[HN Gopher] The Dome: A simple violation of determinism in Newto...
___________________________________________________________________
The Dome: A simple violation of determinism in Newtonian mechanics
(2005)
Author : chmaynard
Score : 105 points
Date : 2023-08-05 14:29 UTC (8 hours ago)
(HTM) web link (sites.pitt.edu)
(TXT) w3m dump (sites.pitt.edu)
| dang wrote:
| Related:
|
| _The Dome: A Simple Violation of Determinism in Newtonian
| Mechanics_ - https://news.ycombinator.com/item?id=26507118 -
| March 2021 (2 comments)
| aidenn0 wrote:
| ELI5: why do they go to the trouble of constructing a dome when a
| simple cone would have the same properties? It seems to me that
| the motion of the object is not infinitely differentiable in
| either case, and the dome shape only serves to obscure this fact.
| amluto wrote:
| Because the cone is intuitively ridiculous? If I say I'm
| balancing an object (presumably of finite size) perfectly
| centered on a cone, then the obvious question is "you did
| _what_ "? It just seems more absurd than balancing an object on
| a continuous surface.
| eigenket wrote:
| The easiest way to see it is to consider the time-reversed
| version. You chuck the ball up the shape so it perfectly stops
| balanced at the top. It turns out this is possible for the
| weird dome in question, but not for pretty much any other shape
| - including a cone. Going back to time running normally it
| turns out that this means that a ball balanced perfectly on a
| cone only has one option consistent with Newton's laws, it'll
| stay balanced there forever, while there are multiple
| trajectories consistent with Newton's laws for the dome.
| aidenn0 wrote:
| Why is it not possible for a cone? Slide it up the
| (frictionless) cone with kinetic energy equal to mgh and it
| should stop on the point.
| scatters wrote:
| It's the reverse of a point mass sliding up a hemispherical
| dome. It takes infinite time to reach the top.
| eigenket wrote:
| It never reaches the top in finite time on a cone, it just
| keeps going up and up getting slower and slower.
| aidenn0 wrote:
| That makes sense; thanks.
| kp1197 wrote:
| I don't buy the argument that merely because solutions exist,
| they can spontaneously be "chosen" by the system, even if such
| choosing doesn't break rules. We are talking about an idealized
| setup using disproven physics, so it's weird to talk about
| correctness. But I think the author is invoking a variety of
| Murphys law: Anything that can happen, might happen
| spontaneously. If that's true in idealized Newtonian physics
| world, maybe we should "fix" that? Or maybe the question of what
| to do about weird multiple solution sotuations is simply not in
| the scope of Newtons theory.
| sebzim4500 wrote:
| This is just what determinism means: that for a given intial
| condition there is only a single solution obeying the rules.
|
| The article demonstrates that newtonian mechanics is not
| deterministic, which is surprising at least to me.
| mvaliente2001 wrote:
| I think parent comment (by @sebzim4500) is the clearest most
| concise summary of the argument stated in the article.
| Cushman wrote:
| Hoping I'm not too late to head off the usual confusion: this is
| an interesting result in _philosophy of_ physics, not in physics.
|
| We already know that classical mechanics is non-physical. This
| result (and others) show that it is not even internally
| consistent-- that is, you shouldn't need any empirical evidence
| to know that there's something else going on.
|
| That's interesting to philosophers and historians, but since you
| and I already know empirically that it's non-physical, it
| shouldn't come as much of a surprise.
|
| Anyway, if you enjoyed learning about the dome, you may also look
| up the lesser-known "space invaders", in which arbitrary objects
| can appear at infinity, with infinite velocity, and then be
| brought to rest at any time T. But again, don't look for reasons
| that doesn't actually happen-- it means the theory is wrong.
| kergonath wrote:
| > We already know that classical mechanics is non-physical.
| This result (and others) show that it is not even internally
| consistent-- that is, you shouldn't need any empirical evidence
| to know that there's something else going on.
|
| I don't think this is a right approach. We already know that no
| theory is complete and perfect, so we can say the same thing
| about any theory. Even worse than that, we can make the
| philosophical point that any theory, being conceived within our
| limited brains, physically cannot be anything other than models
| and approximations. The logical conclusion of this argument is
| then that we should throw our hands in the air and stop
| discussing anything.
|
| It's also wrong in this case specifically because there is
| absolutely no reason why this thought experiment cannot be
| proven or disproven within Newtonian mechanics.
| drdeca wrote:
| > We already know that no theory is complete and perfect, so
| we can say the same thing about any theory.
|
| If you mean this as an appeal to Godel's incompleteness
| results, the things those show can't happen aren't the same
| kinds of things that a "fully complete theory of physics"
| would have to satisfy.
|
| That's not to say that I expect that we will ever (in this
| world) have a complete description of the physics of this
| world,
|
| But I'm quite confident that Godel's incompleteness theorems
| do not pose a fundamental barrier to the laws of physics of a
| world being perfectly known by entities in that world.
| Cushman wrote:
| > I don't think this is a right approach.
|
| You're welcome not to think so! I'm just pointing out the
| relevant context. This isn't an argument in a vacuum, there's
| an academic discipline that has thoroughly engaged with it.
|
| Part of that is a vast body of literature that in turns
| agrees and disagrees with your observations-- to take a side,
| I'd start clicking links from "theory-ladenness".
|
| But the dome is rather boring. P1, this math describes a
| deterministic system; P2, here is a non-deterministic result;
| QED, P1 is false.
