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.