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The three body problem lacks a closed form solution. How does that mean it's unpredictable, though? I thought that numerical methods can be used to make n-body

The three body problem lacks a closed form solution. How does that mean it's unpredictable, though? I thought that numerical methods can be used to make n-body predictions to arbitrary precision. Are these simulations less accurate than I am thinking? How do engineers and scientists working on space probes plan their trajectories and such?

JumpCrisscross 2y ago
> numerical methods can be used to make n-body predictions to arbitrary precision Arbitrary precision, not arbitrary length. Even "from [a] mathematical viewpoint, given an exact initial condition, we can gain mathematically reliable trajectories of chaotic dynamic systems" to only a "finite...interval" [1]. (This is due to "numerical noises, i.e. truncation and round-off error, where truncation error is determined by numerical algorithms and round-off error is due to the limited precision of numerical data, respectively.") For a physical system like the weather, uncertainty "mainly comes from limited precision of measurement," though there is also the "inherently uncertain/random property of nature, caused by such as thermal fluctuation, wave-particle duality of de Broglie’s wave, and so on." [1] https://www.sciencedirect.com/science/article/abs/pii/S1007570413003109 https://www.sciencedirect.com/science/article/abs/pii/S10075...
- echoangle 2y ago
  Finite interval doesn’t mean it can’t be arbitrary. I’m not saying it can be in this example but your counterpoint doesn’t follow from the quote. For example, I can calculate the Fibonacci sequence to an arbitrary length but not infinite.
  - JumpCrisscross 2y ago
    > Finite interval doesn’t mean it can’t be arbitrary Skim the paper. Numerical noise means you cannot calculate the 3-body problem to an arbitrary length. There is a finite, mathematical limit even with perfect knowledge of initial conditions.
    - echoangle 2y ago
      Isn’t the paper about the uncertainties that inherently exist with physical systems? There isn’t any claim that mathematically exact starting values can’t be propagated with arbitrary precision to arbitrary length, and I would claim that this is possible (but not practical due to compute being limited, of course). But there’s no hard limit of precision and length where a simulation can’t be made if the starting conditions are exact. The point of the paper is that starting conditions are never exact which limits the length you can propagate.
      - JumpCrisscross 2y ago
        > Isn’t the paper about the uncertainties that inherently exist with physical systems? It talks about that. Which is relevant when we're talking about the weather. But it opens by discussing the hard mathematical limits to numerical methods. > there’s no hard limit of precision and length where a simulation can’t be made if the starting conditions are exact Wrong. Read. The. Paper. Numerical methods for chaotic systems are inherently, mathematically uncertain. Beyond a certain number of steps, adding precision doesn't yield a more precise answer, it just produces a different one. At a certain point, the difference between the different answers you get with more precision covers the entire solution space.
strls 2y ago
You can solve to arbitrary precision but you can't measure and specify initial conditions to arbitrary precision, making the solution wrong outside of a small time interval.
- JumpCrisscross 2y ago
  > can solve to arbitrary precision but you can't measure and specify initial conditions to arbitrary precision Even with perfect knowledge of initial conditions, numerical noise limits the forecast interval.

JumpCrisscross 2y ago

> numerical methods can be used to make n-body predictions to arbitrary precision Arbitrary precision, not arbitrary length. Even "from [a] mathematical viewpoint, given an exact initial condition, we can gain mathematically reliable trajectories of chaotic dynamic systems" to only a "finite...interval" [1]. (This is due to "numerical noises, i.e. truncation and round-off error, where truncation error is determined by numerical algorithms and round-off error is due to the limited precision of numerical data, respectively.") For a physical system like the weather, uncertainty "mainly comes from limited precision of measurement," though there is also the "inherently uncertain/random property of nature, caused by such as thermal fluctuation, wave-particle duality of de Broglie’s wave, and so on." [1] https://www.sciencedirect.com/science/article/abs/pii/S1007570413003109 https://www.sciencedirect.com/science/article/abs/pii/S10075...

echoangle 2y ago
Finite interval doesn’t mean it can’t be arbitrary. I’m not saying it can be in this example but your counterpoint doesn’t follow from the quote. For example, I can calculate the Fibonacci sequence to an arbitrary length but not infinite.
- JumpCrisscross 2y ago
  > Finite interval doesn’t mean it can’t be arbitrary Skim the paper. Numerical noise means you cannot calculate the 3-body problem to an arbitrary length. There is a finite, mathematical limit even with perfect knowledge of initial conditions.
  - echoangle 2y ago
    Isn’t the paper about the uncertainties that inherently exist with physical systems? There isn’t any claim that mathematically exact starting values can’t be propagated with arbitrary precision to arbitrary length, and I would claim that this is possible (but not practical due to compute being limited, of course). But there’s no hard limit of precision and length where a simulation can’t be made if the starting conditions are exact. The point of the paper is that starting conditions are never exact which limits the length you can propagate.
    - JumpCrisscross 2y ago
      > Isn’t the paper about the uncertainties that inherently exist with physical systems? It talks about that. Which is relevant when we're talking about the weather. But it opens by discussing the hard mathematical limits to numerical methods. > there’s no hard limit of precision and length where a simulation can’t be made if the starting conditions are exact Wrong. Read. The. Paper. Numerical methods for chaotic systems are inherently, mathematically uncertain. Beyond a certain number of steps, adding precision doesn't yield a more precise answer, it just produces a different one. At a certain point, the difference between the different answers you get with more precision covers the entire solution space.

strls 2y ago

You can solve to arbitrary precision but you can't measure and specify initial conditions to arbitrary precision, making the solution wrong outside of a small time interval.

JumpCrisscross 2y ago
> can solve to arbitrary precision but you can't measure and specify initial conditions to arbitrary precision Even with perfect knowledge of initial conditions, numerical noise limits the forecast interval.