|
| Since we know that P1 doesn't describe the universe, it
| should be hard to have a strong opinion unless you're
| invested in a philosophical position about pre-modern
| scientific practice.
|
| (I'm not invested, so I don't have an opinion other than that
| it isn't trivially refutable as stated.)
| kergonath wrote:
| > You're welcome not to think so! I'm just pointing out the
| relevant context.
|
| It isn't, though. The argument is not "lol Newtonian
| physics are dumb" and you'll see nothing of the sort in the
| Norton argument. The argument is instead "Newtonian physics
| can be non-deterministic", which is something we can
| demonstrate regardless of the validity range of Newtonian
| physics.
|
| > This isn't an argument in a vacuum, there's an academic
| discipline that has thoroughly engaged with it.
|
| Sure. How is that linked to your point?
|
| > But the dome is rather boring. P1, this math describes a
| deterministic system; P2, here is a non-deterministic
| result; QED, P1 is false.
|
| Indeed.
|
| > Since we know that P1 doesn't describe the universe, it
| should be hard to have a strong opinion unless you're
| invested in a philosophical position about pre-modern
| scientific practice.
|
| That's a strange position to take from a philosophical
| point of view. Why things are wrong matters more than
| whether they are because, again, everything is wrong to
| some extent. So, of course in some abstract epistemological
| sense anything that can be formulated within Newtonian
| mechanics is wrong. But then nothing is right, so who
| cares?
|
| The initial question is much more interesting from a
| philosophical point of view: does the old Newtonian
| mechanics, which is still the closest to our daily
| experience, contains seeds of non-determinism? But then the
| logic is flawed: the issue is not that Newtonian mechanics
| are wrong, it is that, in your formalism, P2 does not
| follow from P1 and is actually wrong.
| Cushman wrote:
| Mmm, I promise I haven't taken any position. Is that a
| position? In which case, mea culpa, I really don't care
| if the dome holds. Even if I cared, I wouldn't care,
| because space invaders already gets us all that plus
| sound effects.
|
| Not for nothing, even historians and philosophers of
| physics broadly don't care if the dome holds. Some do, of
| course, but unless you're thinking of a specific paper
| it'll probably be less frustrating for everyone to leave
| it there-- with no hard feelings!
| Cushman wrote:
| I'll take my lumps for saying this: If you've downvoted me I'd
| love to know why!
|
| Did I sound preachy? I really don't mean to! I'm not an
| educator, and it's hard to communicate a discipline's "common
| knowledge" without coming off a bit patronizing.
|
| It's just, every time the dome comes up people want to talk
| about the _physics_ of it-- but the author isn't a physicist,
| the journal isn't for physicists, it's not making any claims in
| physics...
|
| It's a (famous) philosophy paper, specifically philosophy of
| science, specifically philosophy of physics. If you aren't a
| philosopher, physics is annoyingly irrelevant here.
|
| There's so much more to say-- space invaders! It's way weirder
| than this! The math still checks out! Philosophy is cool
| actually! Sorry to take up your time!
| Animats wrote:
| > this is an interesting result in philosophy of physics, not
| in physics.
|
| Yes. Although it has consequences. It's a demonstration that
| some physical variables have to be quantized or probabilistic
| to avoid divide by zero errors in reality.
|
| This becomes clear when you do idealized Newtonian physics with
| impulses. An impulse is an infinite force applied over zero
| time with finite energy transfer. That's not something that can
| exist in the physical universe. It's just asking for divide by
| zero problems. It's also why impulse-constraint physics engines
| for games have some rather strange semantics.
| crazygringo wrote:
| > _you may also look up the lesser-known "space invaders"_
|
| Can you provide a pointer/link? I'm googling with various other
| keywords and can't find anything that isn't the video game.
| Cushman wrote:
| Oh, of course! SEP's summary is pretty readable.
| https://plato.stanford.edu/entries/determinism-
| causal/#ClaMe... Past that you'll probably need to read the
| papers, it's very niche.
|
| Sorry, I should have remembered that's not actually an easy
| google :)
| crazygringo wrote:
| Thanks for that! What a fascinating set of problems.
|
| It is interesting that all of the ones listed in that
| section involve either infinities or infinitesimally small
| points -- except for Norton's dome. Which really makes it a
| great example for that reason.
| ajkjk wrote:
| This seems like a mathematical trick -- pretending like perfectly
| smooth functions exist in reality -- which is then extrapolated
| from in bizarre and unilluminating ways.
|
| The only reasonable reading of Newton's laws (and descriptions of
| e.g. physically-constructed curves like this dome) is that they
| are true up to some small epsilon length scale. No matter how
| small the epsilon, as long as it is not literally zero, this
| doesn't work.
|
| (for instance if things aren't perfectly smooth then there is
| some small force proportional to the discrepancy ~O(e^2) or O(sin
| e) or whatever, which moves the ball off the dome)
|
| If I was teaching physics from sceatch, I would state on day one:
| physics is the practice of building models whose low-order
| approximations give correct predictions about reality. There is
| no such thing as a perfect model to infinite decimal places.
|
| (That one 9 decimal place calculation from QFT doesn't invalidate
| this: given a model, the calculations may be perfectly accurate!
| But the model is still fuzzy because of fuzzy inputs. In that
| case, it has been possible to make it very very not fuzzy.)
| aidenn0 wrote:
| > This seems like a mathematical trick -- pretending like
| perfectly smooth functions exist in reality -- which is then
| extrapolated from in bizarre and unilluminating ways.
