3 ms·

>We know that exactly the same initial conditions will lead to the exact same trajectory. Do we have a proof of this? Or is it that we just assume this?

throwaway81523 5y ago
Yes we have a proof (in the classical case). It is called the Picard–Lindelöf theorem. The "dome" in your example is called an unstable equilibrium: the ball starts rolling because of an unpredictable perturbation. Small motions are amplified according to the Lyapunov exponent, which in the unstable case is large. Hirsch and Smale and Devaney's book on differential equations and dynamical systems is a good place to learn about this. In classical mechanics we model a zero-diameter point at the top of the dome, but a real physical dome would be made of molecules vibrating randomly and that starts the ball rolling. At the end of the day, classical mechanics is nonphysical: it doesn't describe nature except as an approximation. Quantum mechanics on the other hand is non-deterministic. Is there a deeper, so-called superdeterministic reality underneath quantum mechanics? Some people think yes, and it has not been disproved, but it is pretty far out of the mainstream from what I can tell.
- deltasixeight 5y ago
  No read the dome example again. You are wrong. Nortons dome is literally demonstrating there is proof for the non deterministic case. I've discussed this in other places but the theorem you reference only applies to functions that are Lipschitz continuous. Not all functions have this property globally and the dome and also the three body problem are two examples of things that are not Lipschitz continuous. Read it carefully. Nortons Dome is saying that the mathematical model described by newtons laws of classical mechanics is in itself non-deterministic. > Is there a deeper, so-called superdeterministic reality underneath quantum mechanics? Some people think yes, and it has not been disproved, but it is pretty far out of the mainstream from what I can tell. Never made this claim. Not even Einstein made this claim. Simply put, it felt wrong to Einstein simply because probability is a sort of bayesian outlook on the world. It's an admission that we lack knowledge about a system. Such is the case for much of science but not quantum mechanics? Hence the quote by Einstein: "God does not play dice with the universe." Either way not saying that quantum mechanics is crap and wrong but this is a possibility the mainstream definitely considers in a very speculative philosophical fashion. The quantum model works extraordinarily well but at the same time it's inconsistent with relativity and you gotta admit something is a bit off here when considered from the Bayesian angle.
  - throwaway81523 5y ago
    Oh I see what you mean, the dome is a little bit pointy at the top, so the first derivative is not continuous. I missed that at first. Sorry. Still though, this seems like a typical situation in physics where you get superposed solutions to a DE and then throw out the nonphysical ones. Someone else just created a new thread about the dome problem, and web search shows a vast literature about it (like there is for the Monty Hall problem). I guess you're already familiar with the literature so there's not much I can add. Superdeterminism isn't classical determinism, it's a specific idea in the interpretation of quantum mechanics. It's based on the idea that not only is it fully determined which box the particle ends up in, but it's also determined which box the experimenter will look in, and those forced choices conspire to make the experimenter think the particle is actually following the Born rule. At least that's my best understanding of it: physics isn't my thing. See: https://en.wikipedia.org/wiki/Superdeterminism https://en.wikipedia.org/wiki/Superdeterminism Besides the people mentioned in that article, I believe Gerard 't Hooft is an adherent. He has a bunch of articles on his site about a possible classical mechanism underneath QM. But, I think most physicists think that is unlikely. You might like John Baez's article "Struggles with the continuum", which is about various annoying singularities that come up in areas of physics including classical mechanics: https://math.ucr.edu/home//baez/continuum.pdf https://math.ucr.edu/home//baez/continuum.pdf

throwaway81523 5y ago

Yes we have a proof (in the classical case). It is called the Picard–Lindelöf theorem. The "dome" in your example is called an unstable equilibrium: the ball starts rolling because of an unpredictable perturbation. Small motions are amplified according to the Lyapunov exponent, which in the unstable case is large. Hirsch and Smale and Devaney's book on differential equations and dynamical systems is a good place to learn about this. In classical mechanics we model a zero-diameter point at the top of the dome, but a real physical dome would be made of molecules vibrating randomly and that starts the ball rolling. At the end of the day, classical mechanics is nonphysical: it doesn't describe nature except as an approximation. Quantum mechanics on the other hand is non-deterministic. Is there a deeper, so-called superdeterministic reality underneath quantum mechanics? Some people think yes, and it has not been disproved, but it is pretty far out of the mainstream from what I can tell.

deltasixeight 5y ago
No read the dome example again. You are wrong. Nortons dome is literally demonstrating there is proof for the non deterministic case. I've discussed this in other places but the theorem you reference only applies to functions that are Lipschitz continuous. Not all functions have this property globally and the dome and also the three body problem are two examples of things that are not Lipschitz continuous. Read it carefully. Nortons Dome is saying that the mathematical model described by newtons laws of classical mechanics is in itself non-deterministic. > Is there a deeper, so-called superdeterministic reality underneath quantum mechanics? Some people think yes, and it has not been disproved, but it is pretty far out of the mainstream from what I can tell. Never made this claim. Not even Einstein made this claim. Simply put, it felt wrong to Einstein simply because probability is a sort of bayesian outlook on the world. It's an admission that we lack knowledge about a system. Such is the case for much of science but not quantum mechanics? Hence the quote by Einstein: "God does not play dice with the universe." Either way not saying that quantum mechanics is crap and wrong but this is a possibility the mainstream definitely considers in a very speculative philosophical fashion. The quantum model works extraordinarily well but at the same time it's inconsistent with relativity and you gotta admit something is a bit off here when considered from the Bayesian angle.
- throwaway81523 5y ago
  Oh I see what you mean, the dome is a little bit pointy at the top, so the first derivative is not continuous. I missed that at first. Sorry. Still though, this seems like a typical situation in physics where you get superposed solutions to a DE and then throw out the nonphysical ones. Someone else just created a new thread about the dome problem, and web search shows a vast literature about it (like there is for the Monty Hall problem). I guess you're already familiar with the literature so there's not much I can add. Superdeterminism isn't classical determinism, it's a specific idea in the interpretation of quantum mechanics. It's based on the idea that not only is it fully determined which box the particle ends up in, but it's also determined which box the experimenter will look in, and those forced choices conspire to make the experimenter think the particle is actually following the Born rule. At least that's my best understanding of it: physics isn't my thing. See: https://en.wikipedia.org/wiki/Superdeterminism https://en.wikipedia.org/wiki/Superdeterminism Besides the people mentioned in that article, I believe Gerard 't Hooft is an adherent. He has a bunch of articles on his site about a possible classical mechanism underneath QM. But, I think most physicists think that is unlikely. You might like John Baez's article "Struggles with the continuum", which is about various annoying singularities that come up in areas of physics including classical mechanics: https://math.ucr.edu/home//baez/continuum.pdf https://math.ucr.edu/home//baez/continuum.pdf