Violation of Determinism in Newtonian Mechanics by J Norton

[zwierz]

http://www.pitt.edu/~jdnorton/Goodies/Dome/

I think that most wonderful point in this story is that the person who writes such texts is Distinguished Professor of University of Pittsburgh.

Nevertheless I believe that the question he stated up can confuse an undergraduate student and thus deserves to be discussed at classes.

Any opinions?

 

[mfb]

The link is broken.
I guess this is about the shape where the object can roll down, but doesn't have to, and the time-reversed system behaves differently?

 

[zwierz]

The link is broken.

strange, I opened it moment ago

 

[stevendaryl]

It works for me. It's very interesting.

 

[mfb]

Now it works for me as well.
Yeah, that's the system I had in mind. A nice setup to discuss, as it has a surprising result.Quantum mechanics doesn't have that problem.

 

[zwierz]

I do not see any problems. First of all you can not place a particle exactly at the top of the dome with exactly zero velocity. So that an effect the article is devoted to is not observable.
On the other hand, everybody knows that there are physical systems and there are mathematical models of those systems. Mathematical models can be correct and can be incorrect. In the last case one should throw off incorrect model and construct a correct one. For example if we take in respect a dry friction between the dome and the particle then all these Mr. Norton's "spontaneous motions" disappear. Moreover, $r^{3/2}$ is also just a mathematical approximation of the real dome. We can approximate the dome by a polynomial , this also removes the "spontaneous motions"

 

[mfb]

Well, it is obvious that real-life systems won't be perfect, but this is a study of Newtonian mechanics and the theoretical implications of the theory, not its real-life approximations.

 

[zwierz]

Well, it is obvious that real-life systems won't be perfect, but this is a study of Newtonian mechanics and the theoretical implications of the theory, not its real-life approximations.

If that is not about real-life systems then why do we refer to the dome and the particle? Let's just write down any second order equation and say that it is Newton's second law. From pure mathematical viewpoint non-uniqueness in non-Lipschitz ODE is completely trivial effect.

 

[stevendaryl]

I do not see any problems. First of all you can not place a particle exactly at the top of the dome with exactly zero velocity. So that an effect the article is devoted to is not observable.

Ah! So if you assume a distribution of values for the initial velocity and initial position, instead of both being exactly zero, then you would find that for almost all initial conditions, the particle will slide down the hill. So the nondeterminism is a weirdness that only applies to a set of measure zero.

Mathematical models can be correct and can be incorrect. In the last case one should throw off incorrect model and construct a correct one. For example if we take in respect a dry friction between the dome and the particle then all these Mr. Norton's "spontaneous motions" disappear. Moreover, r3/2" role="presentation">r3/2 is also just a mathematical approximation of the real dome. We can approximate the dome by a polynomial , this also removes the "spontaneous motions"

Yeah, it's interesting that a lot of paradoxes are resolved by making the conditions more realistic. As if nature abhors a paradox.

 

[zwierz]

Yeah, it's interesting that a lot of paradoxes are resolved by making the conditions more realistic.

not only paradoxes. For example the famous Navier-Stokes Eq. problem http://www.claymath.org/sites/default/files/navierstokes.pdf turns into a simple one if we decline the condition of incompressibility and consider a model that takes in respect thermodynamics