|
| Given that it is common to make certain claims about the model
| of NM (e.g. determinism and reversibility) a mathematical trick
| that demonstrates this is not true is a valid refutation of
| claims about the model.
|
| You can use NM to model other physically impossible systems
| (such as objects with arbitrarily high rigidity or arbitrarily
| low friction), but they are not internal contradictions within
| the NM model, so are less valid for criticism of NM.
| spekcular wrote:
| You may be interested to know that this exact objection has
| been made in the philosophical literature. See "Causal
| Fundamentalism in Physics" by Zinkernagel (2010). Available
| here: https://philsci-archive.pitt.edu/4690/1/CausalFundam.pdf
|
| At the end, the author notes (as you do) that if you consider a
| finite difference equation with small time steps, there are no
| pathological solutions. He also mentions that Newton takes this
| difference equation approach when solving problems in his
| _Principia_.
|
| See also "The Norton Dome and the Nineteenth Century
| Foundations of Determinism" by van Strien:
|
| >> Abstract. The recent discovery of an indeterministic system
| in classical mechanics, the Norton dome, has shown that
| answering the question whether classical mechanics is
| deterministic can be a complicated matter. In this paper I show
| that indeterministic systems similar to the Norton dome were
| already known in the nineteenth century: I discuss four
| nineteenth century authors who wrote about such systems, namely
| Poisson, Duhamel, Boussinesq and Bertrand. However, I argue
| that their discussion of such systems was very different from
| the contemporary discussion about the Norton dome, because
| physicists in the nineteenth century conceived of determinism
| in essentially different ways: whereas in the contemporary
| literature on determinism in classical physics, determinism is
| usually taken to be a property of the equations of physics, in
| the nineteenth century determinism was primarily taken to be a
| presupposition of theories in physics, and as such it was not
| necessarily affected by the possible existence of systems such
| as the Norton dome.
| [deleted]
| MightyBuzzard wrote:
| [dead]
| lisper wrote:
| I don't understand why anyone would think this is even the
| slightest bit weird. The situation described is dynamically
| unstable. The object remaining at rest forever is only possible
| in the Platonic ideal: zero friction, infinitely rigid materials,
| no thermal motion, no external perturbations. As soon as the
| state diverges from the Platonic ideal in any way, positive
| feedback will amplify that divergence. So the prediction for the
| Platonic ideal is exactly what one would expect: the object might
| stay still, or it might start to move at any time without any
| apparent cause.
|
| Of course, this situation can never actually be observed because
| even if you could somehow construct it (and good luck with that),
| you couldn't actually _look_ at it because merely shining a light
| on the object would give it a nudge.
| sebzim4500 wrote:
| Maybe I'm just stupid but I found it very surprising that such
| a shape exists (i.e. stopping a ball at the top in finite time)
| even in a platonic ideal.
| amluto wrote:
| The force at the top is not differentiable, which allows
| surprising things to happen -- see my other comment.
| michael1999 wrote:
| Consider how many pop-sci singularitans think their god in a
| box could predict the future through mere calculation. Many
| people do have a naive clockwork-universe model of causality.
| nabla9 wrote:
| This is just one example of non-uniqueness condition. There are
| first-order differential equations that don't have unique
| solution for a given initial condition.
| divs1210 wrote:
| > merely shining a light on the object would give it a nudge
|
| so you're saying you would change the outcome by measuring the
| system?
| bee_rider wrote:
| Sure, but more in the "I knocked the plank off the desk by
| accident while measuring it, by hitting it really hard with
| the ruler" sense, and less in the quantum physics is weird
| sense.
| kp1197 wrote:
| The article is asking us to consider an idealized situation
| lampiaio wrote:
| In an idealized situation though, why would the ball _not_
| stay at rest?
| tsimionescu wrote:
| The whole point is that Newtonian mechanics doesn't
| uniquely predict the motion of the ideal ball on that ideal
| shape. The ball could stay there forever, but it could also
| start moving down along the shape at any point in time -
| both are valid possibilities in the idealized model. This
| is the unintuitive part.
| orangecat wrote:
| _The ball could stay there forever, but it could also
| start moving down along the shape at any point in time_
|
| Only if the fourth derivative spontaneously changes from
| zero to nonzero. It doesn't seem any more surprising than
| the conditions f(0)=f'(0)=f''(0)=0 not uniquely
| determining f(x) for all x.
| tsimionescu wrote:
| The condition imposed by the construction of the problem
| and the laws of motion is that f''(t) = sqrt(t), and that
| f''(t) = 0 => F(t) = 0. The function given as an example
| in the article, f(t) = {(1/144) (t-T)^4, t >= T | 0, t <
| T}, obeys both laws, just as much as f(t) = 0 does.
|
| I'm not sure what the fourth derivative has to do with
| this argument.
| romwell wrote:
| The point is that it's unstable _even in the Platonic ideal_.
|
| That's the surprising part.
| lisper wrote:
| Well, yeah, but my point is that it shouldn't be surprising.
| If you think about it, if it is even _possible_ to bring a
| particle to rest for a finite time in the Platonic ideal then
| that plus time reversal necessarily entails non-determinism.
| So non-determinism should be no more surprising than the fact
| that it is actually possible to bring a particle to rest for
| a finite time.
|
| I think the only reason this example surprises people is that
| everyone just _assumes_ that bringing a particle to rest is
| possible /easy without really thinking through what this
| would actually require in the Platonic ideal. It's actually
| very challenging to stop things from moving without friction.
