5 ms·
The Dome (2005)
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- kerblang 2y agoEgads, a determinism crisis, with math! We must squarsh the ornery willfulness of this wee ball, but how?
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- foxglacier 2y agoWith the time reversal trick, it almost seems like you could say the cause happens in the future and propagates backwards in time? I'm curious that if it doesn't work for a hemispherical dome because of requiring infinite time, does that mean a ball on a hemispherical dome is stable and won't spontaneously roll off?
- abdullahkhalids 2y agoNo. What all this shows is that both cases (hemispherical dome or the other one in OP) of a ball in stable equilibrium are outside the domain of applicability of the theory. You should simply not use the theory as is to make predictions about when and how the ball will roll off.
- tsimionescu 2y agoActually, there is nothing mysterious here. The theory, applied correctly, predicts the expected thing: a particle that starts at rest at the apex will never leave it. The other "solution" is not actually a particle at rest (it has a non-0 6th derivative of position, that is the rate of change of the rate of change of its acceleration is 1/6). So it's actually describing a particle whose acceleration will increase at an increasing rate forever, but that just happens to be 0 at time T.
- abdullahkhalids 2y agoYes. The situation is correctly described by a modified version of standard Newton's theory, in which you start caring about higher derivatives.
- foxglacier 2y agoJust in case I'm completely wrong about reality, both cases are unstable equilibrium, aren't they? Equilibrium in that forces balance and zero acceleration but unstable in that any disturbance will lead to it rolling off rather than returning to where it was.
- bennythomsson 2y agoYes, if it's rolling on a hemispherical dome then it's always been rolling, unless an external force pushed it.
- wat10000 2y agoTime reversible systems make causality fuzzy. There’s nothing special about this example in that respect. You can’t object have ball A hit ball B. A causes B to move. You can then reverse it, end up with an equivalent but time reversed system where B hits A, and B causes A to move.
- abdullahkhalids 2y agoThis is a really neat example. When I was in academia, I wondered if I could teach a advanced undergrad physics course that focused exclusively on breaking all the standard physics theories. There are three ways that theories typically break. - Falsified by experiment. This is for example discussed a lot in the Modern Physics course with regards to classical theories. - Broken math for the theory. This is an example of this, and most of the theories have some weird stuff like this that has to be ignored. - Philosophical unpleasantness. Sometimes theories have weird philosophical consequences that indicate that we should come up with something better.
- ocfnash 2y agoI think it is worth comparing this problem with the question of the behaviour of a particle placed at the apex of a cone. I claim it is clear that in this case, the problem is clearly not well-posed because the apex is a singular point: the slope at the apex is undefined. The singular nature of the slope (first derivative) is the issue. This "dome" is essentially the same issue just with the singularity buried one level deeper: you need to take second derivatives to see it. Indeed a planar cross section containing the vertical axis through its center is a graph of the equation $y^2 = |x|^3$ (up to constants) and this is not twice differentiable at $x = y = 0$. Newtonian mechanics is governed by a second order differential equation, so we need a C^2 regularity assumption to get uniqueness. So for me there is not really any more philosophically interesting than the question about a particle balancing at the apex of a cone.
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- dcrazy 2y agoI am curious about the validity of (re)phrasing of Newton’s 2nd Law as “instantaneous.” The acceleration is non-differentiable at T=0. Can we really combine it with analysis of times t<T and t>T?
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- dventimi 2y agoThere are many good treatments of this supposed loophole. I happen to like this one: https://blog.gruffdavies.com/2017/12/24/newtonian-physics-is https://blog.gruffdavies.com/2017/12/24/newtonian-physics-is... It points out many flaws in Norton's reasoning, some fatal to his argument, some not. Putting it as simply as I can, Norton seems to claim that "Newton's Laws" are non-deterministic. That's not quite right. Rather, they are non-complete. I.e. they are incomplete. They're incomplete insofar as Newton's First Law ("An object at rest remains at rest, and an object in motion remains in motion at constant speed and in a straight line unless acted on by an unbalanced force") establishes first-order and second-order derivatives (momentum and acceleration) as state variables but places no constraints on higher-order derivatives. However, higher-order derivatives are (as many as are needed) among a system's state variables. In many real systems (but far from all), higher-order derivatives are zero and human experience with them is rare, so they're easy to overlook. Norton's (unphysical) Dome is a specific example of a general class of systems where higher-order derivatives are not zero. Given that, the two branches of Norton's equation of motion (for the stable and unstable trajectories) cannot both describe the same system (or the same particle) with the same set of state variables. That's the sleight-of-hand. Again, all credit to Gareth Davies for working this out. I am absolutely not trying to pass off his work for my own. Just reporting it and trying to summarize it.
