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Why is there a consistent theory of continuum mechanics in which one just consider things like differential elements and apply Newtons laws? Is there a deeper reason for it. Is it the nature of newtonian framework that makes it happen or is it somehow related to nature of bodies (topological spaces with borel measure etc)?

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What do you mean why? Why not? –  Marek Feb 21 '11 at 9:10
    
Newtons laws are stated for objects called particles.Now they use the same laws to the objects they call as differential elements and describe the dynamics of entire body.What makes such description possible?..is my question –  Ket Feb 21 '11 at 11:54
    
actually, Newton laws only make sense for extended objects that appear smooth macroscopically. It's possible to treat those as particles in certain cases because you can usually separate center of mass movement and treat the rest as a solid body. E.g. one often can treat planets as point masses. In other words, it's approximations all the way down to quantum turtles. –  Marek Feb 21 '11 at 19:58

3 Answers 3

What evidence do you have that there is a consistent theory of continuum mechanics? Certainly, when looked at through a macroscope, the universe looks like it behaves according to continuum mechanics, but this completely breaks down on the microscopic level. So you can't justify a consistent theory of continuum mechanics by using the universe. There's no reason that good approximations to our universe have to be consistent theories of physics; physicists currently believe that QED is a very accurate approximation to the electromagnetic force, but that it cannot be made consistent at the smallest scales without adding additional physics. And there are papers showing that if you just define your physics using, say, the wave equation without putting some kind of restrictions on the initial conditions, very funny things can happen. Also, Newton's laws of gravitation with point particles have some very unpleasant consequences. So I would say that, unless you're very clever about how you specify it, a theory of continuum mechanics would very likely not be consistent.

If you are very clever, you might be able to make a consistent theory of continuum mechanics, but I don't know of anybody who has actually done this.

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Where can I get one of these "macroscopes", I wonder? –  Tim Goodman Feb 21 '11 at 17:34
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macroscope made my day :D –  Marek Feb 21 '11 at 19:54
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That's a curious answer: certainly we have many consistent theories, quantum and classical, whose basic elements are fields. For just one example - any asymptotically free gauge theory is a consistent and complete quantum mechanical system with no need for completion at short distances. Some people may even say those are the only consistent theories - particles are a problematic concepts unless they are regarded as excitations of some field (infinite self-force, problematic coupling to gravity and other fields, etc.). Not that I know that any of this is relevant to this question... –  user566 Feb 22 '11 at 9:57
    
What an awesome and fascinating link - and a real world example too. Thanks –  WetSavannaAnimal aka Rod Vance Jul 25 '13 at 4:27

I assume you're referring to condensed matter materials. The continuum limit is only an approximation which breaks down at the atomic/molecular scale. However, for materials many many orders of magnitude larger than the interatomic scale, we may use the continuum limit approximation if the system is in quasiequilibrium thermodynamically.

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I guess you are talking about the Banach-Tarski paradox, which is not really an inconsistency in mathematics, it is a result following from the axiom of choice, I dont think only Newtons laws implies the negation of the axiom of choice, so I think Banach-Tarski vs mass conservation is resolved (ie can not appear), by the ways of which laws such as conservation of mass is applied to continuum mechanics.

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Uh, what? B-T is based on existence of non-measurable sets. What this has to do with Newtonian mechanics and conservation of mass is completely beyond me... –  Marek Feb 21 '11 at 9:15
    
@Marek If you could double the volume of a sphere in continuum mechanics, you would violate conservation of mass, I interpret the qustion as asking exactly what in CM prohibits that. If you only start with Newton's laws and some simple definition of continuum you could get B-T paradox. –  user1708 Feb 21 '11 at 9:23
    
Yes, but those constructions are based on non-measurable sets which are obviously unphysical. But I give up, I honestly don't understand what this question and answer has to do with physics... –  Marek Feb 21 '11 at 9:44
    
I don't think I understand what this question is asking. But regarding the Banach-Tarski paradox, continuous matter is just an approximation to the real world. We shouldn't be surprised to get unphysical results if we pretend this approximation holds to arbitrary scales. –  Tim Goodman Feb 21 '11 at 11:53
    
I was not talking about B-T.Physical objects I assume are measurable sets. –  Ket Feb 21 '11 at 12:02

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