6 ms·

Thanks for the reference to the video. I watched it a few weeks ago and was befuddled by it. How can the ball just randomly start rolling in a random direction?

Thanks for the reference to the video. I watched it a few weeks ago and was befuddled by it. How can the ball just randomly start rolling in a random direction? It seemed to me that an obvious explanation would be that there is air flow in the environment and with the ball balanced in an unstable position that some air movement would easily nudge the ball off balance. I understand the diff eq of motion with the singularity but it seems to me that a ball balanced at the apex of any radially symmetric convex surface would eventually commence rolling, due to fluctuations in the air flow.

enlightens 2y ago
Norton’s Dome plays fast and loose with the math. It could be a halfway decent way of modeling a ball that randomly started moving, but that’s not actually how anything works. For example: https://physics.stackexchange.com/questions/39632/nortons-dome-and-its-equation https://physics.stackexchange.com/questions/39632/nortons-do...
- jdhwosnhw 2y ago
  Norton’s dome is a valid paradox, in the sense that the math really does admit two valid equations of motion. The link you provided doesn’t dispute that fact (other than commenters pointing out that you need a proportionality constant to make the units work out). My favorite intuitive explanation for the presence of the paradox is well summarized by the Wikipedia article on the dome: “To see that all these equations of motion are physically possible solutions, it's helpful to use the time reversibility of Newtonian mechanics. It is possible to roll a ball up the dome in such a way that it reaches the apex in finite time and with zero energy, and stops there. By time-reversal, it is a valid solution for the ball to rest at the top for a while and then roll down in any one direction.”
  - pxx 2y ago
    I don't know if you can actually stitch the equations together though because they have different initial values, albeit in higher order derivatives than Newtonian mechanics cares about. see https://blog.gruffdavies.com/2017/12/24/newtonian-physics-is-deterministic-sorry-norton/ https://blog.gruffdavies.com/2017/12/24/newtonian-physics-is... which was linked to from that new question: > If we start at an arc length of 1/144 for example, it will run up the dome and arrive at the apex in 1 second. As we’ve seen, it has zero velocity and zero acceleration at this point, but moves off after anyway because it still has a positive value for snap.
  - deleted 2y ago
    [deleted]
  - Retric 2y ago
    Calling it a valid paradox is questionable, there’s a little mathematical sleight of hand required for the particle to actually stop in finite time. It doesn’t work for say particle sliding up a sphere. To work the curvature of the dome is infinite is at the apex, which then breaks many things. There’s a lot of disagreements around this paradox and much older related examples because Newtonian physics is somewhat ill defined: https://philsci-archive.pitt.edu/8833/1/dome_v3.pdf https://philsci-archive.pitt.edu/8833/1/dome_v3.pdf
    - ASalazarMX 2y ago
      Since many thought experiments allow for objects to be composed of infinite one-dimensional points that somehow form higher-dimensional bodies, an infinite curvature could be interpreted as the apex being a single point supporting the point at the bottom of the ball. Both points should be perfectly aligned and in perfect equilibrium. It has no reason to roll unless the placement was uneven, and if it was uneven, it would not break determinism.
      - deleted 2y ago
        [deleted]
  - adrian_b 2y ago
    Norton's dome is not a valid paradox, it just exploits in an overly complicated way the fact that almost no handbook of physics bothers to present a complete set of axioms for the classical mechanics (and even less for relativistic or quantum mechanics). So the "paradox" is based on the sloppy teaching of physics from a mathematical point of view. The form of the Norton's dome does not matter. The so-called paradox is just a random example of the fact that there exist multiple functions of a variable that have in the origin the same values for the function and for the first 2 derivatives, e.g. various pairs of polynomials of the 4th order. Therefore if you accept any function of time as describing a possible motion, you can always find motions that at some moment in time have the same position, velocity and acceleration. This is not an example of indeterminacy in classic mechanics, because one of the axioms of the classic mechanics is that all the forces that exist in nature are such that the state of a mechanical system is completely determined by the positions, velocities and accelerations of its components (in other words, a mechanical system must be described by a system of differential equations of the second degree that has a unique solution). There is no difficulty of imagining other kinds of forces, for which this assumption is not true, but a theory where such forces exist is no longer the Newtonian mechanics, in the same way as any geometry where Euclid's axiom of parallels is not true is no longer an