4 ms·

For a two-body problem the discrepancies between your model and reality increase gradually with time, but it's still possible to predict eclipses decades in adv

For a two-body problem the discrepancies between your model and reality increase gradually with time, but it's still possible to predict eclipses decades in advance with a precise albeit imperfect measurement of initial conditions. With a three-body problem any slight shift causes a wildly different trajectory, bearing no resemblance to the original so your measurements of the initial condition have to be perfect.

Grustaf 5y ago
No that's not it. The two body problem has a closed, analytical solution, the three body one doesn't, so you need to simulate it. It's a fundamentally different approach.
- extropy 5y ago
  Both of you are correct.
- l332mn 5y ago
  That's not really the issue. The three body problem does have an analytical solution in the form of a power series, but the problem is that it converges so slowly to be of any practical use.
  - Grustaf 5y ago
    It is though, if you don’t have a closed form solution you need to use an iterative process to calculate the positions, meaning errors will accrue over time. For a closed form solution that wouldn’t be the case. Thanks for mentioning the existence of an analytical solution at all though, I wasn't aware of that.
    - johncolanduoni 5y ago
      There are iterative methods/systems that stabilize over time. For example, symplectic integrators on tame problems oscillate lightly around the true energy of the system over time. The issue here is the properties of the underlying problem, not the set of solution methods.
    - chestervonwinch 5y ago
      > meaning errors will accrue over time This is not universally true. Error behavior is a function of the particular problem, the algorithm used to approximate its solution, and the properties of input data. A large subtopic of numerical analysis is concerned with this kind of stuff. See [1] or [2] to get a flavor. [1]: https://en.wikipedia.org/wiki/Numerical_stability https://en.wikipedia.org/wiki/Numerical_stability [2]: https://en.wikipedia.org/wiki/Lax_equivalence_theorem https://en.wikipedia.org/wiki/Lax_equivalence_theorem
ekianjo 5y ago
arent eclipses three body problems?
- nwatson 5y ago
  For all practical purposes eclipses are two two-body problems ... Sun vs Earth-Moon pair, and Earth vs Moon. The scale of the Sun is so vastly larger and distances so great from sun to earth there's no need to do 3-body. If the three had roughly the same mass would be different story.

Grustaf 5y ago

No that's not it. The two body problem has a closed, analytical solution, the three body one doesn't, so you need to simulate it. It's a fundamentally different approach.

extropy 5y ago
Both of you are correct.
l332mn 5y ago
That's not really the issue. The three body problem does have an analytical solution in the form of a power series, but the problem is that it converges so slowly to be of any practical use.
- Grustaf 5y ago
  It is though, if you don’t have a closed form solution you need to use an iterative process to calculate the positions, meaning errors will accrue over time. For a closed form solution that wouldn’t be the case. Thanks for mentioning the existence of an analytical solution at all though, I wasn't aware of that.
  - johncolanduoni 5y ago
    There are iterative methods/systems that stabilize over time. For example, symplectic integrators on tame problems oscillate lightly around the true energy of the system over time. The issue here is the properties of the underlying problem, not the set of solution methods.
  - chestervonwinch 5y ago
    > meaning errors will accrue over time This is not universally true. Error behavior is a function of the particular problem, the algorithm used to approximate its solution, and the properties of input data. A large subtopic of numerical analysis is concerned with this kind of stuff. See [1] or [2] to get a flavor. [1]: https://en.wikipedia.org/wiki/Numerical_stability https://en.wikipedia.org/wiki/Numerical_stability [2]: https://en.wikipedia.org/wiki/Lax_equivalence_theorem https://en.wikipedia.org/wiki/Lax_equivalence_theorem

ekianjo 5y ago

arent eclipses three body problems?

nwatson 5y ago
For all practical purposes eclipses are two two-body problems ... Sun vs Earth-Moon pair, and Earth vs Moon. The scale of the Sun is so vastly larger and distances so great from sun to earth there's no need to do 3-body. If the three had roughly the same mass would be different story.