4 ms·

Chaotic dynamics was taught before Norton’s Dome. Three body problem, articulated pendulum, etc. Are those out of vogue as examples of non-determinism?

yorwba 2y ago
Chaotic dynamics aren't necessarily non-deterministic. Chaos is about small changes to initial conditions causing large changes in the future, but the exact same initial conditions can still deterministically lead to the exact same future state. Norton's Dome is an example where multiple solutions exist that have the exact same initial conditions but still develop differently.
- postalrat 2y ago
  I haven't seen a convincing explaination why Norton's dome is non-deterministic other than people wanting it.
  - 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...
  - pontus 2y ago
    Here's how I think about it: For any dome-like shape, you can start a marble at the bottom and roll it up with some initial speed. If you roll it with insufficient initial speed it'll turn around and come back down. If you roll it too hard, it'll overshoot the peak. By continuity, there must exist some initial condition where it stops at the top. Now, here's the thing that makes Norton's dome special: For a typical dome shape it'll take an infinite amount of time before that marble stops at the top. If you plot the position as a function of time it'll have some type of sigmoid-like shape. However, for the special case of Norton's dome, you can make it settle at the top in a finite amount of time where it'll sit for the rest of eternity. In other words, if you plot the position as a function of time, there will be some critical time after which its position is constant. Now, the clever thing to do now is to realize that Newton's laws are time reversal symmetric which means that any motion forward in time could equally well happen backwards in time. So, you're allowed to take any position plot and flip it horizontally; this is also going to be a valid trajectory. For any typical dome shape this is not a problem. For a typical dome shape you have a sigmoid-like solution which, when flipped, is still sigmoid shaped. In particular this means that there is no finite time at which you can place the marble at the top of the dome and have it roll off. At any finite time, the marble will be slightly off the top and have a small nonzero speed. Norton's dome is different. If you flip its trajectory horizontally you'll see that there are many moments in time where you can start the marble at the top to have it abruptly start rolling off the top at some later time. This is the paradox. You can choose to have it sit at the top for one second and then start rolling or sit at the top for one minute and then start rolling. Unlike other domes, Norton's dome seems to violate our intuition for how initial conditions work. In all cases the marble starts at the top with zero initial speed and yet falls off the top att different moments.
  - deleted 2y ago
    [deleted]
  - Certhas 2y ago
    Pontus gave a good explanation of the physics. Mathematically the statement is simply that there are different trajectories with the same initial conditions that solve the same differential equation. This is easy to calculate and has nothing to do with "people wanting it". It's simply a mathematical fact. What do you not find convincing here?
  - mannykannot 2y ago
    The answer given in Stack Exchange and the Wikipedia article linked to in one of its comments provide explanations, though whether you find them convincing is, of course, up to you! The Wikipedia article says there are solutions to the classical equations of motion: one in which the ball remains stationary forever, and then all those where, after an arbitrary period during which the ball is stationary, it rolls off the dome in an arbitrary direction. What makes this indeterminate is that the analysis of a single initial state yields multiple possible outcomes. The article goes on to say "Notice in the second case that the particle appears to begin moving without cause and without any radial force being exerted on it by any other entity, apparently contrary to both physical intuition and normal intuitive concepts of cause and effect, yet the motion is still entirely consistent with the mathematics of Newton's laws of motion so cannot be ruled out as non-physical." This raises the question of what we mean by 'physical', and whether theories of physics define the physical or describe something existing independently. I will leave that to the more philosophically minded; for myself, I will just note that as there are cases (and physically realizable ones at that) where classical physics gives answers that are not merely indeterminate but outright wrong (the ultraviolet catastrophe being a canonical example), I don't think anything of consequence hangs on this particular case. https://en.wikipedia.org/wiki/Norton%27s_dome#Solutions_to_the_equations_of_motion https://en.wikipedia.org/wiki/Norton%27s_dome#Solutions_to_t...