Euclidean geometry. If Newtonian mechanics were a correct model for the World, a ball would remain forever on the top of the dome, without ever falling. In reality, even assuming the validity of Newtonian mechanics, the main reason why any attempt to test this experimentally would fail is the thermal motion, due to which a ball can never be at rest, so it would always start immediately to fall in a random direction. The violation of the axioms is why the so-called different solutions are not solutions within Newtonian mechanics. On the other hand the argument that the initial state could be obtained by launching the ball towards the top, and then time reversal would demonstrate a valid solution, it is also wrong, because the so-called solution cannot be obtained by time reversal. If the ball is launched with only enough energy to reach the top, so it will come to rest, then it requires an infinite time to reach the top. Reversing the time means that the ball will remain on the top for an infinite time, without falling, as expected.
  - ASalazarMX 2y ago
    The real paradox is why and how a stationary object in a perfect world that obeys Newton's laws suddenly started moving. There's only mass and gravity, not even thermal or atomic effects allowed. The best justification the author gives is that it happened and this thought experiment doesn't explain why, how, or even when. TL;DR: Magic breaks Newton's laws
ajross 2y ago
> How can the ball just randomly start rolling in a random direction? Because that's legal according to the laws of motion. The intuitive answer is that it's the time reversed situation to a ball being carefully rolled UP the dome so that it stops and comes to rest on the apex. The shape function of the dome was carefully constructed so that this process takes finite time. So if it's legal in one direction it must be legal in the other. Obviously this is a statement about math and not physics (since the underlying physical theory here is, after all, wrong!) What we thought were a bunch of well-constructed rules for classical dynamics turn out to have some holes.
- moralestapia 2y ago
  >The intuitive answer is that it's the time reversed situation to a ball being carefully rolled UP the dome so that it stops and comes to rest on the apex. That's nonsense. The arrow of entropy always goes forward. Sure, the ball comes to the top of the dome to rest but it also carries direction, momentum and a lot of other properties that you have to put in as well in your hypothetical entropy-arrow-now-goes-back scenario. This is high-school grade physics, come on. It's surprising some people still take John Norton seriously, not because of the dome, but because of his many other "controversial" takes on physics that fail miserably on their foundations.
  - ajross 2y ago
    > The arrow of entropy always goes forward. The arrow of what now?[1] This is classical dynamics we're doing. I repeat, this is a math result, not an argument about physical systems. [1] Edit as this was clearly missed: THIS IS SARCASM. Thermodynamics and statistical mechanics are excellent theories and worth studying as they tell us deep and profound things about the natural world. This particular novelty is a result from classical dynamics where they don't apply. The "arrow of time" in Newtonian mechanics is absolutely reversible, and there is no Newtonian idea of "entropy".
    - moralestapia 2y ago
      https://en.wikipedia.org/wiki/Entropy_as_an_arrow_of_time https://en.wikipedia.org/wiki/Entropy_as_an_arrow_of_time Read. Then post. >I repeat, this is a math result, not an argument about physical systems. Did you even care to read the title of the post?
josh-sematic 2y ago
The point is that this is not discussing a physically realizable situation, but an idealized one. The engineering and manufacturing precision that would be required to actually achieve this setup are infinite and unattainable. In the idealized setup there is no air, no surface imperfections, no deviations from central positioning, etc.. And yet despite this idealized perfection, a case can still be made that under this construction the ball might spontaneously move in an undetermined direction. The discussion of whether that case “holds water” and what that means if so is an abstract philosophy discussion rather than one with any obvious practical implications.
flatline 2y ago
If you watched the video, you would have the answer. Well, an answer. Which is that there does not need to be a cause! Even in an idealized dome with no air, friction, or external forces. As it does for you for different reasons, this also matches my lay intuition of physics: sometimes things just spontaneously occur, and a system in dynamic equilibrium simply will not hold still forever.
Asooka 2y ago
It's an entirely nonsense argument. Akin to arguing that algebra is nondeterministic with "zero divided by zero is a random number, because any number times zero is zero". In the case of classical physics, we come to a singularity in which there are several solutions for how the system resolves. This doesn't make classical physics nondeterministic, this simply means if you come to such a solution, then classical physics have no answer for what happens next.

