Norton's Dome

[swansont]

Brought up here; I'd never heard of this conundrum

 

http://en.wikipedia.org/wiki/Norton's_dome

http://www.pitt.edu/~jdnorton/Goodies/Dome/

 

Basically, there is a dome shape and the top is a point of unstable equilibrium. There is a solution where an object at the top can spontaneously move off of the top, and once it does, continues accelerating. This is offered as an example of a violation of determinism, as the motion is purportedly spontaneous.

 

But I have a quibble with (at least) one of the arguments

 

But one may still worry that spontaneous acceleration somehow violates Newton's First Law:

In the absence of a net external force, a body remains at rest or in a state of uniform motion in a straight line.

It is natural to visualize "uniform motion in a straight line" over some time interval, but we will need to apply the law at an instant. At just one instant, the law corresponds to motion with zero acceleration. So the instantaneous form of Newton's First Law is:

In the absence of a net external force, a body is unaccelerated.

 

 

I don't think that creating a new version of the first law means that you can claim this is consistent with Newton's laws. The emphasis on "uniform motion in a straight line" misses the important caveat of "remains", i.e. an object at rest remains at rest — that, to me, has a clear implication that you can't apply this instantaneously. It is a condition that must be true over an arbitrary interval of time. So an instantaneous form of the law that isn't violated is a bit of a straw man.

Edited June 3, 2015 by swansont
fix typo

[studiot]

Did I somehow not post my reply?

 

I was not impressed with the inadequate analysis of the mechanics of the dome.

 

What do you expect if you divide a singularity in space by a discontinuity in time?

[michel123456]

If I understand clearly from the set up

The dome of Figure 1a sits in a downward directed gravitational field, with acceleration due to gravity g

 

So everything in this case is in a state of acceleration. There is no object exactly "at rest", only objects that have the same acceleration, and thus feel like "at rest" i.e. in constant motion.

 

--------------

IOW the mass at the apex is constantly under acceleration. Even when it looks "at rest".

Edited June 3, 2015 by michel123456

[swansont]

If I understand clearly from the set up

So everything in this case is in a state of acceleration. There is no object exactly "at rest", only objects that have the same acceleration, and thus feel like "at rest" i.e. in constant motion.

 

--------------

IOW the mass at the apex is constantly under acceleration. Even when it looks "at rest".

 

It's a Newtonian physics issue, so not being exactly at rest doesn't come into play. And the normal force cancels that of gravity, so F=0 at the start.

[Delta1212]

Brought up here; I'd never heard of this conundrum

 

http://en.wikipedia.org/wiki/Norton's_dome

http://www.pitt.edu/~jdnorton/Goodies/Dome/

 

Basically, there is a dome shape and the top is a point of unstable equilibrium. There is a solution where an object at the top can spontaneously move off of the top, and once it does, continues accelerating. This is offered as an example of a violation of determinism, as the motion is purportedly spontaneous.

 

But I have a quibble with (at least) one of the arguments

 

 

I don't think that creating a new version of the first law means that you can claim this is consistent with Newton's laws. The emphasis on "uniform motion in a straight line" misses the important caveat or "remains", i.e. an object at rest remains at rest — that, to me, has a clear implication that you can't apply this instantaneously. It is a condition that must be true over an arbitrary interval of time. So an instantaneous form of the law that isn't violated is a bit of a straw man.

 

I agree. You might as well argue that you are a vegetarian by defining an instantaneous boundary point between eaten and uneaten meat, so that it has either been eaten or not been eaten and at any one instant you are not technically "eating" meat.

[studiot]

Does the OP not agree with me that the proffered analysis commences with fallacy?

 

Small wonder it leads immediately to a singularity, we would ordinarily be able to ignore, but cannot because of the shape of the dome.

 

I like delta1212's analogy of the discontinuity in time that is also presented. +1

 

I see no conflict with Newton's Laws, properly applied.

Edited June 3, 2015 by studiot

[swansont]

Does the OP not agree with me that the proffered analysis commences with fallacy?

 

I'm focusing on a different issue, but that may very well be contributing to the problem of defining when the acceleration started.