| crazygringo wrote:
| > _then that plus time reversal necessarily entails non-
| determinism._
|
| No. Generally speaking in the "Platonic ideal", we assume
| that if we reversed time, the particle would move from rest
| in a deterministic way, that there's only one solution for
| its movement. The surprising part here is that there are
| _multiple_ valid solutions. Which takes a very-specially
| designed "dome" to demonstrate -- which is very surprising
| indeed.
|
| > _It 's actually very challenging to stop things from
| moving without friction._
|
| No it's not. All it takes is a billiard ball in motion
| hitting another billiard ball at rest. The first billiard
| ball will now have stopped moving. Which is kind of the
| canonical example of how we _expect_ things to be
| deterministically reversible.
| lisper wrote:
| Yes, you're right, and this occurred to me after I posted
| but while I was in the shower so I couldn't correct it
| :-) I should have said "if it is possible to bring _all_
| of the constituent particles in a system to rest for a
| non-zero time... "
| ShamelessC wrote:
| You're just reframing the article as though it's obvious
| and then claiming that it is obvious. The problem is - the
| article already does just that.
|
| Further, you have changed your stance from one of "of
| course it moves from some noise in the environment" to one
| more closely resembling the article's main points.
|
| You basically aren't making sense. Your initial abuse of
| "platonic ideals" is the sort of thing that gives
| philosophical arguments a bad reputation.
| tylerneylon wrote:
| The article is pointing out that, if we think of Newtonian
| mechanics as a set of math equations, then there exist simple
| systems in which the future of the system has multiple equally-
| valid solutions to those equations.
|
| Mathematically, it's possible to have a differential equation
| with an initial condition, and still have multiple different
| solutions. That's what the author creates as well.
|
| I don't think this article is about continuous vs discrete
| physics, as some commenters suggest - nor about reality at all
| (not directly). Rather, this is pointing out a surprising
| property of the model of Newtonian mechanics.
| AbrahamParangi wrote:
| This is more of a quirk of which axioms you choose and which you
| discard rather than anything else. For instance, implicit in this
| model is a continuous representation of the universe, but the
| universe is not actually continuous!
| eximius wrote:
| That hasn't been proven, has it?
| romwell wrote:
| Well sure. The post is about the axioms, and the paradox that
| follows from them.
| tsimionescu wrote:
| All current models of space-time (including quantum mechanics,
| QFT, and GR) have space and time as continuous quantities.
| jakeinspace wrote:
| There is no evidence that spacetime is discrete. It's quite
| plausible, but even in a quantum gravity theory it's not a
| given.
| Zamicol wrote:
| Exactly. This makes the assumption that the universe is
| continuous instead of discrete. Quantum physics is all about
| discreteness. Newton himself suspected light was discrete, what
| he named "corpuscles".
|
| Even if continuous, I would still argue against this article.
| In a continuous universe non-determinism, randomness, is not
| needed. In the provided example I would expect no action to
| take place, or acknowledge that a continuous universe infers an
| infinite resolution of information for physical systems.
| kergonath wrote:
| There's some confusion here.
|
| > This makes the assumption that the universe is continuous
| instead of discrete. Quantum physics is all about
| discreteness.
|
| This is irrelevant. The discussion is within the framework of
| classical Newtonian dynamics. The discrete-ness of the
| universe has no effect whatsoever, because the moment we say
| we're using classical mechanics, we assume a flat Euclidean
| continuous spacetime. The argument is made in these terms,
| and in can be proven or disproven in this framework.
|
| > Newton himself suspected light was discrete, what he named
| "corpuscles".
|
| This has absolutely nothing to do with whether spacetime
| itself is quantised or not. You can have a concept of point
| particles in a continuous space, that's not a problem.
|
| > In a continuous universe non-determinism, randomness, is
| not needed. In the provided example I would expect no action
| to take place
|
| That is a good intuition.
|
| > or acknowledge that a continuous universe infers an
| infinite resolution of information for physical systems.
|
| However, this is not. Even in a continuous universe (and,
| until proven otherwise, our understanding is that ours is),
| infinite "information" is not really a thing.
| Zamicol wrote:
| >infinite "information" is not really a thing.
|
| That's the point. That's why a continuous universe is
| absurd from an informational perspective.
| kergonath wrote:
| You missed my point. It's not a thing _in a continuous
| universe_. And again, there is no proof whatsoever that
| ours is not, and we don't need to invoke any spacetime
| quantisation to explain any of the current established
| theories. I am not saying that the universe is not
| discrete (I just don't know), but if it's discrete
| character we're this trivial to observe, it would have
| been settled for a long time.
| sebzim4500 wrote:
| To my knowledge, there are very few concrete proposals
| for what a discrete universe would actually look like.
| Most physicists believe that the universe should be
| symmetric under lorentz transforms, which rules out all
| the obvious ways like splitting the universe into little
| cubes.
| H8crilA wrote:
| The universe is continuous (both spatially and time-wise) in
| Newtonian mechanics.
| azeemba wrote:
| And in quantum mechanics, we don't even know if spacetime is
| quantized right?
|
| People usually talk about planck length as some quantized
| length but it's just a scale where quantum effects are
| significant.
| bjornsing wrote:
| Interesting... If you take the initial conditions of the ball
| sitting still at the top of the dome and integrate Newton's laws
| stepwise forward in time, then surely the ball will just sit
| there forever. But for some reason this is not certain in
| continuous time?
