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

[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.

 

[Dale]

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

No, it isn’t like that at all. The most relevant difference is that with the radioactive decay the other solutions can be experimentally confirmed to be physical, so they cannot be rejected. In the case under consideration, no experiment can possibly show that those discarded solutions are physical, so rejecting them is allowable.

I must also object to your language above suggesting that QM requires indeterminism

I am not knowledgeable enough on QM to refute this so I concede the point. The question in this thread isn’t about QM anyway, so it probably is best to stay out of that quagmire.

 

[A.T.]

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.

What definition of "determinsim" are you using here? Does "determinsim" mean that the same initial condition produces the same specific outcome, or just the same probability distribution for infinitely many possible outcomes?

 

[A.T.]

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

If we just apply Newton's Laws of motion: The net force on the ball is zero, so the ball will stay at zero velocity, and not roll down.

If that contradicts time-reversal-symmetry, then Newton's Laws are apparently not perfectly time-reversal-symmetric. But they do make a unique prediction in this case.

 

[jbriggs444]

If we just apply Newton's Laws of motion: The net force on the ball is zero, so the ball will stay at zero velocity, and not roll down.

If that contradicts time-reversal-symmetry, then Newton's Laws are apparently not perfectly time-reversal-symmetric. But they do make a unique prediction in this case.

No. They do not. If two results are both consistent with Newton's Laws then Newton's laws do not choose between them.

The correct conclusion that can be drawn from Newton's laws is that if the net force on the ball is zero, the second derivative of the ball's position is zero. That property is preserved in the solution where the ball begins moving.

You may be tempted to refer to the first law and suggest that an object subject to no net force must persist in a state of rest. However, the first law is still upheld. At every instant when the ball is accelerated, there is an open interval within which it is subject to a non-zero net force.

 

[hmmm27]

I'm curious (not being highly mathematically educated), if there's a distinction between

a) the ball's position is fixed, and
b) the ball's movement is zero (as the limit of a movement solution).

 

[jbriggs444]

I'm curious (not being highly mathematically educated), if there's a distinction between

a) the ball stays where it is, and
b) the ball's movement is zero (being solely the limit of a movement solution).

a) "the ball stays where it is" I would interpret as: "x(t) = x(0) for all t > 0"

b) "the ball's movement is zero" might be interpreted as: "x'(0) = 0" but could also be interpreted as x'(t) = 0 for all t >= 0".

If we take the former interpretation then a) and b) are not equivalent.

If we take the latter interpretation for b then an easy consequence of the mean value theorem is that a) and b) are equivalent.

 

[Halc]

What definition of "determinsim" are you using here? Does "determinsim" mean that the same initial condition produces the same specific outcome, or just the same probability distribution for infinitely many possible outcomes?

I was using the former definition in that quote.

Let's give some thought to some of the alternatives in a deterministic world:

Suppose the one determined outcome is that the object falls to the NNE after an hour, the same direction every time with that exact initial condition.

This would for one violate symmetry: The situation is identical in every way to the setup in a rotated coordinate system where the object now falls to the south, and the setup thus produces a sort of test for absolute coordinates. This alone is not fatal, but still implies a loss of principle of relativity.
The NNE fall is after an hour, and yet the condition after 50 minutes is in every way identical to the original condition, so by that argument, the solution is nondeterministic, or there is a hidden variable (timer) required to account for the determined behavior.

If the one determined solution is being balanced forever, we lose the time symmetry property of the physics. The object can go to the top and stop there, but the reverse situation cannot happen. That's also true of course of the above scenario where the ball can roll in from the west but cannot roll back to the west, only to the NNE after an hour.

 

[DrStupid]

I don’t see the applicability of the 2nd law of thermo here.

That was an analogy.

The 2nd law of thermo describes the macro state.

There are possible macrostates that would violate the second law. It is for example not impossible but just very unlikely that you die by asphyxidation because all oxygen molecules in your room coincidentally move away from you.

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

Isn't it the very purpose of Norton's dome to ask if these physical grounds are justified?