 

There's also the issue that not every mathematical solution to a problem is physically realizable. i.e. "unphysical" solutions exist. Problems can be quadratic (e.g. ballistics problems being quadratic in t), and have one of the solutions not be within the defined parameters of the problem. We toss negative speeds if solving from a value of KE.

[studiot]

It's not the when that matters, is the wacking great pole in the density distribution and the contact forces that the analysis starts off with.

 

A point mass has infinite density and if balanced on a point support is subject to infinite contact forces in a gravitational field.

 

How's that for a kick-off?

 

Edit : Actually I should have said contact stresses, not contact forces.

Edited June 3, 2015 by studiot

[michel123456]

It's not the when that matters, is the wacking great pole in the density distribution and the contact forces that the analysis starts off with.

 

A point mass has infinite density and if balanced on a point support is subject to infinite contact forces in a gravitational field.

 

How's that for a kick-off?

that's it. The dome is transpierced.

---------------

That reminds me the egg of Colombus.

[swansont]

It's not the when that matters, is the wacking great pole in the density distribution and the contact forces that the analysis starts off with.

 

A point mass has infinite density and if balanced on a point support is subject to infinite contact forces in a gravitational field.

 

How's that for a kick-off?

 

Edit : Actually I should have said contact stresses, not contact forces.

 

Singularities are not necessarily an issue. A infinitesimal volume means infinite density, but you can still have a finite mass, which is what's important.

[michel123456]

 

Singularities are not necessarily an issue. A infinitesimal volume means infinite density, but you can still have a finite mass, which is what's important.

A finite mass applied over a zero surface applies an infinite pressure.

[studiot]

 

Singularities are not necessarily an issue. A infinitesimal volume means infinite density, but you can still have a finite mass, which is what's important

 

Of course, "Not necessarily", in appropraite circumstances.

 

But equally of course, necessarily in other circumstances, including this one.

 

The issue is that the notion of a 'point mass' is contingent upon its definition which includes the phrase "much smaller than the smallest significant dimension of the system under consideration" or words to that effect.

 

Since the smallest significant dimension in this system is zero that is a very severe constraint.

[swansont]

A finite mass applied over a zero surface applies an infinite pressure.

 

Over an infinitely small area, from a finite mass. How is that relevant to the discussion?

The issue is that the notion of a 'point mass' is contingent upon its definition which includes the phrase "much smaller than the smallest significant dimension of the system under consideration" or words to that effect.

 

Since the smallest significant dimension in this system is zero that is a very severe constraint.

 

I'm not familiar with that definition. I am familiar with the definition that it has no spatial extent. Literally a point.

[overtone]

Many thanks for leading me to this paper:

http://quod.lib.umich.edu/p/phimp/3521354.0003.004/1/--causation-as-folk-science?page=root;size=150;view=image#pagenav

 

I'm going to have a citation now, for the arguments in the various "free will" discussions. Citations are very important for those of us unable to make our own arguments.

 

It also helps clarify some of the issues above in this thread - such as the "infinite mass/instantaneous motion/infinitesimal point" one. He's not arguing invalidly from Newton's Laws to acausal motion, but illustrating the nature of cause/effect description. It's secondary, derivative, heuristic.

Edited June 3, 2015 by overtone

[studiot]

 

I'm not familiar with that definition. I am familiar with the definition that it has no spatial extent. Literally a point.

 

 

Here is an discussion due to Glauert that explains it rather well.

 

 

 

 

 

 

 

[swansont]

 

Here is an discussion due to Glauert that explains it rather well.

 

Glauert1.jpg

 

 

 

 

 

 

 

 

That's the definition of particle, though, not point mass. Though I suspect one could come up with a Norton's dome description with a finite sized object, as long as it was symmetric; the contact point would still be a point, and I think that's all that matters.

[studiot]

 

Though I suspect one could come up with a Norton's dome description with a finite sized object, as long as it was symmetric; the contact point would still be a point, and I think that's all that matters.

 

 

 

Of course, the supported object could be quite large, eg an inverted dome and there would be a theoretical contact point of zero dimensions.

 

However you would need to take into account the distribution of mass for any non zero mass point or particle (arguing a difference is a diversion) even an infinitesimal one.