  - TeMPOraL 2y ago
    I looked over the explanation on the Wiki and, thinking about it, I'd say that the surface around the tip of Norton's Dome is not a continuous function, which alone is enough for me to disqualify it. Intuitively, that's a sufficient explanation to me, or at least a sufficient start of one. IANAPhysicist, so I'll ask here: are there any examples of surfaces or phenomena in classical physics that are defined by a discontinuous function, and are something you'd actually expect to see existing in the real world? Things seemingly discontinuous until you zoom in close enough don't qualify.
- chrsig 2y ago
  isn't the point of the chaos that as time progresses the predictability of outcomes severely deteriorates? or maybe a better frame: any simulation the seed would still result in the same answer, since the computation is deterministic, but the system being simulated is not likely to behave the same at that time under the same initial conditions.
  - marcosdumay 2y ago
    > isn't the point of the chaos that as time progresses the predictability of outcomes severely deteriorates? Yes. And that's what the GP said. Predictability deteriorates because small errors in the initial conditions grow in proportion to the total value up to the point they can more than explain the entire value. > but the system being simulated is not likely to behave the same at that time under the same initial conditions. No, that part is wrong. Chaos is not about non-determinism.
  - naasking 2y ago
    Yes, predictability deteriorates because the sensitivity to initial conditions increases over time. We can't measure or create initial conditions to infinite precision so at simulation will be accurate only to our measurement precision.
- cft 2y ago
  I would only call this a thought experiment. We are yet to encounter a physical Norton's Dome: it probably doesn't exist in nature.
  - crazygringo 2y ago
    What do you mean it doesn't exist in nature? Of course it does, you could build one yourself. It's just a dome with a specific shape. Of course the ball will choose a way down based on tiny physical forces which we can't eliminate in the real world. Fundamentally, Norton's Dome is non-deterministic in classical physics, but reality is quantum.
    - cft 2y ago
      >What do you mean it doesn't exist in nature? So, for example, for large enough r, the gravitational force \sqrt(r) will exceed the free fall accelleration g? More importantly, does this additional branch of solutions that satisfies the initial conditions, survive under the small deformations of this dome shape? The perfect Dome shape certainly does not exist.
      - crazygringo 2y ago
        Usually, when we say something doesn't exist in nature, we mean it's fundamentally incompatible with our 3 spatial dimensions as they exist, passes through itself, or is infinite along some dimension, requires infinitely thin surfaces, etc. But by your definition, even something as simple as a cylinder or sphere doesn't exist in nature, because basic mathematical relationships like radius to circumference won't survive "small deformations". I don't know what point you're trying to make. Norton's Dome "exists in nature" as much as a sphere or a cube does. If it doesn't exist in nature, then no geometric form does.
        cft 2y ago
        >I don't know what point you're trying to make In the sense that one cannot create an ideal dome and make an experiment, whether a point particle placed exactly at the top later randomly starts to fall. One has to study if this class of solutions survives deformations of the ideal dome, to make such an experiment (neglecting quantum effects).
choxi 2y ago
Aren’t those examples deterministic? That’s the most interesting aspect of chaotic systems to me, they’re deterministic but still not predictable
HDThoreaun 2y ago
Those examples are deterministic. The issue with them is that we can not precisely measure their starting positions and the locations of the objects diverge quickly with minuscule different starting positions. If we could measure the starting position of either of those problems with 100% precision we can predict what they would do.

