[HN Gopher] Newtonian physics IS deterministic (sorry Norton) (2...
___________________________________________________________________
Newtonian physics IS deterministic (sorry Norton) (2017)
Author : spekcular
Score : 26 points
Date : 2021-08-15 18:59 UTC (4 hours ago)
(HTM) web link (blog.gruffdavies.com)
(TXT) w3m dump (blog.gruffdavies.com)
| [deleted]
| messe wrote:
| 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]
| whatshisface wrote:
| > _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.
| whatshisface wrote:
| 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.
|
| https://en.wikipedia.org/wiki/Norton%27s_dome
| davrosthedalek wrote:
| 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)
| whatshisface wrote:
| 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.
| messe wrote:
| 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)).
| davrosthedalek wrote:
| Wait, no, height h propto r^3/2, so dh/dr propto r^1/2. So
| the slope is well defined, I think. The derivative of the
| slope isn't, but that's fine.
| whatshisface wrote:
| What about F = -k x^2/m?
| messe wrote:
| What about it? x^2 is a C^\infty function, differentiable
| and continuous everywhere.
___________________________________________________________________
(page generated 2021-08-15 23:02 UTC)