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zvwner
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I was wondering what happens if you put a perfect sphere (a ball) on the top of a perfect pyramid. To which side will the ball fall and why? It is random? An if it is, does a pattern emerge after many attempts?
hmmm27 said:Why would it fall ? Unless the placement isn't perfect, or there's a wind... or a bird flies into it, etc.
If you placed it perfectly then stability doesn’t matter. It is balanced. Stability only matters if there is some imperfection.zvwner said:But if it is a perfect pyramid, I assume it is perfectly pointed. Therefore it can't remain stable.
Of course, Dale's posts about physical systems not being mathematically perfect are valid points. A physical system cannot be perfectly symmetrical since there are always moving atoms or forces applied by random objects. Nothing stands still in reality. So the above example is a classical mathematical one, not one composed of real quantum fundamentals with no exact state.stanford said:Finally, an elegant example of apparent violation of determinism in classical physics has been created by John Norton (2003). As illustrated in Figure 4, imagine a ball sitting at the apex of a frictionless dome whose equation is specified as a function of radial distance from the apex point. This rest-state is our initial condition for the system; what should its future behavior be? Clearly one solution is for the ball to remain at rest at the apex indefinitely.
But curiously, this is not the only solution under standard Newtonian laws. The ball may also start into motion sliding down the dome—at any moment in time, and in any radial direction. This example displays “uncaused motion” without, Norton argues, any violation of Newton's laws, including the First Law. And it does not, unlike some supertask examples, require an infinity of particles.
Figure 4: A ball may spontaneously start sliding down this dome, with no violation of Newton's laws.
(Reproduced courtesy of John D. Norton and Philosopher's Imprint)
It makes it fun as a party puzzle. In many situations, the perfect case is asymptotically close the the imperfect case, so it doesn't matter much. But in this case it matters a lot.Dale said:If you placed it perfectly then stability doesn’t matter. It is balanced. Stability only matters if there is some imperfection.
Halc said:The ball must fall after an undetermined amount of time (which has a calculable half life).
It is the reverse of the solution of a ball rolling (sliding actually) up a hill to a point bringing to exactly the point where its kinetic energy runs totally out. The ball rolls up there and stays put (for a while), and by time symmetry, it is valid physics to play that video in reverse, which is the situation you're describing.
True, which makes it a lot like the pencil example, no?hmmm27 said:But, the ball never actually reaches the apex.
In the time reversed scenario (on a Norton dome) the ball does reach the apex and stops at the apex in finite time. That is precisely the point. If the ball reaches the apex and stops for a time, it can re-start and fall back down at any time, and in any direction, non-deterministically, without ever violating Newton's laws.Halc said:It seems that the only important part of it is that it is locally level near the top, which yes, prevents the ball from ever reaching the apex. Even a ball on a pointed cone would never reach the apex.
Yes, but this doesn’t say the same thing you did. It even says specifically “one solution is for the ball to remain at rest at the apex indefinitely”, so your statement “The ball must fall” (emphasis added) is not supported.Halc said:
Say you place the ball on the dome and, after an unspecified time, the ball starts rolling down. As you say, in the time-reversed scenario, the ball will roll up and stop for a time. But then what? If you continue playing the time-reversed movie until its end, the ball will not roll back down but eventually your hand will pick it up and take it away. Classical determinism is safe. How long must one wait after the ball is placed on the dome before deciding that there is no causal connection between the subsequent motion of the ball and the manner of its placement on the dome?jbriggs444 said:In the time reversed scenario (on a Norton dome) the ball does reach the apex and stops at the apex in finite time. That is precisely the point. If the ball reaches the apex and stops for a time, it can re-start and fall back down at any time, and in any direction, non-deterministically, without ever violating Newton's laws.
Who says it will not roll back down? The laws of physics are silent on the question. Likely I am missing your point.kuruman said:If you continue playing the time-reversed movie until its end, the ball will not roll back down
No time at all. As long as position and velocity are both zeroed at the apex, the causal link is broken. Those boundary conditions are consistent with multiple continuations, either into the future or into the past.How long must one wait after the ball is placed on the dome before deciding that there is no causal connection between the subsequent motion of the ball and the manner of its placement on the dome?
Halc said:I cannot find the original article about the pencil, which computed that a perfect balanced pencil shaped object has a 50% chance of falling before 30 seconds at Earth gravity.
I would not describe that as "perfectly balanced" then, but rather as "uncertainly balanced".jbriggs444 said:I suspect that the calculation of the 30 second half-life (for the inverted pendulum case) has to do with an [overly?] simplistic application of quantum mechanics. The pencil will have uncertainty in position or momentum or both.
The recorded movie played backwards. You will see your hand pick it up and take it away if you play the reversed movie until its end. I don't mean to be facetious or funny with this but to point out that the hand specifies the initial conditions, the ball's position and momentum. If you eliminate the hand in the time-reversed movie, the ball will return to the same position with reversed momentum. What will happen then (without the hand to remove the ball) as time keeps on moving backwards? Can't Newton's laws with said initial conditions make a prediction? If "no", why not?.jbriggs444 said:Who says it will not roll back down? The laws of physics are silent on the question.
Or accept them and predict antimatter.Dale said:Philosophers may think that spurious solutions are deeply meaningful but scientists tend to simply reject them and move on.
Dale said:Yes, but this doesn’t say the same thing you did. It even says specifically “one solution is for the ball to remain at rest at the apex indefinitely”, so your statement “The ball must fall” (emphasis added) is not supported.
No. They cannot. The burden is not on me to explain why the laws of physics make no prediction. It is on you to explain why you think they can.kuruman said:Can't Newton's laws with said initial conditions make a prediction? If "no", why not?.
