3 ms·

This is the first time I'm encountering Norton's dome, and I'm not particularly academic, just really like learning about math -- so I'm hoping a friendly HNer

by chrsig 2y ago

This is the first time I'm encountering Norton's dome, and I'm not particularly academic, just really like learning about math -- so I'm hoping a friendly HNer can help me out here. Is norton's dome essentially describing a saddle point? Is the only reason it's nondeterministic because at that point things go to infinity? If we're in the world of mechanics, wouldn't it be up to the machine to determine what to do at that point? Implementation defined, one might say?

kadoban 2y ago
Not really. It's a dome/hill shape, not a saddle, and nothing in particular goes to infinity. The only thing special about the shape is that it's constructed such that there exists a way to kick a ball straight up the slope and (if you're impossibly precise), it'll come to rest at the top in finite time. This in itself is fine, but starts feeling real weird once you are familiar with the time-reversability of physical systems. If you time-reverse this system, you end up with a ball that sits at the top for an arbitrary period of time, and then suddenly just rolls down for no deterministic reason. If you considered a bunch of different runs of this, some where the ball starts at the top, stationary, and some where it's kicked up to stop at the top (from various locations, at various times), they all start at the exact same conditions in the time-reversed system. So why do they do different, unpredictable things?
- chrsig 2y ago
  I'm a bit out of my depth, but I think I recall Gerald Sussman talking about dealing with a similar problem using an operator that indicated that evaluated to exactly one out of a finite set of elements but can be evaluated to any in the set. If I'm recalling, it was ok because each of the elements of the set were differentiable functions, but the operator produces a set of possible results, but doesn't specify which. Sort of like a monad, it seemed. I want to say he was referring to a quadrature, but I honestly can't recall and wont be able to spare the time to hunt down the talk until later. Coincidentally, I just got a copy of structure and interpretation of classical mechanics just the other day, so hopefully I'll get some more appreciation for the problem. [0] https://mitpress.mit.edu/9780262553452/structure-and-interpretation-of-classical-mechanics/ https://mitpress.mit.edu/9780262553452/structure-and-interpr...

by chrsig 2y ago

kadoban 2y ago

Not really. It's a dome/hill shape, not a saddle, and nothing in particular goes to infinity. The only thing special about the shape is that it's constructed such that there exists a way to kick a ball straight up the slope and (if you're impossibly precise), it'll come to rest at the top in finite time. This in itself is fine, but starts feeling real weird once you are familiar with the time-reversability of physical systems. If you time-reverse this system, you end up with a ball that sits at the top for an arbitrary period of time, and then suddenly just rolls down for no deterministic reason. If you considered a bunch of different runs of this, some where the ball starts at the top, stationary, and some where it's kicked up to stop at the top (from various locations, at various times), they all start at the exact same conditions in the time-reversed system. So why do they do different, unpredictable things?

chrsig 2y ago
I'm a bit out of my depth, but I think I recall Gerald Sussman talking about dealing with a similar problem using an operator that indicated that evaluated to exactly one out of a finite set of elements but can be evaluated to any in the set. If I'm recalling, it was ok because each of the elements of the set were differentiable functions, but the operator produces a set of possible results, but doesn't specify which. Sort of like a monad, it seemed. I want to say he was referring to a quadrature, but I honestly can't recall and wont be able to spare the time to hunt down the talk until later. Coincidentally, I just got a copy of structure and interpretation of classical mechanics just the other day, so hopefully I'll get some more appreciation for the problem. [0] https://mitpress.mit.edu/9780262553452/structure-and-interpretation-of-classical-mechanics/ https://mitpress.mit.edu/9780262553452/structure-and-interpr...