- ttoinou 2y agoReading the original article I immediately thought about higher order derivative, which made me wonder what laws apply to them, that I’ve never studied that, that’s odd
- ThePhysicist 2y agoI think one can simply use the Euler-Lagrange method which is able to account for the constraint forces acting on the ball. Haven't worked that out for this particular problem but it should be relatively easy. Davies argument is a bit overcomplicated I think, the main challenge here is correctly accounting for the geometric constraints in the movement of the particle. I find the argument about the higher-order derivatives a bit weird as well, the system can be fully described using its potential and kinetic energy which are scalar (possibly time-dependent) fields and implicitly contain all forces, given some initial conditions (position and momentum) we can solve the equation of motion of the system with that.
- peeters 2y agoInteresting that this is making the rounds...maybe OP also had this recent video show up in their Youtube feed? It includes interview clips with John Norton, the author of this paper. https://www.youtube.com/watch?v=EjZB81jCGj4 https://www.youtube.com/watch?v=EjZB81jCGj4
- stronglikedan 2y agoUp and Atom on YT recently released a nice video explaining this paradox, in case you prefer to learn about it this way like I do. https://www.youtube.com/watch?v=EjZB81jCGj4 https://www.youtube.com/watch?v=EjZB81jCGj4
- Retric 2y ago“Instead of imagining the mass starting at rest at the apex of the dome, we will imagine it starting at the rim and that we give it some initial velocity directed exactly at the apex. If we give it too much initial velocity, it will pass right over the apex to the other side of the dome. So let us give it a smaller initial velocity. We produce the trajectory T1:” As acceleration, jerk, snap, crackle, pop, … all must approach 0, does it ever actually reach the apex with zero velocity in finite time? That seems like the most obvious solution here where if you start with non zero velocity it never quite reaches the apex, and if you start with zero velocity at the apex it never leaves it.
- brudgers 2y agoPrevious discussion: https://news.ycombinator.com/item?id=37012347 https://news.ycombinator.com/item?id=37012347
- dventimi 2y agohttps://sites.pitt.edu/~jdnorton/papers/DomePSA2006.pdf https://sites.pitt.edu/~jdnorton/papers/DomePSA2006.pdf
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- dventimi 2y agoFrom the second paragraph: A mass sits on a dome in a gravitational field. After remaining unchanged for an arbitrary time, it spontaneously moves in an arbitrary direction, with these indeterministic motions compatible with Newtonian mechanics Well, no. "It" does not spontaneously move in an arbitrary direction. It remains in place forever.
- PeterWhittaker 2y agoNot on Norton's Dome: it is the classic example of indeterminism in classical mechanics. While QM is sufficient for indeterminism, it is not necessary, as this example shows. Even in classical mechanics, physics is weirder than our intuition allows.
- tsimionescu 2y agoNo, that is just a mistake that Norton makes. The only physical trajectory for a particle starting at the apex at rest is that it will remain at rest. The other equation that Norton comes up with is not a physical description of a particle at rest, it is a description of a particle which has complex motion (the fourth derivative of its position(t), sometimes called crackle, is 1/6). Basically that means that in every second, its acceleration increases by (1/12s³). An equation of motion that has any non 0 derivative of time is not describing a particle at rest. Not to mention, branching functions are also not valid equations of motion. You can't take two valid equations of motions and stitch them at some arbitrary time. The same problem will appear on a perfectly flat surface if you do that. I could say that the function f(t) = 0 for t < T, 7t for t >= T is a valid solution to the equations of motion, by the same logic (its second derivative is 0, equal to the net force acting on the particle). This doesn't prove that Netwonian mechanics is non-deterministic, it shows that you can't use functions that arbitrarily change as equations of motion (they just violate Newton's first law).