[ydoaPs]

The idea behind the dome is to try to show that there are cases of indeterminism in Newtonian physics. There are previous known examples (such as space invaders), but they break energy conservation. The claim here is that Norton's dome shows indeterminism without breaking any rules.

 

While I tend to agree with you on this, Friedman and Earman have been said to have shown that the first law is a special case of the second when formulated in a four-dimensional mechanics, so it's redundant. From everything else I've read of Earman (and it's a lot; he's one of my intellectual mancrushes), I'm willing to give him the benefit of the doubt on this until I have time to read the paper to which I just linked.

 

For those who are unpersuaded, Norton provides an appeal to time reversability in pages 11 and 12 of the original paper. Basically, since Newtonian Mechanics is time-reversable, any event happening in one direction through time is also has a possible reversal. Eggs can, though I'm not aware of any reports of this happening, unsplatter and jump back on the counter according to Newtonian metaphysics. Norton here doesn't give a rigorous mathematical treatment of time reversal, but rather a qualitative one simply to pump intuition to his side.

 

Consider the same dome and the same ball, but this time with the ball at the bottom. Now, roll the ball up the dome. If you roll too hard, then the ball reaches the peak then rolls down the other side. If you don't roll the ball hard enough, then it fails to reach the peak and rolls back down. If, however, you roll it in the goldilocks zone, the ball stops exactly on the peak and stays there forever. Running that in reverse, according to Norton, gives the solution that the ball can sit atop the dome for an unspecified amount of time and spontaneously roll down the dome.

 

For those who want to read more about it, Norton has a longer more recent paper here, though it doesn't really address Swansont's issue.

[studiot]

 

Consider the same dome and the same ball, but this time with the ball at the bottom. Now, roll the ball up the dome. If you roll too hard, then the ball reaches the peak then rolls down the other side. If you don't roll the ball hard enough, then it fails to reach the peak and rolls back down. If, however, you roll it in the goldilocks zone, the ball stops exactly on the peak and stays there forever. Running that in reverse, according to Norton, gives the solution that the ball can sit atop the dome for an unspecified amount of time and spontaneously roll down the dome.

 

 

 

 

That just ain't possible within the specification.

 

The contact was specified to be frictionless so you can't roll it up or down the dome.

 

Just because the proposal has a big name behind it doesn't mean we should stop being critical (in the correct analytical sense).

Edited June 4, 2015 by studiot

[overtone]

 

 

The contact was specified to be frictionless so you can't roll it up or down the dome.

OK, slide it. Or spin it. Same as.

 

He's demonstrating that Newtonian physics can be recovered from modern physics, as a useful heuristic or mental shorthand, without thereby establishing the reality of a "cause". The point is that all cause and effect description is from a less technically sound or less rigorous "folk" comprehension of any scientific theory - "cause" is not something that we lost recently when quanta were discovered, but something that was never there except in the folk approach to comprehending the cutting edge science of the day. There never was any force of gravity causing planets to do this or that, for example - and Newton knew that, just as modern physicists know that nothing is bending space-time ("causing" it to bend).

[MigL]

Analyzing 'unstable' behavior, whether a ball on a dome, a pencil balanced on its tip, or even our universe in a non-expanding initial state, is problematical when using idealized conditions.

In reality, it is impossible to isolate the deterministic, macroscopic domain from the indeterministic, quantum domain.

 

The amount of time a system remains, or can remain, in an unstable condition, is ultimately determined by the magnitude of the influence of the quantum domain on the unstable, macroscopic, system. Einstein, in his Brownian motion paper, showed how a system, that should be completely deterministic, of little colliding 'billiard' balls, is in fact totally random and indeterministic due to the major influence of quantum 'jiggling'.

[md65536]

I don't think that creating a new version of the first law means that you can claim this is consistent with Newton's laws. The emphasis on "uniform motion in a straight line" misses the important caveat of "remains", i.e. an object at rest remains at rest — that, to me, has a clear implication that you can't apply this instantaneously. It is a condition that must be true over an arbitrary interval of time. So an instantaneous form of the law that isn't violated is a bit of a straw man.

I disagree. If you're right that you can't apply the first law instantaneously, then there should be no possible case where an instant on its own violates the law (since it can't be applied), so any instant would be vacuously consistent with the law. Perhaps they falsely suggest the analysis supports the first law using their new version (though I don't see it that way), but a law that doesn't apply isn't inconsistent.