yorwba 2y ago

Chaotic dynamics aren't necessarily non-deterministic. Chaos is about small changes to initial conditions causing large changes in the future, but the exact same initial conditions can still deterministically lead to the exact same future state. Norton's Dome is an example where multiple solutions exist that have the exact same initial conditions but still develop differently.

postalrat 2y ago
I haven't seen a convincing explaination why Norton's dome is non-deterministic other than people wanting it.
- 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...
- pontus 2y ago
  Here's how I think about it: For any dome-like shape, you can start a marble at the bottom and roll it up with some initial speed. If you roll it with insufficient initial speed it'll turn around and come back down. If you roll it too hard, it'll overshoot the peak. By continuity, there must exist some initial condition where it stops at the top. Now, here's the thing that makes Norton's dome special: For a typical dome shape it'll take an infinite amount of time before that marble stops at the top. If you plot the position as a function of time it'll have some type of sigmoid-like shape. However, for the special case of Norton's dome, you can make it settle at the top in a finite amount of time where it'll sit for the rest of eternity. In other words, if you plot the position as a function of time, there will be some critical time after which its position is constant. Now, the clever thing to do now is to realize that Newton's laws are time reversal symmetric which means that any motion forward in time could equally well happen backwards in time. So, you're allowed to take any position plot and flip it horizontally; this is also going to be a valid trajectory. For any typical dome shape this is not a problem. For a typical dome shape you have a sigmoid-like solution which, when flipped, is still sigmoid shaped. In particular this means that there is no finite time at which you can place the marble at the top of the dome and have it roll off. At any finite time, the marble will be slightly off the top and have a small nonzero speed. Norton's dome is different. If you flip its trajectory horizontally you'll see that there are many moments in time where you can start the marble at the top to have it abruptly start rolling off the top at some later time. This is the paradox. You can choose to have it sit at the top for one second and then start rolling or sit at the top for one minute and then start rolling. Unlike other domes, Norton's dome seems to violate our intuition for how initial conditions work. In all cases the marble starts at the top with zero initial speed and yet falls off the top att different moments.
- deleted 2y ago
  [deleted]
- Certhas 2y ago
  Pontus gave a good explanation of the physics. Mathematically the statement is simply that there are different trajectories with the same initial conditions that solve the same differential equation. This is easy to calculate and has nothing to do with "people wanting it". It's simply a mathematical fact. What do you not find convincing here?
- mannykannot 2y ago
  The answer given in Stack Exchange and the Wikipedia article linked to in one of its comments provide explanations, though whether you find them convincing is, of course, up to you! The Wikipedia article says there are solutions to the classical equations of motion: one in which the ball remains stationary forever, and then all those where, after an arbitrary period during which the ball is stationary, it rolls off the dome in an arbitrary direction. What makes this indeterminate is that the analysis of a single initial state yields multiple possible outcomes. The article goes on to say "Notice in the second case that the particle appears to begin moving without cause and without any radial force being exerted on it by any other entity, apparently contrary to both physical intuition and normal intuitive concepts of cause and effect, yet the motion is still entirely consistent with the mathematics of Newton's laws of motion so cannot be ruled out as non-physical." This raises the question of what we mean by 'physical', and whether theories of physics define the physical or describe something existing independently. I will leave that to the more philosophically minded; for myself, I will just note that as there are cases (and physically realizable ones at that) where classical physics gives answers that are not merely indeterminate but outright wrong (the ultraviolet catastrophe being a canonical example), I don't think anything of consequence hangs on this particular case. https://en.wikipedia.org/wiki/Norton%27s_dome#Solutions_to_the_equations_of_motion https://en.wikipedia.org/wiki/Norton%27s_dome#Solutions_to_t...
- TeMPOraL 2y ago
  I looked over the explanation on the Wiki and, thinking about it, I'd say that the surface around the tip of Norton's Dome is not a continuous function, which alone is enough for me to disqualify it. Intuitively, that's a sufficient explanation to me, or at least a sufficient start of one. IANAPhysicist, so I'll ask here: are there any examples of surfaces or phenomena in classical physics that are defined by a discontinuous function, and are something you'd actually expect to see existing in the real world? Things seemingly discontinuous until you zoom in close enough don't qualify.
chrsig 2y ago
isn't the point of the chaos that as time progresses the predictability of outcomes severely deteriorates? or maybe a better frame: any simulation the seed would still result in the same answer, since the computation is deterministic, but the system being simulated is not likely to behave the same at that time under the same initial conditions.
- marcosdumay 2y ago
  > isn't the point of the chaos that as time progresses the predictability of outcomes severely deteriorates? Yes. And that's what the GP said. Predictability deteriorates because small errors in the initial conditions grow in proportion to the total value up to the point they can more than explain the entire value. > but the system being simulated is not likely to behave the same at that time under the same initial conditions. No, that part is wrong. Chaos is not about non-determinism.
- naasking 2y ago
  Yes, predictability deteriorates because the sensitivity to initial conditions increases over time. We can't measure or create initial conditions to infinite precision so at simulation will be accurate only to our measurement precision.
cft 2y ago
I would only call this a thought experiment. We are yet to encounter a physical Norton's Dome: it probably doesn't exist in nature.
- crazygringo 2y ago
  What do you mean it doesn't exist in nature? Of course it does, you could build one yourself. It's just a dome with a specific shape. Of course the ball will choose a way down based on tiny physical forces which we can't eliminate in the real world. Fundamentally, Norton's Dome is non-deterministic in classical physics, but reality is quantum.
  - cft 2y ago
    >What do you mean it doesn't exist in nature? So, for example, for large enough r, the gravitational force \sqrt(r) will exceed the free fall accelleration g? More importantly, does this additional branch of solutions that satisfies the initial conditions, survive under the small deformations of this dome shape? The perfect Dome shape certainly does not exist.
    - crazygringo 2y ago
      Usually, when we say something doesn't exist in nature, we mean it's fundamentally incompatible with our 3 spatial dimensions as they exist, passes through itself, or is infinite along some dimension, requires infinitely thin surfaces, etc. But by your definition, even something as simple as a cylinder or sphere doesn't exist in nature, because basic mathematical relationships like radius to circumference won't survive "small deformations". I don't know what point you're trying to make. Norton's Dome "exists in nature" as much as a sphere or a cube does. If it doesn't exist in nature, then no geometric form does.
      - cft 2y ago
        >I don't know what point you're trying to make In the sense that one cannot create an ideal dome and make an experiment, whether a point particle placed exactly at the top later randomly starts to fall. One has to study if this class of solutions survives deformations of the ideal dome, to make such an experiment (neglecting quantum effects).

choxi 2y ago

Aren’t those examples deterministic? That’s the most interesting aspect of chaotic systems to me, they’re deterministic but still not predictable

HDThoreaun 2y ago

Those examples are deterministic. The issue with them is that we can not precisely measure their starting positions and the locations of the objects diverge quickly with minuscule different starting positions. If we could measure the starting position of either of those problems with 100% precision we can predict what they would do.