| selimthegrim wrote:
| I'm pretty sure this is a failure to understand virtual work and
| virtual displacement.
| gus_massa wrote:
| I agree, ut I think your comment needs more details. I'll try:
|
| The idea is that the Newton's Laws an be rewritten and get
| equivalent equations that look very different. One of the
| rewrites is "Lagrangian Mechanics". Instead of a differential
| equation, you have a calculus of variations. In this new
| formulation it's more clear that the solution is not unique,
| and you get the particle that rest on top of the dome, the one
| that falls down, and even some solutions where the particle
| waits a little and then "decides" to fall down without an
| eternal trigger. For this formulation, two of the man concepts
| are virtual work and virtual displacement. More details in:
| https://en.wikipedia.org/wiki/Lagrangian_mechanics#D'Alember...
| amluto wrote:
| Off the top of my head (haven't verified carefully):
|
| What they're really saying is that they have an initial value
| problem in classical mechanics that does not have a unique
| solution.
|
| Fortunately, the theory of such things is very well established.
|
| From the article, the dome results in motion like this:
|
| > d2r/dt2 = sqrt(r) [reformatted as text]
|
| This function is interesting at r=0 -- it's not differentiable,
| and it's not even locally Lipschitz. So one would not generally
| expect solutions to the initial value problem to be unique, and
| math and classical mechanism still work.
|
| You can read more about the theory here:
|
| https://en.m.wikipedia.org/wiki/Initial_value_problem
|
| (The differential equation in question is second order and is in
| two spatial dimensions. The standard transformation to an
| ordinary vector differential equation applies, and you end up
| with four variables (x, v_x, y, v_y or however you like to name
| them) plus time, and the time derivatives of v_x and v_y as
| functions of everything else are not Lipschitz in any open set
| containing the origin.)
|
| And, intuitively, what's going on is that the acceleration of the
| mass is exquisitely sensitive to position around the top of the
| dome (it varies infinitely quickly with a small displacement),
| which is precisely what breaks the uniqueness of the initial
| value problem.
| koryk wrote:
| I am not a physicist, but the "no probabilities" problem could
| just be solved with a constant offset right? Or just say that
| whichever direction the marble rolls is the zero degree mark.
|
| I don't see how "no cause" problem is fully true either. The
| system has potential energy due to the marble being places at the
| top of the dome, so that was the cause.
| romwell wrote:
| To see "no probabilities" easier: there's no probability
| distribution for _when_ that ball starts rolling.
| JohnDeHope wrote:
| "...with acceleration due to gravity g... A point-like unit mass
| slides frictionlessly over the surface under the action of
| gravity."
|
| If you make up funny physics problems that don't look like the
| real world, then the math works out funny. What about heat,
| light, the motion of the larger entire system? There are a lot of
| forces they are just hand waving away. I think it's disingenuous
| to hand wave away much of the complexity of the real world and
| still expect your algebra problem to not be overly simplistic.
| kergonath wrote:
| It's frustrating as the text does not go anywhere near
| demonstrating any "simple violation of determinism in Newtonian
| mechanics". The core issue of the discussion linked in the story
| is that there are mathematical solutions to Newton's second law
| that are inconsistent with other bits of classical physics (in
| this case that a particle at rest in a given Galilean frame of
| reference cannot just start moving without something else
| applying some force to it). That is entirely true, interesting,
| and there is nothing wrong with this.
|
| What this does not tell us is how it somehow violates any kind of
| determinism.
|
| > Then there are many solutions of Newton's law F = ma. In one
| the ball remains at rest on top of the dome. But in others, it
| starts to roll down the dome in some arbitrary direction!
| Moreover it can start rolling at any time.
|
| That is just not going to happen in (classical) reality, though.
| Because once you properly set the initial state of the ball
| (force=velocity=0, or any other values), then the solution
| becomes unique and that's it, there is one possible trajectory.
| The ball starting to move without anything acting on it would
| violate other principles of classical mechanics. It's not going
| to happen regardless of whether that trajectory is consistent
| with Newton's second law.
| SlySherZ wrote:
| The article is hard to follow for me, but if I understood it
| correctly, this is not true:
|
| > That is just not going to happen in (classical) reality,
| though. Because once you properly set the initial state of the
| ball (force=velocity=0, or any other values), then the solution
| becomes unique
|
| If you set velocity = velocity = 0, then the ball staying at
| the top is a valid solution, AND the ball rolling down the hill
| (in any direction) is also a valid solution.
|
| If this sounds confusing (it did for me), look at the example
| at the end, it's possible to do the reverse - send the ball
| rolling up the hill with perfect velocity, such that it stops
| at the very top after time T. And if _that_ is possible, the
| opposite is also possible because NM is time reversible.
| kergonath wrote:
| > The article is hard to follow for me, but if I understood
| it correctly, this is not true
|
| You are right, I was missing some conditions. The higher
| order derivatives need to be zero as well.
|
| > If you set velocity = velocity = 0, then the ball staying
| at the top is a valid solution, AND the ball rolling down the
| hill (in any direction) is also a valid solution.
|
| It is a valid solution to the f=ma equation. It is not a
| valid trajectory in Newtonian physics because it violates
| other principles. It is a "gotcha" only if you think that
| Newton's second law is the entirety of classical mechanics.
|
| > If this sounds confusing (it did for me), look at the
| example at the end, it's possible to do the reverse - send
| the ball rolling up the hill with perfect velocity, such that
| it stops at the very top after time T.