enlightens 2y ago

Norton’s Dome plays fast and loose with the math. It could be a halfway decent way of modeling a ball that randomly started moving, but that’s not actually how anything works. For example: https://physics.stackexchange.com/questions/39632/nortons-dome-and-its-equation https://physics.stackexchange.com/questions/39632/nortons-do...

jdhwosnhw 2y ago
Norton’s dome is a valid paradox, in the sense that the math really does admit two valid equations of motion. The link you provided doesn’t dispute that fact (other than commenters pointing out that you need a proportionality constant to make the units work out). My favorite intuitive explanation for the presence of the paradox is well summarized by the Wikipedia article on the dome: “To see that all these equations of motion are physically possible solutions, it's helpful to use the time reversibility of Newtonian mechanics. It is possible to roll a ball up the dome in such a way that it reaches the apex in finite time and with zero energy, and stops there. By time-reversal, it is a valid solution for the ball to rest at the top for a while and then roll down in any one direction.”
- pxx 2y ago
  I don't know if you can actually stitch the equations together though because they have different initial values, albeit in higher order derivatives than Newtonian mechanics cares about. see https://blog.gruffdavies.com/2017/12/24/newtonian-physics-is-deterministic-sorry-norton/ https://blog.gruffdavies.com/2017/12/24/newtonian-physics-is... which was linked to from that new question: > If we start at an arc length of 1/144 for example, it will run up the dome and arrive at the apex in 1 second. As we’ve seen, it has zero velocity and zero acceleration at this point, but moves off after anyway because it still has a positive value for snap.
- deleted 2y ago
  [deleted]
- Retric 2y ago
  Calling it a valid paradox is questionable, there’s a little mathematical sleight of hand required for the particle to actually stop in finite time. It doesn’t work for say particle sliding up a sphere. To work the curvature of the dome is infinite is at the apex, which then breaks many things. There’s a lot of disagreements around this paradox and much older related examples because Newtonian physics is somewhat ill defined: https://philsci-archive.pitt.edu/8833/1/dome_v3.pdf https://philsci-archive.pitt.edu/8833/1/dome_v3.pdf
  - ASalazarMX 2y ago
    Since many thought experiments allow for objects to be composed of infinite one-dimensional points that somehow form higher-dimensional bodies, an infinite curvature could be interpreted as the apex being a single point supporting the point at the bottom of the ball. Both points should be perfectly aligned and in perfect equilibrium. It has no reason to roll unless the placement was uneven, and if it was uneven, it would not break determinism.
    - deleted 2y ago
      [deleted]
- adrian_b 2y ago
  Norton's dome is not a valid paradox, it just exploits in an overly complicated way the fact that almost no handbook of physics bothers to present a complete set of axioms for the classical mechanics (and even less for relativistic or quantum mechanics). So the "paradox" is based on the sloppy teaching of physics from a mathematical point of view. The form of the Norton's dome does not matter. The so-called paradox is just a random example of the fact that there exist multiple functions of a variable that have in the origin the same values for the function and for the first 2 derivatives, e.g. various pairs of polynomials of the 4th order. Therefore if you accept any function of time as describing a possible motion, you can always find motions that at some moment in time have the same position, velocity and acceleration. This is not an example of indeterminacy in classic mechanics, because one of the axioms of the classic mechanics is that all the forces that exist in nature are such that the state of a mechanical system is completely determined by the positions, velocities and accelerations of its components (in other words, a mechanical system must be described by a system of differential equations of the second degree that has a unique solution). There is no difficulty of imagining other kinds of forces, for which this assumption is not true, but a theory where such forces exist is no longer the Newtonian mechanics, in the same way as any geometry where Euclid's axiom of parallels is not true is no longer an Euclidean geometry. If Newtonian mechanics were a correct model for the World, a ball would remain forever on the top of the dome, without ever falling. In reality, even assuming the validity of Newtonian mechanics, the main reason why any attempt to test this experimentally would fail is the thermal motion, due to which a ball can never be at rest, so it would always start immediately to fall in a random direction. The violation of the axioms is why the so-called different solutions are not solutions within Newtonian mechanics. On the other hand the argument that the initial state could be obtained by launching the ball towards the top, and then time reversal would demonstrate a valid solution, it is also wrong, because the so-called solution cannot be obtained by time reversal. If the ball is launched with only enough energy to reach the top, so it will come to rest, then it requires an infinite time to reach the top. Reversing the time means that the ball will remain on the top for an infinite time, without falling, as expected.
- ASalazarMX 2y ago
  The real paradox is why and how a stationary object in a perfect world that obeys Newton's laws suddenly started moving. There's only mass and gravity, not even thermal or atomic effects allowed. The best justification the author gives is that it happened and this thought experiment doesn't explain why, how, or even when. TL;DR: Magic breaks Newton's laws

