5 ms·

No physics expert but isn't this unpredictable (based on what I saw in series) ? Amd this does seem predictable, I saw this for almost a minute

jgchaos 11mo ago
The link points to one of the stable solutions, and there are actually quite a few of those. The problem is that there’s no general closed form that tells us exactly where the bodies will be in the future, so we rely on numerical methods to approximate the motion. If you hit Reset All a few times or add more bodies, you’ll start to see the chaos
- Pulcinella 11mo ago
  There actually is an analytical solution using a power series that actually converges (Karl Sundman's work). Unfortunately, the universe still mocks our attempts. Though the series converges, it does so incredibly slowly. From Wikipedia: The corresponding series converges extremely slowly. That is, obtaining a value of meaningful precision requires so many terms that this solution is of little practical use. Indeed, in 1930, David Beloriszky calculated that if Sundman's series were to be used for astronomical observations, then the computations would involve at least 10^8000000 terms.
  - Nevermark 11mo ago
    > the computations would involve at least 10^8000000 terms. Well we could speed up that simulation pretty easily, just arrange the actual masses and velocities somewhere... Then I thought, is there a way to scale the distances, masses and velocities to create a system with the same, but proportionally faster behavior? One guess as to perhaps why not: As distances get small, normal matter bodies will get close enough to actually collide. Perhaps some tiny primordial black holes would be useful.
- taneq 11mo ago
  When you say 'stable' here, do you mean 'periodic' or are these solutions actually stable in the face of small perturbations (as opposed to the sensitive dependence on initial conditions that we'd expect from a chaotic system)?
- sbrorson 11mo ago
  An interesting corollary to this is that even if the future trajectory of a general 3-body orbit is predictable in theory using numerical methods (and infinite precision calculations), in practice the use of finite-precision floating point means that after some time the trajectory predicted by an ODE solver will diverge from the mathematically-true trajectory. Even symplectic integrators have this problem. More details on the general case of chaos are provided by this insightful blog post: https://www.stochasticlifestyle.com/how-chaotic-is-chaos-how-some-ai-for-science-sciml-papers-are-overstating-accuracy-claims/ https://www.stochasticlifestyle.com/how-chaotic-is-chaos-how...
lutusp 11mo ago
> No physics expert but isn't this unpredictable (based on what I saw in series) ? A three-body orbital problem is an example of a chaotic system, meaning a system extraordinarily sensitive to initial conditions. So no, not unpredictable in the classical sense, because you can always get the same result for the same initial conditions, but it's a system very sensitive to initial settings. > Amd this does seem predictable, I saw this for almost a minute The fact that it remains calculable indefinitely isn't evidence that it's predictable in advance -- consider the solar system, which technically is also a chaotic system (as is any orbital system with more than two bodies). For example, when we spot a new asteroid, we can make calculations about its future path, but those are just estimates of future behavior. Such estimates have a time horizon, after which we can no longer offer reliable assurances about its future path. You mentioned the TV series. The story is pretty realistic about what a civilization would face if trapped in a three-solar-body system, because the system would have a time horizon past which predictions would become less and less reliable. I especially like the Three Body Problem series because, unlike most sci-fi, it includes accurate science -- at least in places.
- adastra22 11mo ago
  There are stable solutions. See: Earth’s Moon (or any other planetary moon in the solar system).
  - lutusp 11mo ago
    > There are stable solutions. See: Earth’s Moon (or any other planetary moon in the solar system). Those are not stable solutions. Remember that Earth's moon only came into existence because of a collision with a protoplanet in the past, and if a large enough body passed close by in the future, we might lose our moon -- all because of the complexity of orbital systems with more than two members. > (or any other planetary moon in the solar system) There are any number of examples of planets gaining and/or losing moons because of multi-body orbital complexity.
    - adastra22 11mo ago
      If you are presupposing external perturbations or collisions, it's not an N=3 system... we're talking about the three body problem. A tidally locked system with periodic resonance is permanently stable in the absence of external forces.
adastra22 11mo ago
No, and please don’t try to learn anything from that science-free series. The author doesn't even have a Wikipedia-level understanding of what he is writing about. N-body problems for N>3 do not have exact, closed form solutions. For N=2 the solution is an ellipse. For N=3+ there is no equation you can write down that you can just plug in t and get any future value for the state of the system. But that is NOT the same as saying it is unpredictable. It is perfectly predictable. You just have to use one of the many numerical solutions for integrating ODEs.
constantcrying 11mo ago
The math in the 3 body problem was made up. Computing the trajectory of a 3 body problem is a comparatively simple task. The two grains of truth are that the solutions for most starting conditions are not analytic, roughly meaning that they can not be expressed in terms of functions. The other being that the numerical solution to an ODE diverges exponentially.