 

[Dale]

Isn't it the very purpose of Norton's dome to ask if these physical grounds are justified?

Then I don't think that Norton accomplished his purpose. You just get the same assumptions you started with.

If you start with the assumption that Newton's laws are causal then you choose the causal solution and discard the others and wind up with a single perfectly valid causal solution. If you start with the assumption that Newton's laws are acausal then you keep everything and wind up with a perfectly valid set of acausal solutions.

Both approaches are theoretically valid and there is no experiment that can inform either choice so you are free to stick with your prior assumptions either way.

 

[Dale]

If that contradicts time-reversal-symmetry, then Newton's Laws are apparently not perfectly time-reversal-symmetric. But they do make a unique prediction in this case.

To me this is a key point. You can pick time reversal symmetry or causality, not both. Since we have many experiments with causes and effects and remarkably few experiments reversing time, I tend to favor causality.

However, both assumptions are valid assumptions to make and both are consistent with Newton's laws.

 

[phyzguy]

To me this simple example shows the absurdity of determinism. In order for the ball to remain at the top of the pyramid forever, the position and velocity of the ball have to be controlled to infinite precision. While this is mathematically possible, it is physically absurd.

 

[Dale]

To me this simple example shows the absurdity of determinism. In order for the ball to remain at the top of the pyramid forever, the position and velocity of the ball have to be controlled to infinite precision. While this is mathematically possible, it is physically absurd.

Why is that an absurdity of determinism rather than an absurdity of the problem setup?

 

[DrStupid]

If you start with the assumption that Newton's laws are causal then you choose the causal solution and discard the others and wind up with a single perfectly valid causal solution. If you start with the assumption that Newton's laws are acausal then you keep everything and wind up with a perfectly valid set of acausal solutions.

That means that neither assumption results from Newton's laws. So where do the come from instead? In physics you shouldn't assume something just for fun. There should be a reason or a benefit.

 

[jbriggs444]

That means that neither assumption results from Newton's laws. So where do the come from instead? In physics you shouldn't assume something just for fun. There should be a reason or a benefit.

Aesthetics?

Today I come to the situation with the desire to treat it as a math problem. So I want the broadest solution set that is consistent with the specification of the problem.

Tomorrow I might want to be able to make a sweeping statement about determinism without worrying about corner cases that have 0% probability of arising, even with careful effort.

"Don't care" cases are useful when you are trying to craft a theory. Or a logic circuit. You are free to choose whichever behavior gives the most elegant theory. Or simplest circuit.

 

[Dale]

That means that neither assumption results from Newton's laws. So where do the come from instead? In physics you shouldn't assume something just for fun. There should be a reason or a benefit.

The reason is “because it usually works” (for either assumption). That is the same reason why we assume Newton’s laws to begin with. In science the only stronger reason is “because it always works”.

 

[DrStupid]

Aesthetics?

That's a matter of taste.

The reason is “because it usually works” (for either assumption).

If both assumption work than scientists actually don't have to use some criterion to reject some of the solutions. They can also keep them all.

 

[Dale]

They can also keep them all.

Certainly. I never said otherwise. My whole objection is to the claim that the ball “must” fall. You are allowed to keep them all, but you are also allowed to reject the acausal ones. As you say, it is a matter of taste

 

[A.T.]

Let's give some thought to some of the alternatives in a deterministic world:

Suppose the one determined outcome is that the object falls to the NNE after an hour, the same direction every time with that exact initial condition.

This would for one violate symmetry: The situation is identical in every way to the setup in a rotated coordinate system where the object now falls to the south, and the setup thus produces a sort of test for absolute coordinates. This alone is not fatal, but still implies a loss of principle of relativity.
The NNE fall is after an hour, and yet the condition after 50 minutes is in every way identical to the original condition, so by that argument, the solution is nondeterministic, or there is a hidden variable (timer) required to account for the determined behavior.If the one determined solution is being balanced forever, we lose the time symmetry property of the physics. The object can go to the top and stop there, but the reverse situation cannot happen. That's also true of course of the above scenario where the ball can roll in from the west but cannot roll back to the west, only to the NNE after an hour.

