Norton's Dome Paradox

[Swammerdami]

The Picard–Lindelöf theorem guarantees that certain differential equations (with boundary conditions) have single solutions; this makes Newtonian systems deterministic. But that theorem doesn't apply to  Norton's dome because that Dome violates  Lipschitz continuity. Thus this severely challenges the idea that Newton's Laws are deterministic. A simple way to derive this is to recall that Newton's Laws are reversible. Since a ball rolled up the particular shape of Newton's Dome at precisely a specific velocity, will come to rest at the very top, it follows by reversibility that a stationary ball at the very top may spontaneously make a descent, in any direction!

The preceding paragraph would have been utterly meaningless to me yesterday. But I just watched the YouTube "The Dome Paradox: A Loophole in Newton's Laws" by a charming presenter. I don't know if you will find the YouTube incredible or just tedious, but the Dome Paradox IS interesting.

I once decided that a simultaneous collision, e.g. the cue ball striking two objects at EXACTLY the same time, could have MULTIPLE solutions. Is this correct?

 

[Bronzeage]

If I read the Wiki article correctly, I don't see a paradox.

"... and then after an arbitrary period of time starts to slide down the dome in an arbitrary direction. The apparent paradox in this second case is that this would seem to occur for no discernible reason."

There are a lot of things that happen for no discernable reason, but that is a lack of discernment on my part. I may not know why something happened, but that doesn't mean it violated some fundamental law of physics.

 

[Bomb#20]

The Picard–Lindelöf theorem guarantees that certain differential equations (with boundary conditions) have single solutions; this makes Newtonian systems deterministic. But that theorem doesn't apply to  Norton's dome because that Dome violates  Lipschitz continuity. Thus this severely challenges the idea that Newton's Laws are deterministic. A simple way to derive this is to recall that Newton's Laws are reversible. Since a ball rolled up the particular shape of Newton's Dome at precisely a specific velocity, will come to rest at the very top, it follows by reversibility that a stationary ball at the very top may spontaneously make a descent, in any direction!

The idea that Newton's Laws are reversible seems pretty dubious to me. In the first place, the Norton's Dome argument appears to be based on Newton's 2nd Law; Newton had some others. The 2nd Law no doubt permits a stationary ball at the very top to spontaneously make a descent in any direction, but the 1st Law excludes that. "An object at rest remains at rest, or if in motion, remains in motion at a constant velocity unless acted on by a net external force."

In the second place, the Painlevé conjecture has been proven. The N-body problem has Newtonian solutions in which objects accelerate to infinite speed in a finite time. Try reversing one of those.

I once decided that a simultaneous collision, e.g. the cue ball striking two objects at EXACTLY the same time, could have MULTIPLE solutions. Is this correct?

Were you assuming the objects were infinitely rigid? If you model the objects as sets of point-particles held together in solid state by finite classical forces, I don't think you get multiple solutions.

 

[Swammerdami]

The Picard–Lindelöf theorem guarantees that certain differential equations (with boundary conditions) have single solutions; this makes Newtonian systems deterministic. But that theorem doesn't apply to  Norton's dome because that Dome violates  Lipschitz continuity. Thus this severely challenges the idea that Newton's Laws are deterministic. A simple way to derive this is to recall that Newton's Laws are reversible.

The idea that Newton's Laws are reversible seems pretty dubious to me.

I do NOT think that reversibility claim is contested. The notion was introduced by Laplace and apparently Newton himself! (Although Newton himself imagined God intervening to keep orbits stable IIRC.) Maxwell's Laws are also reversible. (This reversibility fact, and the closely associated principle of determinism, may be challenged by some models of quantum mechanics.)

The infamous Second Law of Thermodynamics (SLT) is IIUC the ONLY "Law" of physics which is NOT reversible. But rather than being a "Law", the SLT is just a fact of statistics and reverses if one sets the boundary conditions in the future instead of the past!

In the first place, the Norton's Dome argument appears to be based on Newton's 2nd Law; Newton had some others. The 2nd Law no doubt permits a stationary ball at the very top to spontaneously make a descent in any direction, but the 1st Law excludes that. "An object at rest remains at rest, or if in motion, remains in motion at a constant velocity unless acted on by a net external force."

The YouTube discusses this objection toward its end. I didn't understand it but it seemed to be that the irregularity of the motion functions led to a zero initial acceleration despite the change in motion.

