4 ms·

This example assumes a physically impossible infinitely-sharp corner and an Zeno-paradoxical infinitely small jump from the initial position to some other posit

by gowld 4y ago

This example assumes a physically impossible infinitely-sharp corner and an Zeno-paradoxical infinitely small jump from the initial position to some other position after time T. It shows that Newtonian mechanics is only an approximation of the real world.

deleted 4y ago
[deleted]
courgette 4y ago
Thanks to pointing out the relevance
jfengel 4y ago
It requires a lot of not actually constructable elements, but that's not really the issue. The issue is that it's not reversible: a ball rolls up but never rolls down. If you play a tape of it backwards, it rolls off in a seemingly random direction, with no source of randomness and no initial kick of energy. It doesn't need an infinitely sharp corner. It's weird right at the very top. We do know that Newton is only an approximation, but general relativity doesn't seem to help here. Quantum mechanics might, but the problem itself doesn't seem to hint at it. It seems to nod in the direction of involving entropy at a deep level, but it's unclear how.
- naasking 4y ago
  Seems to me that this actually points out a fundamental inconsistency of classical theories, namely the use of infinitely divisible quantities (real numbers) which leads you to paradox. I think Norton's dome is unproblematic if you model it in a discrete theory.
  - fsh 4y ago
    As far as we know, time and space are continuous. Various aspects of quantum field theories would not work in a discretized universe.
    - naasking 4y ago
      > As far as we know, time and space are continuous. Our models are continuous. The map is not the territory. > Various aspects of quantum field theories would not work in a discretized universe. If you're referring to symmetries, that's not true [1]. Discretized theories have received barely any attention because physicists have simply built upon the formalisms they're used to from classical physics. We see people frequently saying that we should abandon classical intuitions now that we have quantum mechanics, but we haven't really even tried to abandon classical formalisms, which are just another form of classical intuition. IMO, baking continuity and uncountable infinities is what's responsible for many of the difficulties we've had, and various forms of discretization have resolved these issues in the past [2]. My conjecture is that one of the next big advances in physics will be to take discretization seriously. [1] https://arxiv.org/abs/1902.08997 https://arxiv.org/abs/1902.08997 [2] https://arxiv.org/abs/1609.01421 https://arxiv.org/abs/1609.01421
      - crazygringo 4y ago
        > Our models are continuous. The map is not the territory. Because all our evidence is that it is continuous. Our models are the best description of the evidence we observe -- i.e. the territory. Unless you have evidence of time and space being non-continuous?
        naasking 4y ago
        > Because all our evidence is that it is continuous. No, we haven't really tested this to any appreciable degree, and existing tests rule out only certain types of discrete models just like they rule out certain continuous models. Our models are continuous because continuous models are what we've been using for centuries. I'm not sure why anyone still believes that science is a purely evidence-driven endeavour. History has clearly shown it's subject to fads, celebrity and inertia. > Unless you have evidence of time and space being non-continuous? General relativity produces unphysical nonsense like singularities, which is a form of internal inconsistency similar to Norton's dome. This isn't evidence that spacetime is discrete so much as that our continuous model is incomplete, but it seems quite clear that a discrete analogue of GR would not suffer from this problem. Furthermore, some new approaches to gravity are discrete theories, like loop quantum gravity, so clearly some physicists are already thinking along these lines.
alecst 4y ago
You can have a sharp corner and you still won’t get a Norton’s dome in every case. There’s definitely something there, but it’s not obvious to me what it is. It does sometimes happen in physics that we throw away valid solutions to differential equations that are “physically impossible”, and I guess this falls in that category. Other examples are not immediately coming to mind unfortunately. But yea I don’t think this shows anything general about Newtonian mechanics.
- onos 4y ago
  Maybe you’re thinking about throwing out exponential growth in differential equations, where only decay makes sense physically. Eg in quantum well analysis this is done cause the former would give a wave that is not normalizeable.
crazygringo 4y ago
The issue here isn't how it relates to the real world even. It's that it doesn't even make sense within Newtonian mechanics. It's a much more fundamental issue to do with math itself, really. How can math that is reversible in time suddenly not be deterministically reversible in this one very special situation? Usually math deals with perfectly sharp corners and infinitesimal distances just fine.
- fsh 4y ago
  The mathematical explanation is given on the wikipedia page. The Picard-Lindelöf theorem states that initial value problems have unique solutions if the right hand side function is Lipschitz continuous. The force created by Norton's dome is not Lipschitz continuous since it has a kink. I find it quite interesting that Newtonian mechanics only works if this condition is satisfied. Of course you cannot make a non-Lipschitz continuous force in practice, since the tip radius will be limited to the size of atoms, but theoretical physics is usually not concerned with such trivialities.
  - jfengel 4y ago
    Is Lipschitz continuity a symmetry that could be evaluated with Noether's theorem? Does it lead to a conserved quantity that could illuminate this?
NotYourLawyer 4y ago
> This example assumes a physically impossible infinitely-sharp corner The sharp corner isn’t the problem, and there are plenty of setups with sharp corners that work just fine. > Zeno-paradoxical infinitely small jump from the initial position to some other position after time T Every acceleration does that. The issue here is the way that the ball gets to the apex in finite time instead of approaching it asymptotically as t goes to infinity.

