What happens if you put a sphere (ball) on the top of a pyramid?

[zvwner]

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]

Why would it fall ? Unless the placement isn't perfect, or there's a wind... or a bird flies into it, etc.

 

[zvwner]

Why would it fall ? Unless the placement isn't perfect, or there's a wind... or a bird flies into it, etc.

But if it is a perfect pyramid, I assume it is perfectly pointed. Therefore it can't remain stable. (I am just guessing tho).

 

[hmmm27]

Realistically no, it isn't stable ; but - also realistically - there's no such thing as "perfect".

If you like splitting hairs, the rotation of the Earth will bring the system through a slight shift in the direction of gravity as the relative position of the Sun and Moon changes. Then, it will fall off.

 

[zvwner]

I just thought that if this is a perfect system, it's basically the same that if we place the ball on a flat surface. Because if you place a theoretical perfect sphere on a theoretical perfect surface, they should only touch in one single point. Therefore If that single point is upon a flat surface or rather a pyramid, Nothing changes! (Of course, all this is in a hypothetically perfect world).

 

[Dale]

But if it is a perfect pyramid, I assume it is perfectly pointed. Therefore it can't remain stable.

If you placed it perfectly then stability doesn’t matter. It is balanced. Stability only matters if there is some imperfection.

 

[Halc]

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.

 

[Dale]

The perfectly balanced ball staying perfectly balanced is also a perfectly valid solution and can also be played in reverse for a valid solution.

 

[Halc]

I see some skepticism on my reply. The example is discussed in pitt.edu and stanford, quoting below the bottom of section 4.1

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.

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)

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.

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.

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. The pencil scenario is a poor example since in the time-reversed situation, the pencil is always approaching the vertical position but never actually reaches it, hence I cannot conceive of how it could fall in as little as 30 seconds.

 

[anorlunda]

If you placed it perfectly then stability doesn’t matter. It is balanced. Stability only matters if there is some imperfection.

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.

 

[hmmm27]

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.

But, the ball never actually reaches the apex.

 

[Halc]

But, the ball never actually reaches the apex.

True, which makes it a lot like the pencil example, no?
Why was Norton so specific about the function of the slope of his hill? 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.

So what if we devised a force map that was zero at the apex but some finite nonzero value everwhere else. I can think of some physcial situations that meet this (such as a pair of slippery shallow cones balanced point-to-point), so why didn't Norton use something like that as his example? I suppose I'd have to digest his reasoning to see why he made his choice of that particular cone. I think it has a property of always being wider than a parabola so that the ball sliding on it can never leave the surface.

 

[jbriggs444]

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.

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.

Norton chose the shape he did so that he could get the central force law that yielded a differential equation where the boundary conditions at t=0 are inadequate to constrain the solution. Many years ago in my first course on differential equations we were exposed to such a situation. I cannot now remember that equation.

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. Either way, that gives you a starting error which will be magnified over the course of 30 seconds and may result in a topple.

 

[Dale]

I see some skepticism on my reply. The example is discussed in pitt.edu and stanford, quoting below the bottom of section 4.1

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.

What they correctly say is that this is not the only valid solution. Solutions with the ball falling are also valid. So they support only the weaker claim that “the ball could fall” and be consistent with the equations of motion resulting in uncaused motion.

Scientists often have multiple solutions to an equation of motion. They have to use some criterion to reject some of the solutions. Since it could fall and it could stay (and since it is idealized to the point of non physicality) it makes sense to reject the falling solutions as they are non-causal.

Note that your references are both philosophy references. Philosophers may think that spurious solutions are deeply meaningful but scientists tend to simply reject them and move on.

 

[kuruman]

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.

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]

If you continue playing the time-reversed movie until its end, the ball will not roll back down

Who says it will not roll back down? The laws of physics are silent on the question. Likely I am missing your point.

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?

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.

Edit: Trying to avoid a philosophical discussion about determinism.

 

[A.T.]

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 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.

I would not describe that as "perfectly balanced" then, but rather as "uncertainly balanced".

 

[kuruman]

Who says it will not roll back down? The laws of physics are silent on the question.

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?.

Of course one could claim that a perfect hand places a perfect ball on the perfect dome with perfect initial conditions. In that case we go back to @Dale's post #6.

 

[phyzguy]

If you accept the uncertainty principle, then as @jbriggs444 said, you can show that the ball will roll down one side of the pyramid fairly quickly, since it cannot be placed at the top with perfectly know position and perfectly known momentum.

 

[A.T.]

Philosophers may think that spurious solutions are deeply meaningful but scientists tend to simply reject them and move on.