|
| This paragraph is confusing. And does not demonstrate much of
| anything, instead asserting facts that we are supposed to
| believe.
|
| In the time-reversal "experiment", where the particle comes
| from the rim towards the apex, it ends up at the apex with a
| non-zero fourth derivative, because of the pathological shape
| of the dome. It cannot stay on the apex for any length of
| time, even with a velocity of 0. It is completely different
| from a particle starting at rest on the apex.
|
| > And if that is possible, the opposite is also possible
| because NM is time reversible.
|
| It is not.
| sgregnt wrote:
| > It is a valid solution to the f=ma equation. It is not a
| valid trajectory in Newtonian physics because it violates
| other principles. It is a "gotcha" only if you think that
| Newton's second law is the entirety of classical mechanics
|
| Could you please elaborate which Newtonian principles it
| does violate?
| kergonath wrote:
| The simplest one is that a particle on its own keeps a
| linear trajectory with a constant speed. A change in that
| (like going from rest to any motion) requires interacting
| with another particle: things do not start moving for no
| reason. This is a generalisation of one of the
| formulations of Newton's first law, which states that
| things that don't move don't start moving without being
| pushed (rough translation).
|
| This is related to another formulation of Newton's first
| law: if there is a force that pushes the ball at some
| time T, it implies that there is another body that felt
| the opposite force.
|
| Another one is a bit more involved, but basically a
| mechanical system cannot change its symmetry by itself.
| In this case, the initial state with a ball at rest has a
| radial symmetry with a centre on the apex of the dome.
| This is not true anymore if the ball moves in one
| direction. This is related to the conservation of
| momentum.
|
| There are a couple of points that can be solved easily,
| but are clearly defects in the original formulation of
| the problem. for example, the height according to the
| equations is not a length, which is not a problem itself
| (we can just multiply by an arbitrary factor with the
| right dimensions) but an indication of sloppy thinking
| and hand waving. Similarly, the force is not bounded in
| the original formulation. Again, this can be fixed by
| restricting the valid range for r, but is rather messy.
| geysersam wrote:
| This is not correct. Momentum is conserved by the
| spurious solution and there's still an equal but opposite
| force on another body (the body producing the
| gravitational force).
|
| I think this example just illustrates a case where the
| Newtonian model of reality simply does not describe
| reality itself
| eigenket wrote:
| Theres a couple of mistakes here. Firstly the particle is
| not on its own, it is being acted on by the dome and by
| gravity.
|
| The thing about symmetry breaking also doesn't make much
| sense. I guess you're trying to appeal to Noether's
| theorem, but Noether's theorem in classical mechanics is
| a consequence of f = ma. You derive the Lagrangian
| formulation of mechanics from f=ma and Noether's theorem
| from that. However the weird solution when then ball
| suddenly randomly falls down the dome after staying put
| for an arbitrary time is completely consistent with f=ma,
| so that can't help you here.
|
| In any case the _radial_ symmetry you 're looking for
| (the system is invariant under rotations around the peak
| of the dome) implies conservation of angular momentum
| about this point, and not about any other point (since
| the setup is manifestly not symmetric under rotations
| about any other point). However (one can easily check)
| that for both the static solution and the randomly starts
| moving solution, the angler momentum about the axis
| through the peak of the dome is always zero.
| tiberious726 wrote:
| Even if it does, that would amount to a contradiction in
| Newtonian mechanics. You don't get to simply ignore that
| the ball starting to roll after arbitrary (non
| deterministic) time T is a solution to these equations.
|
| (Note that the article goes on at length separating
| Newtonian mechanics from the "real world" or whatever)
| lisper wrote:
| > If you set velocity = velocity = 0, then the ball staying
| at the top is a valid solution, AND the ball rolling down the
| hill (in any direction) is also a valid solution.
|
| Yes, that is exactly right. Not only in any direction, but
| beginning at any time.
|
| The easiest way to see this is described at the end: imagine
| the ball is initially in motion and the initial conditions
| are precisely those that bring it precisely to rest at the
| apex of the dome at some time T. (Making this possible is the
| reason the dome has to be a specific shape. Not all shapes
| allow this.) The time-reversal of this motion is the ball
| beginning to move in some arbitrary direction at some
| arbitrary time.
| eesmith wrote:
| > The time-reversal of this motion ... at some arbitrary
| time.
|
| The "ball rolling to the top of the sphere" requires
| infinite time. "Some arbitrary time" is an expression of a
| finite time.
|
| You cannot simple mix ideas of finite and infinite and have
| the result make sense, as anyone who has stayed at the
| Hilbert Hotel knows. https://en.wikipedia.org/wiki/Hilbert'
| s_paradox_of_the_Grand...
| crazygringo wrote:
| > _The "ball rolling to the top of the sphere" requires
| infinite time._
|
| No it doesn't, because it's not a sphere. The dome is
| specifically designed so that it takes finite time.
| There's zero involvement of infinity, or mixing infinity,
| here.
| kergonath wrote:
| > No it doesn't, because it's not a sphere. The dome is
| specifically designed so that it takes finite time.
|
| Can you explain why?
| crazygringo wrote:
| I don't know what kind of answer you're looking for. The
| equation was explicitly chosen/derived to have this
| property. I assume the mathematical proof of that isn't
| something that fits in a few sentences in an HN comment.
| [deleted]
| xyzzyz wrote:
| The article explicitly discusses the fact that this is
| possible specifically because of the shape of the dome,
| and does not work on a hemisphere, precisely for reason
| you bring up.