by gowld 4y ago

deleted 4y ago

[deleted]

courgette 4y ago

Thanks to pointing out the relevance

jfengel 4y ago

It requires a lot of not actually constructable elements, but that's not really the issue. The issue is that it's not reversible: a ball rolls up but never rolls down. If you play a tape of it backwards, it rolls off in a seemingly random direction, with no source of randomness and no initial kick of energy. It doesn't need an infinitely sharp corner. It's weird right at the very top. We do know that Newton is only an approximation, but general relativity doesn't seem to help here. Quantum mechanics might, but the problem itself doesn't seem to hint at it. It seems to nod in the direction of involving entropy at a deep level, but it's unclear how.

naasking 4y ago
Seems to me that this actually points out a fundamental inconsistency of classical theories, namely the use of infinitely divisible quantities (real numbers) which leads you to paradox. I think Norton's dome is unproblematic if you model it in a discrete theory.
- fsh 4y ago
  As far as we know, time and space are continuous. Various aspects of quantum field theories would not work in a discretized universe.
  - naasking 4y ago
    > As far as we know, time and space are continuous. Our models are continuous. The map is not the territory. > Various aspects of quantum field theories would not work in a discretized universe. If you're referring to symmetries, that's not true [1]. Discretized theories have received barely any attention because physicists have simply built upon the formalisms they're used to from classical physics. We see people frequently saying that we should abandon classical intuitions now that we have quantum mechanics, but we haven't really even tried to abandon classical formalisms, which are just another form of classical intuition. IMO, baking continuity and uncountable infinities is what's responsible for many of the difficulties we've had, and various forms of discretization have resolved these issues in the past [2]. My conjecture is that one of the next big advances in physics will be to take discretization seriously. [1] https://arxiv.org/abs/1902.08997 https://arxiv.org/abs/1902.08997 [2] https://arxiv.org/abs/1609.01421 https://arxiv.org/abs/1609.01421
    - crazygringo 4y ago
      > Our models are continuous. The map is not the territory. Because all our evidence is that it is continuous. Our models are the best description of the evidence we observe -- i.e. the territory. Unless you have evidence of time and space being non-continuous?
      - naasking 4y ago
        > Because all our evidence is that it is continuous. No, we haven't really tested this to any appreciable degree, and existing tests rule out only certain types of discrete models just like they rule out certain continuous models. Our models are continuous because continuous models are what we've been using for centuries. I'm not sure why anyone still believes that science is a purely evidence-driven endeavour. History has clearly shown it's subject to fads, celebrity and inertia. > Unless you have evidence of time and space being non-continuous? General relativity produces unphysical nonsense like singularities, which is a form of internal inconsistency similar to Norton's dome. This isn't evidence that spacetime is discrete so much as that our continuous model is incomplete, but it seems quite clear that a discrete analogue of GR would not suffer from this problem. Furthermore, some new approaches to gravity are discrete theories, like loop quantum gravity, so clearly some physicists are already thinking along these lines.

alecst 4y ago

You can have a sharp corner and you still won’t get a Norton’s dome in every case. There’s definitely something there, but it’s not obvious to me what it is. It does sometimes happen in physics that we throw away valid solutions to differential equations that are “physically impossible”, and I guess this falls in that category. Other examples are not immediately coming to mind unfortunately. But yea I don’t think this shows anything general about Newtonian mechanics.

onos 4y ago
Maybe you’re thinking about throwing out exponential growth in differential equations, where only decay makes sense physically. Eg in quantum well analysis this is done cause the former would give a wave that is not normalizeable.

crazygringo 4y ago

The issue here isn't how it relates to the real world even. It's that it doesn't even make sense within Newtonian mechanics. It's a much more fundamental issue to do with math itself, really. How can math that is reversible in time suddenly not be deterministically reversible in this one very special situation? Usually math deals with perfectly sharp corners and infinitesimal distances just fine.

fsh 4y ago
The mathematical explanation is given on the wikipedia page. The Picard-Lindelöf theorem states that initial value problems have unique solutions if the right hand side function is Lipschitz continuous. The force created by Norton's dome is not Lipschitz continuous since it has a kink. I find it quite interesting that Newtonian mechanics only works if this condition is satisfied. Of course you cannot make a non-Lipschitz continuous force in practice, since the tip radius will be limited to the size of atoms, but theoretical physics is usually not concerned with such trivialities.
- jfengel 4y ago
  Is Lipschitz continuity a symmetry that could be evaluated with Noether's theorem? Does it lead to a conserved quantity that could illuminate this?

NotYourLawyer 4y ago

> This example assumes a physically impossible infinitely-sharp corner The sharp corner isn’t the problem, and there are plenty of setups with sharp corners that work just fine. > Zeno-paradoxical infinitely small jump from the initial position to some other position after time T Every acceleration does that. The issue here is the way that the ball gets to the apex in finite time instead of approaching it asymptotically as t goes to infinity.