So to have deterministic roll-down you:
- lose time symmetry
- lose spatial symmetry / relativity
- have to introduce hidden variables

To have deterministic stay-on-top you:
- lose time symmetry

Seems like an easy choice to me.

 

[A.T.]

No. They do not. If two results are both consistent with Newton's Laws then Newton's laws do not choose between them.

The correct conclusion that can be drawn from Newton's laws is that if the net force on the ball is zero, the second derivative of the ball's position is zero. That property is preserved in the solution where the ball begins moving.

You may be tempted to refer to the first law and suggest that an object subject to no net force must persist in a state of rest. However, the first law is still upheld. At every instant when the ball is accelerated, there is an open interval within which it is subject to a non-zero net force.

I would say it depends on how you interpret Newton's Laws:

- If you interpret them as purely relational (stating how forces are related to acceleration), then you indeed have multiple solutions.

- If you interpret them as causal (stating how forces cause acceleration), then you cannot justify the departure from rest by pointing out, that once it starts moving there will be a net force, that caused it to start moving.

Usually the purely relational interpretation is sufficient, because adding the causal part doesn't change the quantitative prediction. This special case however is an example where only the causal part allows you to make/pick a prediction.

 

[Halc]

- If you interpret them as causal (stating how forces cause acceleration), then you cannot justify the departure from rest by pointing out, that once it starts moving there will be a net force, that caused it to start moving.

If you interpret them as causal (stating how forces cause acceleration), then you cannot justify the departure from motion by pointing out a net force that caused it to stop moving. Doesn't this make the uphill case contradict causality just as much?

 

[jbriggs444]

- If you interpret them as causal (stating how forces cause acceleration), then you cannot justify the departure from rest by pointing out, that once it starts moving there will be a net force, that caused it to start moving.

Usually the purely relational interpretation is sufficient, because adding the causal part doesn't change the quantitative prediction. This special case however is an example where only the causal part allows you to make/pick a prediction.

Can you give a definition for "causal" that is not a synonym for "deterministic"?

I agree that if the laws of physics are correct, and always make a deterministic prediction based on the current system state and if the current system state is completely characterized by the position, shape and velocities of all components then, if the ball stays on top for any time at all, the laws of physics must predict that it stays there forever.

I do not agree that Newton's laws alone make a deterministic prediction.

 

[wrobel]

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.

In this example the conditions of the existence and uniqueness theorem for ODE fail. So that there are several solutions with the same initial values. All ODE textbooks contain such examples. It is just another example showing that physics is not the same as mathematical models. But this stupid text by Mr John Norton arises again and again. Especially philosophers like such a stuff.

 

[DrStupid]

I do not agree that Newton's laws alone make a deterministic prediction.

Yes, that's what Norton's dome shows. If you just use Newon's laws you get infinite different solutions. If you want to have a single solution you need something else in addition to Newton's laws. I wonder why we want to have a single solution only. We know that there are systems that cannot be predicted in reality (at least because it is impossible to know the starting conditions with sufficient precision). In case of Norton's dome the prediction that the ball will start sliding after an unknown time is actually closer to reality than the prediction that it will stay forever. If we know that reality is not full predictable why should we implement something like that into the theory?

 

[bahamagreen]

Here is the approach in the tilting pencil, mentioned a ways back...

Tilting Pencils

 

[wrobel]

On the other hand there is a kind of nonuniqueness which is encountered in reality indeed.

Consider a match box on the table. Perhaps this match box has been laying here from yesterday or perhaps I pushed it here from another place a minute ago.

Systems of classical mechanics must meet uniqueness forwards but they are not obliged to have uniqueness backwards

We know that there are systems that cannot be predicted in reality (at least because it is impossible to know the starting conditions with sufficient precision).

yes and it is well described in terms of deterministic ODE