 

If you insist on non-zero durations, there is none in which the mass doesn't remain at rest but is in the absence of a net external force. So it is consistent with the first law still. The only way it could be violated is in an instant.

 

 

Analyzing 'unstable' behavior, whether a ball on a dome, a pencil balanced on its tip, or even our universe in a non-expanding initial state, is problematical when using idealized conditions.

In reality, it is impossible to isolate the deterministic, macroscopic domain from the indeterministic, quantum domain.

In this case they're using an unrealistic simplified model (whether they're omitting the cause of the movement off balance, or indeterministic effects). The explanation of any movement lies outside of the analysis.

 

But I think that's fine since they're not trying to predict/explain what happens and why, they're only showing that their simplified model on its own doesn't violate Newton's laws.

Edited June 4, 2015 by md65536

[swansont]

I disagree. If you're right that you can't apply the first law instantaneously, then there should be no possible case where an instant on its own violates the law (since it can't be applied), so any instant would be vacuously consistent with the law. Perhaps they falsely suggest the analysis supports the first law using their new version (though I don't see it that way), but a law that doesn't apply isn't inconsistent.

 

I don't think that follows. Saying the first law must hold over an interval does not permit instantaneous violations. An object at rest over some interval must remain at rest unless acted upon by an external force. Any instantaneous movement violates that condition — it's not at rest over the entire interval — thus a force must have been exerted.

[ydoaPs]

That just ain't possible within the specification.

 

The contact was specified to be frictionless so you can't roll it up or down the dome.

 

Just because the proposal has a big name behind it doesn't mean we should stop being critical (in the correct analytical sense).

As overtone suggested, your nitpick isn't really relevant. Slide it. Whatever. As it slides, it loses KE to PE. If the KE to start is exactly equal to m_ballgh_peak, then the sliding ball will remain at rest on top of the dome. Ran backward, this is still the ball sitting on top of the dome for an unspecified amount of time and spontaneously sliding down. Furthermore, since the dome is radially symmetric, this can happen at any angle (once you define a zero angle). There are infinitely many solutions such that the ball ends up on top, so there are infinitely many solutions, so the argument goes, that have the ball spontaneously sliding down the dome.

I don't think that follows. Saying the first law must hold over an interval does not permit instantaneous violations. An object at rest over some interval must remain at rest unless acted upon by an external force. Any instantaneous movement violates that condition — it's not at rest over the entire interval — thus a force must have been exerted.

I'll go ahead and read that paper of Earman's tonight, but I probably can't tell you anything about it until monday since I'll be away at a wedding all week. If you're interested in what he said, the paper is linked in my initial post in this thread.

[md65536]

I don't think that follows. Saying the first law must hold over an interval does not permit instantaneous violations. An object at rest over some interval must remain at rest unless acted upon by an external force. Any instantaneous movement violates that condition — it's not at rest over the entire interval — thus a force must have been exerted.

But no one is suggesting an instantaneous violation of the law, you're simply assuming there must be one if the mass moves while no additional external force is introduced (besides gravity, since it has no effect at the mass's initial rest location).

 

Intuitively, the law says what you're saying, that the force has to come first before the object is no longer at rest. However, if it only applies to non-zero durations, then that is not necessarily so. The law does not explicitly make a statement about causality.

 

Consider any duration. If the mass remains at rest over the duration, then it remains in a spot where there is no net force, and there is no violation of the first law. If the mass does not remain at rest, it is at some other location at some point in the duration, at where there is a net force due to the dome shape and gravity, so the first law is not violated in that duration either. Intuitively we want to think of a duration in which the mass is no longer at rest, but before any force allows it to slide off the dome, but mathematically there is none.

 

If anything maybe there's some violation of causality, but regardless, it seems to me that if you simply take out a cause (ie force) to get the mass off balance, you don't end up with a violation of the first law. The law is on the condition, "unless acted upon by an external force," which seems to suggest causality... but in this case, any duration in which the mass is not at rest includes an external force acting on it, so the condition is satisfied.

Edited June 4, 2015 by md65536