then we can not say that the centrifugal force is a non-existent or pseudoforce. It produces acceleration in one reference frame, therefore it must be a force. There can be no fundamental distinction between an inertial and a non-inertial frame. If however we take N2L to be a law dealing with forces defined via external force laws, then we can say centrifugal force is non-existent since there are no physical interactions that can give rise to this force.

    It is not true that there are no definitions in physics.
     
  9. Sep 27, 2006 #8

    ZapperZ

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    OK, since you care that much about definitions, tell me what is a "law", and how is this different than a "theory"? Can a theory somehow "graduate" into a law? Where has this happened in physics?

    There ARE definitions in physics, but these definitions are based on mathematical formalism, rather than callling it with labels such as "law", "axioms", etc. There is no ambiguity when you concentrate on what matters the most - the mathematical formalism. It becomes an exercise in futility like this if you only care about semantic labels!

    If you are bored and have nothing to do, look this up: figure out the conservation laws for energy and momentum and why they are important via the Noether theorem. Then see why such conservation laws results in ALL of the classical mechanics that you are familiar with.

    Zz.
     
  10. Sep 28, 2006 #9
    The mathematical formalism is what I was talking about, not linguistics. In mathematics there is a clear distinguition between definitions and axioms. In the formalism of newtonian mechanics do we take force to be defined via N2L or do we treat it as an already defined quantity that is related to the already defined concepts of motion via the N2L?
     
  11. Sep 28, 2006 #10
    Hi Loom91,

    You and I are of like mind. I am keenly interested in formalizing physics. I participate in this groups because they force me to sharpen my communication and reasoning skills.

    You would like Bertrand Russells 'Foundations of Mathematics'. He discusses such logical issues as the tautological definition of force (F=MA) and the nearly religious fanaticism with which people square velocity (i.e. kinetic energy!)

    Of course, I just got this thread banned to the philosophy channel, but I found a kindred spirit!
     
  12. Sep 28, 2006 #11

    radou

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    What religious fanaticism? There is a reason for which there is a 'squared velocity' term in the experssion for kinetic energy.
     
  13. Sep 28, 2006 #12
    Yes, it eliminates time as a parameter. No argument from me. Sometimes that is useful. But in my personal opinion, I think a little too much is made of the concept.
     
  14. Sep 28, 2006 #13

    radou

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    How does it eliminate time?
     
  15. Sep 28, 2006 #14
    Energy is always conserved. If you know the location of an object, you know the velocity of the object. But you don't know when.

    ax = .5 v*v
     
  16. Sep 28, 2006 #15

    radou

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    Velocity is time dependent.
     
    Last edited: Sep 28, 2006
  17. Sep 28, 2006 #16

    ZapperZ

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    Are you doing this as a HISTORY project, or do you actually want to know start-of-the-art knowledge on these things? For example, if I tell you that practically everything that you encounter in first year Intro Physics is nothing more than various manifestation of conservation of energy (or mass+energy) and conservation of momentum (both linear and angular), would you believe that? Would such a statement, which I have made before when I briefly taught undergrad physics, would cause you to realize that everything you see is a consequence of those two principles?

    And would you be able to connect, via Noether's theorem, on why those two conservation laws are merely reflections on two different symmetry principles that we observe for the world we live in?

    So yes, I have already given you more than enough to show you why Newton's laws, for example, are simply manifestation of more fundamental descriptions of the universe that we know. This directly addresses your very first post.

    Zz.
     
  18. Sep 29, 2006 #17
    That's right. But in the Energy picture, you sacrifice that information and replace velocity as a function of time with velocity as a function of space.
     
  19. Sep 30, 2006 #18

    samalkhaiat

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  20. Sep 30, 2006 #19
    Try V.I. Arnold's book on Mathematical Methods of Classical Mechanics.

    Although I have ask, you mentioned that the Lagrangian version seemed more rigorous. Why is this?

    You should also consider what ZapperZ is saying (if somewhat without tact). Mechanics is largely the result of symmetries.

    However, if you like you devle into logic and model theory and all kinds of stuff I've only read about that gets into the nitty gritty aspects of what makes a theory and why the theory of classical mechanics is fundamentally different than the theory of quantum mechanics, and from a logical point of view not merely the obvious physical differences.

    Cheers,

    Kevin
     
  21. Oct 9, 2006 #20
    I know all of that and am quite aware of Noether's theorem and we have to prove Newton's laws from conservation principles, but this still does not answer my question: Are Newton's laws definitions or postulates?
     
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