Or accept them and predict antimatter.

 

[DrStupid]

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.

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. Norton's dome is just theoretical, but we are still talking about physics and not about mathematics. All cases and almost all cases are not distinguishable in physics.

 

[jbriggs444]

Can't Newton's laws with said initial conditions make a prediction? If "no", why not?.

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.

 

[DaveE]

Sorry, in this universe you can't have perfect.

If you assume perfect objects, then I would argue you can also redefine most physical laws to get whatever answer you like. The whole scenario is fictitious. At the deepest, most pedantic, level the question is meaningless unless you fully describe your perfect universe as well.

For example, what is a smooth surface if it is composed of individual atoms? What external forces act on your objects? What about the uncertainty principle; is your ball both exactly balanced at the point and perfectly still?

 

[Vanadium 50]

A classical pyramid (assuming made out of a continuous material and not atoms) will have an infinitely sharp point. When the sphere is placed on it, the pyramid point will penetrate it (because the pressure is infinite) to a depth determined by Young's modulus. The sphere will not roll without a push.

If your response is "this doesn't count", I would respond that the question needs to be better defined then.

 

[kuruman]

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.

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.

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. Outside Newton's laws one can add the uncertainty principle, zero point motion, absorption and emission of radiation, Brownian motion, you name it. Because this system is so susceptible to perturbations, thare can be no meaningful experiment to verify what will happen. I don't believe that the laws of physics are silent. This question asks one to start the system at a point where chaotic behavior is the only outcome. I say move on.

 

[jbriggs444]

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

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.

 

[Dale]

Or accept them and predict antimatter.

Yes, that is a known risk of rejecting non-physical solutions. Sometimes our judgement of what is non-physical is flat out wrong!

Nevertheless, I think that it is clear that some solutions of some equations of motion are in fact non-physical. So, while I could be wrong about those specific solutions, the general idea of discarding non-physical solutions is necessary and can legitimately be used here to discard the acausal solutions.

 

[Dale]

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.

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.

Norton's dome is just theoretical, but we are still talking about physics and not about mathematics.

Right. Which is why we are justified in discarding some of the mathematical solutions on physical grounds.

 

[Dale]

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.

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.

 

[Dale]

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.

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.

 

[Halc]

In the time reversed scenario (on a Norton dome) the ball does reach the apex and stops at the apex in finite time.

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.

OK, that series converges to a finite number, so I stand corrected on that. My computation assumed the 'ball' is actually a point mass. Not sure if it matters, so perhaps it should also be considered for the case of a large ball sliding up there.

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.

Totally agree, and that was the gist of my first post in this thread.

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.

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.
Yes, a real pencil (or ball) is not on a mathematical hill, but is in fact a collection of particles with no real position until measured. A real pencil will fall for this reason. Norton's example, as I've stated, is strictly a mathematical one. Still, I have no idea where a meaningful time calculation like '30 seconds' can be derived from a mathematical situtation, so the pencil example must not have been a mathematical one. Wish I could find it. I agree with your assessment.

 

[DaveC426913]

Even if you specified a "perfect" balance of "perfect" shapes, the sphere will still fall, because the scenario is not static - it is in a gravitational field that is evolving continually.

 

[A.T.]

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.

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.

Or you define "perfect placing" via symmetry, which a ball rolling to one side would violate. In other words, if you have infinitely many solutions, but only one of them preserves the symmetry of the initial conditions, pick that one.

 

[Dale]

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.

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]

Been investigating the use of a ball instead of a point mass.
The OP's suggests the use of a pyramid instead of a roundish surface as Norton does: There is no simple solution for the ball sliding or rolling up there since it would have to break contact with the peak as it reaches it. Not sure how that would prevent the ball from falling one way or the other, but it would be airborne and thus we'd have to have it bouncing or something, and the reverse solution needs to reverse-bounce it up there.

The whole subject was nicely discussed in depth (as a ball rolling or sliding off a table edge) in another thread here. Post 19 in that thread says that even if the table/pyramid had a very shallow slope, but was not rounded at all, the ball would momentarily go airborn before contacting the slope again, thus the sliding or rolling ball could never stop at the apex.

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.

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.

Also, there are interpretations of QM that are entirely deterministic, so I must also object to your language above suggesting that QM requires indeterminism. In fact, depending on your definition of 'deterministic', a good percentage of them do in that they don't at any point invoke either outside influence (Wigner) or true randomness (God throwing dice).

I favor RQM myself, and I protest wiki listing it as indeterministic since the designation is meaningless given the full relational view.