[HN Gopher] The Dome (2005)
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The Dome (2005)
Author : Tomte
Score : 40 points
Date : 2025-01-03 08:20 UTC (14 hours ago)
(HTM) web link (sites.pitt.edu)
(TXT) w3m dump (sites.pitt.edu)
| dventimi wrote:
| https://sites.pitt.edu/~jdnorton/papers/DomePSA2006.pdf
| dventimi wrote:
| From 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 wrote:
| Not 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.
| dventimi wrote:
| > Not on Norton's Dome
|
| Really? Are you sure about that? Have you tried it? Where
| is this "Newton's Dome" so that the experiment can be
| replicated?
| mrkeen wrote:
| [flagged]
| dventimi wrote:
| I'll take that as a "no".
| PeterWhittaker wrote:
| Not Newton's Dome, Norton's:
| https://en.m.wikipedia.org/wiki/Norton%27s_dome
| dventimi wrote:
| That was a typo from a phone keyboard.
| tsimionescu wrote:
| No, 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).
| [deleted]
| kerblang wrote:
| Egads, a determinism crisis, with math! We must squarsh the
| ornery willfulness of this wee ball, but how?
| foxglacier wrote:
| With 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 wrote:
| No. 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 wrote:
| Actually, 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.
| bennythomsson wrote:
| Yes, if it's rolling on a hemispherical dome then it's always
| been rolling, unless an external force pushed it.
| abdullahkhalids wrote:
| This 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 wrote:
| I 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.
| dcrazy wrote:
| I 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?
| dventimi wrote:
| There are many good treatments of this supposed loophole. I
| happen to like this one:
| 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 wrote:
| Reading 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 wrote:
| I 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.
| selimthegrim wrote:
| I think Norton completely ignores virtual work and
| D'Alembert's theorem
| peeters wrote:
| Interesting 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
| stronglikedan wrote:
| Up 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
| Retric wrote:
| "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 wrote:
| Previous discussion:
| https://news.ycombinator.com/item?id=37012347
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