OK, perhaps I have a blind spot and it wouldn't be the first time or the last. I was under the impression that if one knows the potential and the initial conditions, then one can write a 2nd order diff. eq, using Newton's laws and solve it to predict the evolution of the system. Because this the case under a whole lot of circumstances, I assumed that it would be the case in all circumstances. I see the exception of "perfect" placing with the perfect ball right at the apex of the perfect dome with perfectly zero initial momentum. Within the framework of Newtonian mechanics we predict that the ball will stay in place forever. That's a prediction.jbriggs444 said:No. They cannot. The burden is not on me to explain why the laws of physics make no prediction. It is on you to explain why you think they can.
You can write the 2nd order differential equation, yes. But that equation is not predictive in this case -- in the sense that more than one solution is compatible with the equation and the very specific initial conditions.kuruman said:OK, perhaps I have a blind spot and it wouldn't be the first time or the last. I was under the impression that if one knows the potential and the initial conditions, then one can write a 2nd order diff. eq, using Newton's laws and solve it to predict the evolution of the system
Yes, that is a known risk of rejecting non-physical solutions. Sometimes our judgement of what is non-physical is flat out wrong!A.T. said:Or accept them and predict antimatter.
I don’t see the applicability of the 2nd law of thermo here. The 2nd law of thermo describes the macro state. The 2nd law is valid for the macro state, even when the micro state evolves deterministically. Here we are discussing a perfect micro state, so the deterministic rules are the relevant ones.DrStupid said:That would mean that the second law of thermodynamics (just as an example) is not supported because there remains a very small probability of violations.
Right. Which is why we are justified in discarding some of the mathematical solutions on physical grounds.DrStupid said:Norton's dome is just theoretical, but we are still talking about physics and not about mathematics.
Exactly. Because of this, the discussion is inherently non-scientific. There is no possible way to test it and find out. Any claim about the behavior of the system is unfalsifiable. It will fall, but did it fall because the world is non-deterministic/acausal or because of some imperfection in placement or shape? It is impossible to tell.kuruman said:Within the framework of Newtonian mechanics we predict that the ball will stay in place forever. That's a prediction.
What if the prediction is not borne out by experiment and the ball rolls off as it is most likely to do? Still within the framework of Newton's laws we can explain this behavior in terms of imperfect placement of the ball, imperfect zeroing of the its momentum, air currents, gravitational perturbations from passing cars, the Sun and Moon, etc, etc.
There are lots of cases where the equations of motion predict multiple solutions, some of which must be rejected to make a prediction. This scenario isn’t unique in that sense.jbriggs444 said:You can write the 2nd order differential equation, yes. But that equation is not predictive in this case -- in the sense that more than one solution is compatible with the equation and the very specific initial conditions.
In the vast majority of the cases we encounter in everyday life, Newton's laws are predictive. Given perfect knowledge of the initial conditions, unlimited computational power and a classical world, a single unique outcome is almost always dictated. We might hope that the subset of situations where theoretical predictability is lost are of measure zero relative to the set of all situations whatsoever. If so, we are "almost certain" never to encounter a situation where predictability is lost.
This is not accessible to experiment. We do not have the ability to set up this situation with perfect accuracy.
Based on the picture which gives height as a function of radius, one can compute mechanical energy as a function of radius, and from that get the kinetic energy. So I computed the time needed to go 3/4 of the way to the center: R=16, energy is 16**(3/2) = 64, so speed is proportional to 8 (I'm ignoring the constants, working only with proportions). Go from there to R=4 which is some amount of time proportional to 1.5. Going from there to 1 you get energy 8, speed 2.83 which takes 1.06 units of time, a ratio of sqrt(2), so the next one is going to take 0.75 units of time.jbriggs444 said:In the time reversed scenario (on a Norton dome) the ball does reach the apex and stops at the apex in finite time.
Totally agree, and that was the gist of my first post in this thread.That is precisely the point. If the ball reaches the apex and stops for a time, it can re-start and fall back down at any time, and in any direction, non-deterministically, without ever violating Newton's laws.
It was meant to illustrate a classic (non-quantum) example of an uncaused effect, thus providing evidence that it isn't just QM that allows indeterminism in physics.I suspect that the calculation of the 30 second half-life (for the inverted pendulum case) has to do with an [overly?] simplistic application of quantum mechanics.
Classically, you can also define "perfect placing" as the placing, for which the ball will stay in place forever. If the ball falls to one side, then the placing wasn't perfect, per definition.kuruman said:I see the exception of "perfect" placing with the perfect ball right at the apex of the perfect dome with perfectly zero initial momentum. Within the framework of Newtonian mechanics we predict that the ball will stay in place forever. That's a prediction.
The difference is that Newtonian physics allows determinism. Unlike QM (and unlike your post I objected to) classical mechanics does not require indeterminism and even in this scenario there exists a solution that respects causality.Halc said:It was meant to illustrate a classic (non-quantum) example of an uncaused effect, thus providing evidence that it isn't just QM that allows indeterminism in physics.
My wording did indicate that it must fall, and it indeed doesn't follow from the fact that falling is a valid solution. Still, it being possibly totally deterministic allows not just the one solution, but any of them. Insisting that the one symmetric solution is THE answer is like saying that determinsim must result in a radioactive nucleus never decaying in any amount of time despite its millisecond half life.Dale said:The difference is that Newtonian physics allows determinism. Unlike QM (and unlike your post I objected to) classical mechanics does not require indeterminism and even in this scenario there exists a solution that respects causality.