I agree it all sounds absurd; that's partly why I posted. Presenter made it sound like this is a hotly debated paradox well-known to top theoreticians!

In the second place, the Painlevé conjecture has been proven. The N-body problem has Newtonian solutions in which objects accelerate to infinite speed in a finite time. Try reversing one of those.

Could this be a paradox similar to Norton's Dome, also induced by the irregularity of the Newtonian solutions?

I once decided that a simultaneous collision, e.g. the cue ball striking two objects at EXACTLY the same time, could have MULTIPLE solutions. Is this correct?

Were you assuming the objects were infinitely rigid? If you model the objects as sets of point-particles held together in solid state by finite classical forces, I don't think you get multiple solutions.

Yes, I was taking the most simplified (and therefore detached from the actual physical world) case. I think this, the Norton's Dome, and perhaps your N-body example, do NOT arise in the real world. After all, we all know a ball won't remain motionless on top of a pinnacle.

 

[Jarhyn]

The Picard–Lindelöf theorem guarantees that certain differential equations (with boundary conditions) have single solutions; this makes Newtonian systems deterministic. But that theorem doesn't apply to  Norton's dome because that Dome violates  Lipschitz continuity. Thus this severely challenges the idea that Newton's Laws are deterministic. A simple way to derive this is to recall that Newton's Laws are reversible. Since a ball rolled up the particular shape of Newton's Dome at precisely a specific velocity, will come to rest at the very top, it follows by reversibility that a stationary ball at the very top may spontaneously make a descent, in any direction!

The preceding paragraph would have been utterly meaningless to me yesterday. But I just watched the YouTube "The Dome Paradox: A Loophole in Newton's Laws" by a charming presenter. I don't know if you will find the YouTube incredible or just tedious, but the Dome Paradox IS interesting.

I once decided that a simultaneous collision, e.g. the cue ball striking two objects at EXACTLY the same time, could have MULTIPLE solutions. Is this correct?

To me this merely challenges the idea that reality allows these kinds of infinitely fine and yet infinitely precise positions.

Newton's laws are problematic to determinism because they allow structures to be described as infinitely fine, when the universe has a limit to the meaningful finitude of its structures.

There's also another issue here: at the observable bounds of reality, we are constantly making new contact with freshly seen "perturbative elements": in each moment, far beyond the opaque cosmic microwave background, at temperatures so low as to be thought in insignificant fractions of absolute zero, new matter is coming into our gravitational horizon, but in each new moment, this will rather unpredictably change where the gravitational center of the universe is.

"Down", as a function of reality, shakes unpredictably, but predictably *almost not-at-all*.

I've been thinking about this a lot... And wouldn't this make the direction determined by what the unseen matter is over that horizon in the "block universe"?

I wonder if this is also the source of the unpredictability of many phenomena.

 

[bilby]

After all, we all know a ball won't remain motionless on top of a pinnacle.

Sure; But we also all know that an object in motion will eventually slow down and stop, and that nature abhors a vacuum.

A ball on top of a pinnacle won't remain motionless if acted upon by even the smallest of forces, and there are always myriad small forces.

A ball atop a pinnacle that is observed, to determine whether it has (or has not) remained motionless, will fall if only due to the impact of the photons that allow us to observe it.

According to Schrödinger, it will be in a superimposed state of both fallen and not fallen, until that observation is made.

 

[Bomb#20]

The Picard–Lindelöf theorem guarantees that certain differential equations (with boundary conditions) have single solutions; this makes Newtonian systems deterministic. But that theorem doesn't apply to  Norton's dome because that Dome violates  Lipschitz continuity. Thus this severely challenges the idea that Newton's Laws are deterministic. A simple way to derive this is to recall that Newton's Laws are reversible.

The idea that Newton's Laws are reversible seems pretty dubious to me.

I do NOT think that reversibility claim is contested. The notion was introduced by Laplace and apparently Newton himself!

Not sure that's a good reason to believe it. Newton and Laplace weren't aware of these pathological cases that were only discovered in the late 20th century. Einstein rather infamously was taken by surprise when he found out his equation didn't allow a steady-state universe.

(Although Newton himself imagined God intervening to keep orbits stable IIRC.) Maxwell's Laws are also reversible. (This reversibility fact, and the closely associated principle of determinism, may be challenged by some models of quantum mechanics.)