jgchaos 11mo ago

The link points to one of the stable solutions, and there are actually quite a few of those. The problem is that there’s no general closed form that tells us exactly where the bodies will be in the future, so we rely on numerical methods to approximate the motion. If you hit Reset All a few times or add more bodies, you’ll start to see the chaos

Pulcinella 11mo ago
There actually is an analytical solution using a power series that actually converges (Karl Sundman's work). Unfortunately, the universe still mocks our attempts. Though the series converges, it does so incredibly slowly. From Wikipedia: The corresponding series converges extremely slowly. That is, obtaining a value of meaningful precision requires so many terms that this solution is of little practical use. Indeed, in 1930, David Beloriszky calculated that if Sundman's series were to be used for astronomical observations, then the computations would involve at least 10^8000000 terms.
- Nevermark 11mo ago
  > the computations would involve at least 10^8000000 terms. Well we could speed up that simulation pretty easily, just arrange the actual masses and velocities somewhere... Then I thought, is there a way to scale the distances, masses and velocities to create a system with the same, but proportionally faster behavior? One guess as to perhaps why not: As distances get small, normal matter bodies will get close enough to actually collide. Perhaps some tiny primordial black holes would be useful.
taneq 11mo ago
When you say 'stable' here, do you mean 'periodic' or are these solutions actually stable in the face of small perturbations (as opposed to the sensitive dependence on initial conditions that we'd expect from a chaotic system)?
sbrorson 11mo ago
An interesting corollary to this is that even if the future trajectory of a general 3-body orbit is predictable in theory using numerical methods (and infinite precision calculations), in practice the use of finite-precision floating point means that after some time the trajectory predicted by an ODE solver will diverge from the mathematically-true trajectory. Even symplectic integrators have this problem. More details on the general case of chaos are provided by this insightful blog post: https://www.stochasticlifestyle.com/how-chaotic-is-chaos-how-some-ai-for-science-sciml-papers-are-overstating-accuracy-claims/ https://www.stochasticlifestyle.com/how-chaotic-is-chaos-how...

lutusp 11mo ago

> No physics expert but isn't this unpredictable (based on what I saw in series) ? A three-body orbital problem is an example of a chaotic system, meaning a system extraordinarily sensitive to initial conditions. So no, not unpredictable in the classical sense, because you can always get the same result for the same initial conditions, but it's a system very sensitive to initial settings. > Amd this does seem predictable, I saw this for almost a minute The fact that it remains calculable indefinitely isn't evidence that it's predictable in advance -- consider the solar system, which technically is also a chaotic system (as is any orbital system with more than two bodies). For example, when we spot a new asteroid, we can make calculations about its future path, but those are just estimates of future behavior. Such estimates have a time horizon, after which we can no longer offer reliable assurances about its future path. You mentioned the TV series. The story is pretty realistic about what a civilization would face if trapped in a three-solar-body system, because the system would have a time horizon past which predictions would become less and less reliable. I especially like the Three Body Problem series because, unlike most sci-fi, it includes accurate science -- at least in places.

adastra22 11mo ago
There are stable solutions. See: Earth’s Moon (or any other planetary moon in the solar system).
- lutusp 11mo ago
  > There are stable solutions. See: Earth’s Moon (or any other planetary moon in the solar system). Those are not stable solutions. Remember that Earth's moon only came into existence because of a collision with a protoplanet in the past, and if a large enough body passed close by in the future, we might lose our moon -- all because of the complexity of orbital systems with more than two members. > (or any other planetary moon in the solar system) There are any number of examples of planets gaining and/or losing moons because of multi-body orbital complexity.
  - adastra22 11mo ago
    If you are presupposing external perturbations or collisions, it's not an N=3 system... we're talking about the three body problem. A tidally locked system with periodic resonance is permanently stable in the absence of external forces.

adastra22 11mo ago

No, and please don’t try to learn anything from that science-free series. The author doesn't even have a Wikipedia-level understanding of what he is writing about. N-body problems for N>3 do not have exact, closed form solutions. For N=2 the solution is an ellipse. For N=3+ there is no equation you can write down that you can just plug in t and get any future value for the state of the system. But that is NOT the same as saying it is unpredictable. It is perfectly predictable. You just have to use one of the many numerical solutions for integrating ODEs.

constantcrying 11mo ago

The math in the 3 body problem was made up. Computing the trajectory of a 3 body problem is a comparatively simple task. The two grains of truth are that the solutions for most starting conditions are not analytic, roughly meaning that they can not be expressed in terms of functions. The other being that the numerical solution to an ODE diverges exponentially.