Please include me also in contacts. I developed anosmia about 8 years ago (well before COVID). I truly wish there were some sort of 'cure' that would restore even a small amount of my sense of smell.
And it is an important correction because Fourier was immediately able to use his technique to solve partial differential equations, but it was many decades later before it was shown how it all works with the rigorous foundation of measure theory and functional analysis.
This is good news considering that the Galapagos Turtle is endangered. The mom has been in the Philadelphia Zoo since 1932 and has never had offspring till now. Here's to wishing the children a long and healthy life and many children.
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.
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.
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.”
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
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.
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.
> 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.
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.
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.
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.
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.
> 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.
>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.
Seriously, you don't seem to know much about things you speak confidently about.
Norton's dome is a surprising mathematical situation in very conventional classical mechanics. It doesn't matter what else Norton has done, this observation is trivial to verify for every undergrad maths/physics student.
This has absolutely nothing to do with entropy or the arrow of time.
The mathematical situation is of no practical relevance because it's "density zero": Generic deviations will destroy this peculiar behaviour.
>you don't seem to know much about things you speak confidently about
Good one, chap! How about you argue with substance instead ...
Explain, what makes the ball suddenly start rolling down the dome? Do not hand-wave, just give a direct answer to this question, based on your purported understanding of the problem.
> Good one, chap! How about you argue with substance instead ...
Considering they were replying to a post that was, effectively, arguing "nuh uh!", their response seems reasonable.
> Explain, what makes the ball suddenly start rolling down the dome?
That's _literally_ the entire point. Nothing does. There is nothing that causes the ball to start rolling. But the Newtonian laws of physics indicate it will.
> But the Newtonian laws of physics indicate it will.
Pedantically: they indicate it can. The situation where the ball spontaneously starts rolling[1] at any specific moment in time, without any application of force or interaction with any other part of the system, are perfectly legal and well-defined by the laws of motion. They just can't be predicted determinically.
[1] FWIW it's not even a ball in this case, as the rotational mechanics of a sphere with non-zero moment of inertia would destroy the very carefully constructed function required for the potential energy field.
Nothing. The Newton equations predict that, given the Norton potential, there are two possible trajectories that solve them.
The next state is not uniquely determined by the prior state, so asking what makes the ball roll shows that you don't understand the claim (non-determinism ) at hand.
I'm trying to wrap my head around your logic. So, I'll go step by step to make sure I get it.
If you were able to perfectly balance the ball on a perfectly constructed dome, blah blah, would the ball stay static indefinitely or would it start rolling down some arbitrary path?
The point is that the equations don't tell you what would happen. Both options would be valid according to the equations.
This is completely contrary to our intuition about Newtonian mechanics. The question "given this situation, what would happen?" typically has a unique answer is typical. If it does, we have determinism. The observation of Norton's dome is that mathematically this question does not have a unique answer in all situations.
Are you asking about the real world? Then the answer is that you can not perfectly craft the dome and what happens depends on the imperfections.
Are you asking about a fictional universe governed by the Newton equations and nothing else? Then I can not answer your question because the question builds on a faulty assumption: That this universe is deterministic and that what is determines what will be. Mathematics shows that to not be the case.
The only possible answer to your question in the second case is: It can not be known or predicted what the ball will do.
It is a concrete question! Here's a potential energy function describing a "hill". Here's an object on the hill at this location. How will it move? The question is well formed and complete. And it has more than one answer!
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".
did you? the title and content of this post is about math results. you should really consider the possibility that you're very wrong here.
the discussion is about hypothetical results from classical mechanics, which, along with the rest of physics, is a mathematical model that may be incongruous with observations.
I suggest you read the HN guidelines, you are quite abrasive and aggressive in your posts.
Regarding your post about entropy. The reason it does not apply is because entropy is a concept from statistical mechanics which is about the statistics of ensembles of many (even non-classical) particles. It's a concept invented after Newton dynamics, but does not apply to describing the equations of motion of a single particle (try to define the entropy of the single particle system). Time reversal is a core tenent of Newton dynamics.
Maybe you should read my most, I didn't say that statistical mechanics is nonclassical. I said statistical mechanics does not apply to the discussion of a single particle rolling up or down a slope. Tell me which of the states has more entropy the one with the particle at the top od the Norton dome or the one at the bottom?
And yet, the video in question seems to make it _very_ clear that this has been debated over and over, across various papers and people, and _nobody_ has been able to provide proof as to why it's wrong.
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.
It is more than a little disappointing that a solar plant in the desert cannot survive. While birds, who were killed at the plant by the thousands each year are rejoicing, those of us who seek a cleaner form of energy have been dealt a significant setback. Can industrial scale solar be made to work barring some new more efficient method to turn light into electricity?
It's not efficiency, it's capex. Ivanpah is solar thermal which was soundly defeated by photovoltaics and, now, batteries. Ivanpah's technology was inherently more expensive than PV, and its remaining advantage, ability to store thermal energy to extend generation into the evening, has been rendered less valuable by cheaper batteries.
What am I missing? I can sit in my den and watch SpaceX and now Blue Origin launches, in real time see the telemetry data, see stage separation, reentry burns, etc, etc. As for Saturn V, to quote the Byrds "I was so much older then..." but I don't recall any of that. Doesn't film require taking the exposed product to a lab for after the fact processing? While the Blue Origin images this morning were not nearly as good as the SpaceX, to me the images are absolutely incredible. I am a serious amateur photographer and I do still shoot film on occasion but I see little dynamic range differences now in 2025.
