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A lot of this is missing the point. Norton's dome shows that the mathematical formalism of Newtonian Mechanics admits non-deterministic solutions. That's all. I

by messe 5y ago

A lot of this is missing the point. Norton's dome shows that the mathematical formalism of Newtonian Mechanics admits non-deterministic solutions. That's all. It's only an approximation in the real world after all, so non-physical solutions aren't of real interest in that context, but they are in a mathematical one. The fact that the author quibbles about units, when they've failed to notice the Norton took `g=1`, makes it seem as if they're not familiar with conventions in mathematical physics. The addition of a constant `k` adds no value other than converting between units, which aren't of interest here other than as a bookkeeping mechanism, because we're not dealing with any numerical quantities.

deleted 5y ago
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whatshisface 5y ago
>the mathematical formalism of Newtonian Mechanics admits non-deterministic solutions. That's all. I wouldn't say "that's all." If it actually does admit nondeterministic solutions (I am not convinced that there is no property or axiom buried in the formalism that forbids this case somehow), then the next question is, does quantum mechanics, which implies classical mechanics, admit nondeterministic solutions? If not (wave mechanics has a way of ignoring weird stuff when it only happens at a point), that would make quantum mechanics a consequence of determinism, a fairly shocking possibility.
shonenknifefan1 5y ago
Interestingly, the solution of the ball falling off of the dome when starting at rest is not physical, but since Newtonian Mechanisms is symmetric under time reversal, the the time reverse of that solution should also be a valid solution, and that solution - the ball arriving at the top of the dome from the bottom - is physical.
rhelzerman 5y ago
Dr. Davies got a little carried away in his rhetoric :-) //This is where Norton gets tricky with us. He posits another solution to the equation ... Norton didn't posit this as a solution, it is a solution. //This is baffling. It’s bizarre to stitch two different solutions to an equation Norton never stitches together two solutions. I don't know if Dr. Davies is unfamiliar with a function defined using a case statement, but there's nothing Frankensteinish about it. //but in fact the top equation is not Newtonian at the apex (clearly since it moves despite the absence of a force there). Alas, Dr. Davies has missed the biggest delight of Norton's Dome! The ball can nondeterministically roll off of Norton's dome and yet still obey all of Newton's laws :-). It sure looks like it wouldn't be--after all, how can the ball start to move in absense of a force? But at time t=T, the ball isn't moving yet. Its not even accelerating yet. So at time t=T its obeying Newton's laws. An at every time before and after that it also obeys Newton's laws.

by messe 5y ago

deleted 5y ago

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whatshisface 5y ago

>the mathematical formalism of Newtonian Mechanics admits non-deterministic solutions. That's all. I wouldn't say "that's all." If it actually does admit nondeterministic solutions (I am not convinced that there is no property or axiom buried in the formalism that forbids this case somehow), then the next question is, does quantum mechanics, which implies classical mechanics, admit nondeterministic solutions? If not (wave mechanics has a way of ignoring weird stuff when it only happens at a point), that would make quantum mechanics a consequence of determinism, a fairly shocking possibility.

shonenknifefan1 5y ago

Interestingly, the solution of the ball falling off of the dome when starting at rest is not physical, but since Newtonian Mechanisms is symmetric under time reversal, the the time reverse of that solution should also be a valid solution, and that solution - the ball arriving at the top of the dome from the bottom - is physical.

rhelzerman 5y ago

Dr. Davies got a little carried away in his rhetoric :-) //This is where Norton gets tricky with us. He posits another solution to the equation ... Norton didn't posit this as a solution, it is a solution. //This is baffling. It’s bizarre to stitch two different solutions to an equation Norton never stitches together two solutions. I don't know if Dr. Davies is unfamiliar with a function defined using a case statement, but there's nothing Frankensteinish about it. //but in fact the top equation is not Newtonian at the apex (clearly since it moves despite the absence of a force there). Alas, Dr. Davies has missed the biggest delight of Norton's Dome! The ball can nondeterministically roll off of Norton's dome and yet still obey all of Newton's laws :-). It sure looks like it wouldn't be--after all, how can the ball start to move in absense of a force? But at time t=T, the ball isn't moving yet. Its not even accelerating yet. So at time t=T its obeying Newton's laws. An at every time before and after that it also obeys Newton's laws.