Quantum mechanics is just a different system; this is a problem with the implications of Newtonian physics, never mind whether the universe is observed to be Newtonian. But Newtonian physics itself is trivially nondeterministic: two massive point-particles in a head-on collision can bounce off each other straight back the way they came, or at any arbitrary angle. It's the infinite density that opens up that possibility. Similarly, the Norton's Dome problem depends crucially on the clever way Norton shaped his dome, which has infinite curvature at the apex -- a conventionally shaped dome doesn't exhibit the paradox. It's not really all that surprising that laws we expect to be well-behaved start doing odd things if we put infinities into the settings.

In the first place, the Norton's Dome argument appears to be based on Newton's 2nd Law; Newton had some others. The 2nd Law no doubt permits a stationary ball at the very top to spontaneously make a descent in any direction, but the 1st Law excludes that. "An object at rest remains at rest, or if in motion, remains in motion at a constant velocity unless acted on by a net external force."

The YouTube discusses this objection toward its end. I didn't understand it but it seemed to be that the irregularity of the motion functions led to a zero initial acceleration despite the change in motion.

Not sure what you're referring to by "irregularity" -- you mean the lack of Lipshitz-continuity? It sounded like her solution was to interpret "remains at rest" to mean "zero initial acceleration", which seems to me amounts to saying the 1st law is just a special case of the 2nd so let's discard it. I'm not convinced that's what "remains at rest" means.

I agree it all sounds absurd; that's partly why I posted. Presenter made it sound like this is a hotly debated paradox well-known to top theoreticians!

Yup.

In the second place, the Painlevé conjecture has been proven. The N-body problem has Newtonian solutions in which objects accelerate to infinite speed in a finite time. Try reversing one of those.

Could this be a paradox similar to Norton's Dome, also induced by the irregularity of the Newtonian solutions?

If "irregularity" means something is infinite, could be. As I understand it the N-body trajectories involve particles getting arbitrarily close to each other without ever colliding, which can only happen with infinite density.

I once decided that a simultaneous collision, e.g. the cue ball striking two objects at EXACTLY the same time, could have MULTIPLE solutions. Is this correct?

Were you assuming the objects were infinitely rigid? If you model the objects as sets of point-particles held together in solid state by finite classical forces, I don't think you get multiple solutions.

Yes, I was taking the most simplified (and therefore detached from the actual physical world) case. I think this, the Norton's Dome, and perhaps your N-body example, do NOT arise in the real world. After all, we all know a ball won't remain motionless on top of a pinnacle.

Well, in the real world the N-body particles can't reach infinite speed, per Einstein, and the ball can't have a zero displacement from the peak with zero initial momentum, per Heisenberg. For that matter, you can't make a dome with infinite curvature out of finite-sized atoms, and if you could the infinite pressure would impale the ball.

 

[Bomb#20]

After all, we all know a ball won't remain motionless on top of a pinnacle.

Sure; But we also all know that an object in motion will eventually slow down and stop, and that nature abhors a vacuum.

A ball on top of a pinnacle won't remain motionless if acted upon by even the smallest of forces, and there are always myriad small forces.

A ball atop a pinnacle that is observed, to determine whether it has (or has not) remained motionless, will fall if only due to the impact of the photons that allow us to observe it.

According to Schrödinger, it will be in a superimposed state of both fallen and not fallen, until that observation is made.

In freshman physics, to impress on us the scale on which quantum effects operate, the prof had us consider the problem of a mathematically perfect pencil balanced on its infinitely sharp tip on a perfectly flat infinitely hard table, with not even the smallest outside forces, and invited us to guess how long we should expect it to take for the macroscopic pencil to randomly fall over due to Heisenberg's Uncertainty principle. Our guesses ranged from hours to years; then he worked the problem on the blackboard.

Show

About 5 seconds.

 

[Loren Pechtel]

After all, we all know a ball won't remain motionless on top of a pinnacle.

Sure; But we also all know that an object in motion will eventually slow down and stop, and that nature abhors a vacuum.

A ball on top of a pinnacle won't remain motionless if acted upon by even the smallest of forces, and there are always myriad small forces.

A ball atop a pinnacle that is observed, to determine whether it has (or has not) remained motionless, will fall if only due to the impact of the photons that allow us to observe it.

According to Schrödinger, it will be in a superimposed state of both fallen and not fallen, until that observation is made.

You don't even need the photons. QM will cause it to fall. And it is reversible--QM could bring it to a halt on top of the dome. (Of course it wouldn't stay there as it would get disrupted again.)