| eesmith wrote:
| You are right - I misread it.
|
| The next step would be to verify that the paths always
| stay on the surface. The mathematics shown says the point
| always follows the surface, but I don't see a
| demonstration that that's true.
|
| I no longer have the skills to easily do this
| calculation.
|
| EDIT: Oh man, I used to be a lot better at this. I
| remember the mgh = 1/2 m v^2 and the slope calculation,
| but can't figure out how tell when the falling point mass
| detaches from the slope. If it detaches at h=0 then
| there's no physically viable reversed path on the
| surface.
| kergonath wrote:
| The article explicitly discusses this without
| demonstrating anything. On it's face this argument has
| the weight of these demonstrations.
| wruza wrote:
| If you throw a ball into a bowl, it will also find the
| (anti-)apex. And the time-reversal of that is the ball
| arbitrarily choosing a direction to jump off the center of
| the bowl. So what? Why is it important to mention in case
| of a non-stable equilibrium?
| scatters wrote:
| A ball rolls through the bottom of the bowl and out the
| other side. It doesn't come to rest.
| cma wrote:
| In a time reversal situation where the ball is at maximum
| magnitude momentum, it will just go the opposite way,
| right?
| kergonath wrote:
| > The easiest way to see this is described at the end:
| imagine the ball is initially in motion and the initial
| conditions are precisely those that bring it precisely to
| rest at the apex of the dome at some time T.
|
| This is a red herring. It sounds plausible, but there is no
| trajectory that does this. This is the weakest paragraph in
| the original post, and I am not sure whether this is
| intentional (because the demonstration sounds truthy if you
| don't go too deep in the details) or whether it was not
| entirely thought out. There is some discussion about the
| time-reversal thing here:
| https://blog.gruffdavies.com/2017/12/24/newtonian-physics-
| is... . There isn't much to discuss however, because
| ultimately it is just a distraction.
| hexane360 wrote:
| There's a lot of minor points in that post, but it seems
| like both authors largely agree on the meaning, but are
| using different language. From Dr. Davies' post:
|
| >To remain Newtonian and preserve determinism, we can
| exclude the singular point by constraining the higher
| orders to zero whenever the net force is zero. We lose
| time symmetry for this special case if we do this. If we
| wish to keep that, then we have to accept that Newtonian
| mechanics is incomplete and consider higher order
| differentials.
|
| And from Dr. Norton's article:
|
| >The solutions (3) are fully in accord with Newtonian
| mechanics in that they satisfy Newton's requirement that
| the net applied force equals mass x acceleration at all
| times.
|
| >An important feature of Newtonian mechanics is that it
| is time reversible, or at least that the dynamics of
| gravitational systems invoked here are time reversible.
|
| Dr. Davies is saying that there's three options: a)
| relaxing time-reversal symmetry (at singularities) from
| Newtonian mechanics, by interpreting Newton's First Law
| to apply to higher derivatives; b) considering Newtonian
| mechanics to be incomplete, and make (unspecified)
| choices about what trajectories of higher-order
| derivatives are acceptable; or c) accept non-determinism.
|
| Dr. Norton is defining "Newtonian mechanics" as
| necessarily having time-reversal symmetry, which prevents
| the first solution. He is also defining it as specifying
| acceleration only (which I think is quite reasonable),
| preventing the second solution. Therefore he's concluded
| the third solution: This mathematical stating of
| Newtonian mechanics is non-deterministic.
| kergonath wrote:
| You are entirely right, whether something that depends on
| higher-order derivatives can be called Newtonian is
| debatable. Personally I don't really care either way, as
| this is just a label. Newton did not mention higher-order
| derivatives but on the other hand they are a trivial
| extension to the mathematical framework. It is difficult
| to call a body at rest if any of the derivatives of the
| position is not zero, because then it will start moving
| instantaneously so it is hard to read the first law
| otherwise. And the second law does not care about
| anything other than acceleration. And there certainly
| isn't anything that prevents us from using clever shape
| to roll balls on, as long as the shape make physical
| sense.
|
| What this does not change, however, is that the dome does
| not demonstrate non-determinism. The apparent
| demonstration hinges on logical errors that remain errors
| regardless of the framework used, be it classical or
| quantum mechanics, or relativity.
| archgoon wrote:
| [dead]
| lisper wrote:
| > once you properly set the initial state of the ball
| (force=velocity=0, or any other values), then the solution
| becomes unique and that's it
|
| No, you've missed the point entirely. There are circumstances
| under which the solution is _not_ unique, and the article
| describes such a circumstance.
| [deleted]
| azeemba wrote:
| > Because once you properly set the initial state of the ball
| (force=velocity=0, or any other values), then the solution
| becomes unique and that's it,
|
| The original tumblr post linked to this wikipedia article:
| https://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_t...
| The initial conditions here are _not_ sufficient to pin to a
| unique solution.
|
| > anything acting on it would violate other principles of
| classical mechanics
|
| What other principles of classical mechanics does it violate
| though? I think that's the point of the exercise. To reflect on
| part of our mathematical modeling would prevent this.
| tsimionescu wrote:
| > Because once you properly set the initial state of the ball
| (force=velocity=0, or any other values), then the solution
| becomes unique and that's it, there is one possible trajectory.
|
| As the linked article points out, that is not true. Even for
| force = velocity = 0, there are solutions for which the ball
| starts rolling. Additionally, even adding the first law does
| not help, since the body has F = v = a = 0 at any time before
| T, and it has a > 0 only for times where it also has F > 0. So
| the body spontaneously starting to move is perfectly consistent
| with classical mechanics.
