Norton's Dome | Hacker News Reader

Norton's Dome

en.wikipedia.org

113 pointsmiobrien36 comments

The argument for time reversal assumes that the ball can reach the top of the dome in a finite time, while its speed becomes zero at that limit. Sure, you can calculate the kinetic energy needed to exactly reach the top like described, but would such a ball reach the top in a finite time, as it slows down in the process?

Edit: To answer my own question, the shape of the dome has been specifically chosen to avoid this problem, as described here:

https://www.reddit.com/r/Physics/comments/2cueh3/nortons_dom...

I think there's an easier thought experiment to generate non-determinism in classical mechanics without some continuity assumption:

Take a chaotic system (eg., the moon of one of our solar system planets) and let it evolve for some time, T. Track the position with coordinate X. Let T be large enough that the nth decimal place of X_T is significant to determining X_T+1.

If there is a discontinuity at the nth decimal place, then X_T+1 is not determined by X_T.

For quite observable T, n quickly becomes "sub-quantum". So, if classical mechanics is deterministic, and describes nature, nature must be continuous at arbitary depth.

OR: *classical* mechanics is non-deterministic.

> So, if classical mechanics is deterministic, and describes nature, nature must be continuous at arbitary depth.

Isn't that the classical assumption though? That nature is analogue. To avoid problems with infinities, you can just say let's assume it's continuous up to Graham's number or something.

Or, .. it's continuous but "unknowable" in the sense of becoming increasingly "out of focus" wrt measurement due to the uncertainty principle at and below the order of Planck's constant.

Not every dynamic system is chaotic, and not system with chaotic elements is chaotic everywhere ... but ...

Even quite simple systems can have chaotic regions .. and within those regions two 'particles' (or phase space initial conditions) can startout arbitrarily close (within a fuzzy out of focus impossible to measure Planck distance) and end up nowhere near each other .. (ie not continuous).

This is a mathematical result of dynamic systems and can be arrived either by Lorentz's reasoning ( "the butterfly effect" ) or via Smale's Horshoe Map (taffy folding to infinity!).

This blog post provides an interesting analysis: https://blog.gruffdavies.com/2017/12/24/newtonian-physics-is...

The concluding paragraph:

Position, velocity and acceleration will be zero at t = 0 for every equation of polynomial form of order 3 and above, but non zero everywhere else. Particles following these trajectories move to and from an unstable equilibrium where Newton’s laws fail to be fully descriptive at the singular point t = 0 where the implied force is zero.

The author of the blog post completely misses the point of the problem. The dome is interesting close to the apex. It doesn't matter that the particle would fly off the surface at some point further down on its trajectory. The real issue is that Newtonian physics just doesn't work for forces that are not Lipschitz continuous (which the author brushes over in a half-sentence). The surprising thing is that a simple dome with an easy to draw shape can produce such a force.

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.

Thanks to pointing out the relevance

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.

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.

As far as we know, time and space are continuous. Various aspects of quantum field theories would not work in a discretized universe.

> 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

[2] https://arxiv.org/abs/1609.01421

> 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?

> 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.

We have oodles of everyday evidence that space time is continuous in the general case because we've never found it not to be.

I don't know that you mean that this hasn't been tested. Essentially every experiment ever is consistent with it.

By that logic, every experiment ever conducted is consistent with discrete theories as well.

Experiments that try to distinguish the fine structure of spacetime haven't really been a focus, and are probably impossible in a lab environment. Some studies of distant gamma ray bursts place some limits on some discrete theories, but those limits don't apply to covariant discrete theories that preserve other symmetries like the one I posted above.

Quantum electrodynamics is a continuous theory. Its predictions agree with measurements of the electron g-factor and of energy levels in atomic hydrogen to 12 digits of relative accuracy. It would be interesting if a discretized version (if at all possible to construct) makes the same predictions at that level.

I agree, that would be interesting! The paper I linked above suggests that discrete theories are simpler and require less formal structure (ie. no need to talk about gauges or gauge fixing), that's why I said above that I think a push towards discrete physics would provide many insights.

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.

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.

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.

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.

Is Lipschitz continuity a symmetry that could be evaluated with Noether's theorem? Does it lead to a conserved quantity that could illuminate this?

> 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.

Very interesting. I wonder if something analogous to this could be behind wave function collapse in quantum mechanics.

https://en.wikipedia.org/wiki/Spontaneous_symmetry_breaking

Why can't it just be a triangle? Is there a reason it needs to be curved?

From the article: “ 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. However, the same argument applied to the usual kinds of domes (e.g., a hemisphere) fails, because a ball launched with just the right energy to reach the top and stay there would actually take infinite time to do so.”

I definitely got that part, I was thinking about just usually a linear ramp to get to the top in a certain amount of time. But I didn't think about the fact that the ball kind of has to rotate around at the top in order to be perfectly balanced so now I see why there is some nuance.

It's not a ball, it's an idealized "particle" (0-dimensional point).

The movement of a ball on each shape (triangular, parabolic, Norton's-domic) will be modeled with its own differential equation. Some of these differential equations will have one and only one solution (given the boundary conditions).

One of these differential equations has an infinite number of solutions--even given the boundary conditions :-)

Best way to really understand this is to roll up your sleeves and spend an afternoon setting up and solving the various differential equations.

This is a relevant thread: https://news.ycombinator.com/item?id=28191408

I think the mathematical approach to this paradox would be to line up the reasoning for time-reversal side-by-side with the predictions for Norton's Dome, and find a flaw in either of them. Are we even sure that the reasoning behind time-reversal in Newton's laws is solid?

BTW entropy was mentioned in another thread, but this thought-experiment is frictionless, so if entropy still comes up that would really be interesting.

> Are we even sure that the reasoning behind time-reversal in Newton's laws is solid?

Yes, in fact, even QM is time-charge-parity symmetric. Time symmetry in a mechanical system is also necessary for energy conservation.

Time translation symmetry yields energy conservation; time reversal is a discrete symmetry and does not.

Erm yeah you’re obviously right, not sure what I was thinking but can’t edit anymore.

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