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.
>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.
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.
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.
I've thought about it, and as far as I can tell, there is no need to construct this dome in such a strange way. A trajectory away from from an unstable equilibrium will always start out with zero kinetic energy, and I can't think of any cases where you couldn't stitch together no motion at the equilibrium for t<0 with motion away from it after that to get a solution that obeyed Newton's laws and conservation of energy at all times.
That's not a resolution to the philosophical issue, but it could help us come up with one, given that some other systems are so much simpler but have the same property.
So, if anyone wants to think about this, you can think about x''(x) = -x^2 just as productively, without the extra complexity.
This is a response to part of your comment before you deleted and re-commented. I just thought it was an interesting point to add, and your comment helps give it some context.
> [...] I always thought the only necessary initial conditions were position and velocity or position and momentum, a case is presented where that doesn't happen. It's implied in the exposition of Hamiltonian mechanics that those are the only variables that describe a particle at a point in time, and that's how it is treated in computational simulations.
There's usually an implicit assumption that the functions you're dealing with are sufficiently smooth (differentiable as much as you want at all points). In Norton's dome, the derivative of the height of the surface (the slope) does not exist at r=0 because d/dr sqrt(r) = 1/(2sqrt(r)).
Here's the Wiki article. It points to the lack of Lipschitz continuity at the peak, which contrary to the author, I would call out as a dead straight reason for the behavior to be non-deterministic; because the proof of determinacy depends on Lipschitz continuity.
Can somebody ELI5 how the "simple criticism" works in the Indeterminate derivatives section?
a) h=F/2m deltat^2 only if F is in the direction of h, which it certainly isn't. (Also, if anything, that should be a delta h).
b) F is well defined at r=0 (F=0)
The kinematics don't really impact the argument, all it's saying is that the second derivative of the surface does not exist at the peak, which means the change in force with respect to motion is undefined.
This is slightly off topic, but is an honest question: Would physics and math benefit from a reworking of flat mathematical expressions into something that looks more like code? Additionally, I wonder why more functions aren't given names like sin and cos so that it's all more like an API than constant rewriting the same equation.
For example, Black-Scholes is complete nonsense [1], but the C implementation is pretty easy to parse [2]. Yes, the code is 100 lines longer, but just like in programming, you'd just use it by name.
Not really. If you look at the C code there are a bunch of executable steps that are straightforward to carry out, but it gives no clue at all about why they do what they do. A human after reading them doesn't come out any smarter than a compiler. The equation on the other hand makes reasonable sense if you know something about stochastic PDE's. Of course just looking at the symbols without any context won't help much, but you can read the article:
That in turn might not be easily understandable either, but it's written in terms of well-known concepts that they teach in math classes. You can get textbooks about that stuff and all that sort of thing. At the end of the day, making sense of it will take a level of mental effort comparable (not by coincidence) to taking a couple of years of math classes (starting from basic calculus) and doing all the assignments. Making the equations look more like code won't help with that. It's about developing understanding and facility with the mathematical ideas.
22 comments
[ 3.9 ms ] story [ 79.9 ms ] threadThe 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.
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.
//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.
That's not a resolution to the philosophical issue, but it could help us come up with one, given that some other systems are so much simpler but have the same property.
So, if anyone wants to think about this, you can think about x''(x) = -x^2 just as productively, without the extra complexity.
> [...] I always thought the only necessary initial conditions were position and velocity or position and momentum, a case is presented where that doesn't happen. It's implied in the exposition of Hamiltonian mechanics that those are the only variables that describe a particle at a point in time, and that's how it is treated in computational simulations.
There's usually an implicit assumption that the functions you're dealing with are sufficiently smooth (differentiable as much as you want at all points). In Norton's dome, the derivative of the height of the surface (the slope) does not exist at r=0 because d/dr sqrt(r) = 1/(2sqrt(r)).
https://en.wikipedia.org/wiki/Norton%27s_dome
For example, Black-Scholes is complete nonsense [1], but the C implementation is pretty easy to parse [2]. Yes, the code is 100 lines longer, but just like in programming, you'd just use it by name.
1. https://wikimedia.org/api/rest_v1/media/math/render/svg/d856...
2. https://gist.github.com/codeslinger/472083/0acc95f745def15a3...
[1] https://mitpress.mit.edu/books/structure-and-interpretation-...
https://en.wikipedia.org/wiki/Black-Scholes_equation
That in turn might not be easily understandable either, but it's written in terms of well-known concepts that they teach in math classes. You can get textbooks about that stuff and all that sort of thing. At the end of the day, making sense of it will take a level of mental effort comparable (not by coincidence) to taking a couple of years of math classes (starting from basic calculus) and doing all the assignments. Making the equations look more like code won't help with that. It's about developing understanding and facility with the mathematical ideas.
It looks there's a paper related to the topic.