|
| A better attack on the example is that the shape of the dome
| may not be possible to construct from actual matter. The math
| only works with a shape that has a certain type of infinite
| smoothness with a single bump, which is probably not possible
| even in principle to construct from real matter.
| H8crilA wrote:
| This is a (quite clever) comment on a physics theory that we
| know to be incomplete. Not a comment on reality.
|
| And the math does check out. There really are many solutions to
| this system, i.e. it is nondeterministic under the theory.
| sobellian wrote:
| > That is just not going to happen in (classical) reality,
| though. Because once you properly set the initial state of the
| ball (force=velocity=0, or any other values), then the solution
| becomes unique and that's it, there is one possible trajectory.
|
| Not true, and the trick is very simple, you just contrive a
| force field that admits x''(0) = 0 but some non-zero higher
| derivative. Indeed, if you look at the proposed solution r =
| 1/144 (t - T)^4 for t > T, you'll see that:
|
| r(T) = 0
|
| r'(T) = 0
|
| r''(T) = 0
|
| r'''(T) = 0
|
| r''''(T) is ill-defined (but Newtonian mechanics AFAIK says
| nothing to forbid this)
| tkoolen wrote:
| > but Newtonian mechanics AFAIK says nothing to forbid this
|
| In the Wikipedia article on Newton's laws of motion, the
| first law is stated as "A body remains at rest, or in motion
| at a constant speed in a straight line, unless acted upon by
| a force.". Here it would seem that we leave the state of rest
| not due to a force, but due to some other cause, which the
| first law would forbid. So I think that the particular
| interpretation of Newtonian mechanics used by the author is a
| bit of a strawman.
| sobellian wrote:
| We must ask ourselves what "at rest" means since there are
| multiple trajectories that give x(0) = 0, x'(0) = 0 and it
| seems that some posters believe this cannot fully constrain
| the particle to be either at rest or in motion at t=0.
|
| There are multiple ways to resolve this dissonance. We
| could demand that the higher derivatives are _also_ zero.
| We could derive some elaborate rule excluding non-zero x
| ''(t) in the neighborhood of t=0. Or something else
| altogether. The issue IMO with these resolutions is that
| they're quite complicated and Newton almost certainly did
| not have any of them in mind.
|
| It's much simpler to just content ourselves that NFL is a
| special case of F=ma (where F=0). I'm not sure why we
| should contort ourselves to preserve determinism since we
| know that gets thrown out the window with QM anyway.
| pistachiopro wrote:
| Right, so for his nondeterministic path: r(t)
| = 0 ; t <= T r(t) = (1/144)(t-T)^4 ; t >=
| T
|
| We can see that r''''(T) is either 0 or 1/6, depending on if
| we go with the top or bottom equation. That does look like
| some sort of hidden state change, there.
|
| Interestingly, the spherical dome he mentioned (which doesn't
| yield to this non-determinism) forces all derivatives of r(t)
| to be continuous ...
| WindyLakeReturn wrote:
| >The ball starting to move without anything acting on it would
| violate other principles of classical mechanics.
|
| Isn't gravity acting on the ball? Ideally a ball balanced a the
| tip of a bowl has 0 net force, but only because it has the
| force of gravity pulling it into the bowl and an equal force
| from the bowl pushing back. This would require assuming a
| reality where there is no smallest particle, no atoms making up
| the bowl or ball as they are both perfect mathematical objects
| comprised of infinite points no matter what resolution you look
| at them at.
|
| The issue is that such a system, if it can be created by
| rolling the ball up the bowl with perfect precision from any
| direction, indicates that it is unstable and may reverse at any
| point, but without any obvious reason causing it to do so. So
| either the system has some non-deterministic factor which
| allows for the perfect stability to break at an arbitrary time
| in an arbitrary direction or it isn't time reversible as the
| ability to roll the ball up the bowl cannot be reversed.
|
| Looking at it another way, can you tell the difference between
| a ball I perfectly rolled to the top of the bowl 10 years ago
| and one I did 10 seconds ago?
|
| I do wonder if we've added so many assumptions, with
| mathematically perfect objects and infinite precision forces
| that we have created some sort of paradox. We have hit levels
| of perfectly spherical cows that cause even the perfectly
| spherical cows to complain about unrealistic standards.
| jasonhansel wrote:
| Doesn't this just prove that Newton's First Law is _not,_ in
| fact, equivalent to the claim that "in the absence of a net
| external force, a body is unaccelerated"?
|
| (There's also the question of whether r(t) in (3) is
| differentiable at t=T, right?)
| mathieutd wrote:
| There is an external force here: gravity.
| nabla9 wrote:
| More generally: There are first-order differential equations that
| don't have unique solution for a given initial condition.
|
| This is just one example of non-uniqueness condition in ordinary
| differential equations. Picard-Lindelof theorem tells when there
| are unique solutios
| azeemba wrote:
| The post itself doesn't seem to have much details. The linked
| article seems to have more info:
| https://sites.pitt.edu/~jdnorton/Goodies/Dome/
|
| Edit: Looks like the post has been updated to point here. Just
| for completeness, it used to point to
| https://johncarlosbaez.wordpress.com/2023/08/05/nortons-dome...
| sfink wrote:
| And the article itself is quite short and readable, and does a
| much better job of pointing out the weirdness. I recommend
| reading it.
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