ajross 2y ago

> How can the ball just randomly start rolling in a random direction? Because that's legal according to the laws of motion. The intuitive answer is that it's the time reversed situation to a ball being carefully rolled UP the dome so that it stops and comes to rest on the apex. The shape function of the dome was carefully constructed so that this process takes finite time. So if it's legal in one direction it must be legal in the other. Obviously this is a statement about math and not physics (since the underlying physical theory here is, after all, wrong!) What we thought were a bunch of well-constructed rules for classical dynamics turn out to have some holes.

moralestapia 2y ago
>The intuitive answer is that it's the time reversed situation to a ball being carefully rolled UP the dome so that it stops and comes to rest on the apex. That's nonsense. The arrow of entropy always goes forward. Sure, the ball comes to the top of the dome to rest but it also carries direction, momentum and a lot of other properties that you have to put in as well in your hypothetical entropy-arrow-now-goes-back scenario. This is high-school grade physics, come on. It's surprising some people still take John Norton seriously, not because of the dome, but because of his many other "controversial" takes on physics that fail miserably on their foundations.
- ajross 2y ago
  > The arrow of entropy always goes forward. The arrow of what now?[1] This is classical dynamics we're doing. I repeat, this is a math result, not an argument about physical systems. [1] Edit as this was clearly missed: THIS IS SARCASM. Thermodynamics and statistical mechanics are excellent theories and worth studying as they tell us deep and profound things about the natural world. This particular novelty is a result from classical dynamics where they don't apply. The "arrow of time" in Newtonian mechanics is absolutely reversible, and there is no Newtonian idea of "entropy".
  - moralestapia 2y ago
    https://en.wikipedia.org/wiki/Entropy_as_an_arrow_of_time https://en.wikipedia.org/wiki/Entropy_as_an_arrow_of_time Read. Then post. >I repeat, this is a math result, not an argument about physical systems. Did you even care to read the title of the post?

josh-sematic 2y ago

The point is that this is not discussing a physically realizable situation, but an idealized one. The engineering and manufacturing precision that would be required to actually achieve this setup are infinite and unattainable. In the idealized setup there is no air, no surface imperfections, no deviations from central positioning, etc.. And yet despite this idealized perfection, a case can still be made that under this construction the ball might spontaneously move in an undetermined direction. The discussion of whether that case “holds water” and what that means if so is an abstract philosophy discussion rather than one with any obvious practical implications.

flatline 2y ago

If you watched the video, you would have the answer. Well, an answer. Which is that there does not need to be a cause! Even in an idealized dome with no air, friction, or external forces. As it does for you for different reasons, this also matches my lay intuition of physics: sometimes things just spontaneously occur, and a system in dynamic equilibrium simply will not hold still forever.

Asooka 2y ago

It's an entirely nonsense argument. Akin to arguing that algebra is nondeterministic with "zero divided by zero is a random number, because any number times zero is zero". In the case of classical physics, we come to a singularity in which there are several solutions for how the system resolves. This doesn't make classical physics nondeterministic, this simply means if you come to such a solution, then classical physics have no answer for what happens next.