[HN Gopher] 'Islands' of regularity discovered in the famously c...
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'Islands' of regularity discovered in the famously chaotic three-
body problem
Author : wglb
Score : 85 points
Date : 2024-10-14 18:01 UTC (1 days ago)
(HTM) web link (phys.org)
(TXT) w3m dump (phys.org)
| big-green-man wrote:
| I'd bet a thousand dollars that when graphed, these islands of
| regularity will correspond to the graph of the Mandelbrot set.
| eszed wrote:
| Why Mandelbrot, specifically?
| big-green-man wrote:
| I am going out on a limb and speculating that these stable
| sets of conditions correspond to Julia sets by some
| mathematical relationship and that the underlying fundamental
| principle is the same.
|
| https://en.m.wikipedia.org/wiki/Julia_set
| m3047 wrote:
| The Mandelbrot and Julia sets are the outcomes of specified
| functions and have a relationship with each other. But yeah
| I would speculate that something similar, in the way that a
| Mandelbrot set is self-similar, would be observed in at
| least some circumstances. Gave you a vote for that.
|
| Edit: reading the article I can see that Julia sets are
| defined separately from the Mandelbrot set and the
| corresponding Julia set(s) are an example. So maybe you're
| right. I'm not really a mathematician, I just use math to
| get stuff done.
| moi2388 wrote:
| I'm willing to take that bet.
| big-green-man wrote:
| So what do you think the math is going to look like about the
| starting values of these stable states? I'm speculating that
| they correspond to Julia sets by some fundamental
| relationship.
| didgeoridoo wrote:
| Does anyone know if that visualization is representative or just
| an illustration? I'm struggling with how all these parameters
| (mass, speed, and relative direction for all three objects) could
| map to a fractal on a 2 dimensional plane.
| rcxdude wrote:
| The study didn't plot the entire state space. It's simply a map
| of the end state of a limited space of parameters
| (specifically, the mass of the objects is fixed, two of them
| are in a particular orbit, and basically only the angle and
| timing of when the third object approaches is varied). The plot
| just shows which object gets thrown out of the system for each
| initial configuration.
| didgeoridoo wrote:
| That makes a huge amount more sense. Thanks!
| oolonthegreat wrote:
| AFAIK, these islands of regularity/stability are a common aspect
| of chaotic systems, say the logistic map[1].
|
| [1]: https://en.m.wikipedia.org/wiki/Logistic_map
| Rallen89 wrote:
| Off topic buying haven't seen a bit of code in Wikipedia to run
| to see a visualisation before
| ddejohn wrote:
| I may be mistaken but I seem to recall having also read on
| numerous occasions that ejection of one of the bodies is an
| inevitability in these types of systems.
|
| Given the limited state space of the simulation, I'm not sure I
| see what the big discovery is here.
|
| It's certainly a neat result to see it visualized though.
| pkoird wrote:
| I never really understood the 3 body problem. Am I correct in
| saying that given three bodies that attract one another, there is
| currently no known way of predicting, at arbitrary time, where
| they'll end up at, even though it is possible to simulate to
| reasonable degree of accuracy for a limited time horizon? If that
| is the case, is this an example of what Stephen Wolfram regularly
| calls as computationally irreducible? Also it is so interesting
| to me that the jump from 2 to 3 in different domains herald such
| unfathomable complexity. 2SAT for instance is polynomially solved
| whereas 3SAT is NP Hard. If I were a competent mathematician, I'd
| definitely investigate the link between NP Hardness and
| Computationally irreducible chaotic process like 3Body problem.
| nightowl_games wrote:
| Yes, I believe it is the same as Wolfram's "Computational
| Irreducability". I'll copy paste the question I emailed him
| about it:
|
| > In computers, floating point numbers have a limited
| precision, based on how many bits are used. Does the innate
| 'error' in floating point numbers have implications on your
| theory of 'Computational Irreducability'? Are the 'errors'
| introduced with each floating point operation is fundamentally
| computationally irreducable? Does this have a domino effect
| that causes basic classical mechanics to become computationally
| irreducible? Like the integration of velocity into position, it
| is a O(1) operation to compute the future position given a
| velocity and a starting point, but if a computer is rendering
| each individual frame, it will accumulate error in each frame,
| and the final frame may end up in a different position. Does
| this concept relate to the real world through things like the
| Planck length? Is space perhaps not purely continuous?
| ramesh31 wrote:
| >Is space perhaps not purely continuous?
|
| Pretty sure that's the entire basis of quantum mechanics
| metalliqaz wrote:
| space? or energy?
| telotortium wrote:
| No - space and time are continuous in quantum mechanics,
| despite the fact that other quantities such as angular
| momentum and energy are discrete (quantized). Relevant
| comment from Lubos Motl, a string theorist who produced a
| proof that, under some assumptions, quantum mechanics on
| discrete space and time is inconsistent, echoed by Peter
| Shor: https://physics.stackexchange.com/a/9077
| m3047 wrote:
| I have a partial answer for that. ;-) For at least some
| functions, iterative solutions exist which converge to
| correct solutions. For at least some iterative solutions the
| rate of convergence (the estimated error bounds) is a
| continuous function: it can be seen that convergence proceeds
| normally (I should be more careful with that word) as long as
| the estimated error bounds for the iteration is larger than
| the computationally induced error. I'll hazard a guess that
| with constraints some formulations which model a three body
| problem are solvable to the point where the estimated error
| bounds approaches the computationally induced error due to
| floating point imprecision.
|
| The thing about the three body problem, same as here, is that
| there are regions in the initial position and velocity space
| where very small changes produce extremely large differences
| in function output.
|
| There has been some work / recognition of the value of
| topological understanding to orbital mechanics, although I
| can't find whatever I read a few months ago. Best I could
| find was https://en.wikipedia.org/wiki/Symplectic_geometry
| and this DDG search looks promising: https://duckduckgo.com/?
| q=symplectic+integrators+solar+syste...
| andrewflnr wrote:
| Those are some interesting open questions about physics, but
| the happenstance limitations of modern computer hardware have
| no bearing on them.
| andrewflnr wrote:
| The problem, a bit more precisely, is that very small changes
| in the initial conditions can result in very large changes in
| result. So even if you have a perfect numerical simulation, if
| your inputs are even slightly off the simulation might
| eventually be worthless. Unless you're in one of these islands
| of stability, maybe, I'm fuzzier on that. (And I don't know a
| lot about numerical simulation techniques, but I think that
| sort of situation also makes a "perfect" simulation even harder
| than usual.)
|
| Compare to two-body interactions, where there is a pretty small
| family of closed-form solutions to all the possible
| trajectories. You can just plug in a future time t and (in
| principle) get a mostly correct answer. In practice nothing is
| exactly two-body, the bodies aren't point masses, etc, but the
| small differences have proportionally small effects.
| pradn wrote:
| The paper is at [1], and the summary chart is at [2].
|
| The top chart shows which of the three bodies escapes the system
| given initial starting conditions. The bottom one makes it easier
| to see patterns by reducing noise. It uses the k-nearest-
| neighbors algorithm to find the dominant color near each pixel.
| More than the large pools of stability, what's interesting are
| the bands of stability - like electromagnetic fields.
|
| I have no idea how one codes this up in a computer, with fixed
| precision math and all. Numerical methods is a black box for me.
|
| [1]: https://doi.org/10.1051/0004-6361/202449862 [2]:
| https://www.aanda.org/articles/aa/full_html/2024/09/aa49862-...
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