- #281
vanesch
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Tomaz Kristan said:Seeliger's paradox is a long known one. Infinite mass, distributed over the infinite space.
I also had finite mass...
Tomaz Kristan said:Seeliger's paradox is a long known one. Infinite mass, distributed over the infinite space.
Wrong. It doesn't have a mass center.Tomaz Kristan said:The Hurkyl example is of a finite mass, distributed over the infinite space, but it's not a paradox (yet). Only the mass center has no finite distance form 0.
Why?There is NOTHING like that at all. Everything should work for EVERY finite mass, distributed anyway you want inside a finite amount of space. No doubt about that.
And why, pray dell, do you think you are above such considerations? Why don't you show us the law which permits constructions like yours?Or please, show me the law, which forbids some constructions, like mine.
He did? His famous theorems don't assert such a thing.reilly said:Note Godel says that paradoxes cannot be avoided.
Hurkyl said:Other versions of classical mechanics might permit your configuration, but not insist that the center of mass theorem applies to it.
Tomaz Kristan said:Take a pyramid instead of my initial construct! Then you can imagine, how its points migrate (under system's internal forces only) the way, the initial construct grew.
Morphing of a kind.
Had the construct was immune to the momentum conservation theorem, the pyramid would also be!
What I have nothing heard about.
See now?
StatusX said:If at this point we impose the following axiom (strictly stronger than (3)):
4. [tex]\sum_{\alpha} \sum_{\beta} ||f_{\alpha \beta}|| < \infty[/tex]
Then the sum above converges absolutely, so we can rearrange it so that [itex]f_{\alpha \beta}[/itex] sits next to [itex]f_{\beta\alpha}[/itex], these will cancel by (2), and we have proven conservation of momentum in the model (1,2,4).
However, the system in post 1 does not satisfy (4), so does not fit into a model where conservation of momentum holds.
StatusX said:However, the system in post 1 does not satisfy (4), so does not fit into a model where conservation of momentum holds.
Tomaz Kristan said:A pyramid with the uniform density
vanesch said:The piramid you constructed was not of uniform density, no ? Its density diverged at its vertex ?
StatusX said:Tomaz, what is this pyramid you're talking about?
Tomaz Kristan said:A pyramid with the uniform density would satisfy (4) clearly, and the momentum would be conserved.
Internal forces in the pyramid could rearrange it to become the original construct.
Therefore, the conservation of momentum does not hold for pyramid!?
and THAT is (another form of this) paradox.
StatusX said:Do you have a specific process in mind that would transform a pyramid into the shape from post 1?
Tomaz Kristan said:Then each of those pieces is condensed to a mass point in appropriate places on the pyramid's axe.
Tomaz Kristan said:Looks like not. And that there _is_ an antinomy after all. The conservation of momentum has a weak point.
Then how can you assert that people have been practicing classical mechanics wrongly? :tongue:Tomaz Kristan said:In practice, there may be no problem.