Is there a result showing that something along the lines of the three body problem is undecidable? Or are they known to be decidable or neither?


I mean problems along the lines of the following formulated in some suitable system:


Given masses, velocities and positions in 3 dimensions and a distance d (assume all expresses in rational multiples of G...ie G=1 so no using G as a non-computable oracle) can one decide whether, acting under the influence of Newtonian gravity only,


1. Any of the point masses get within d units of another.
2. Whether any of the masses ever get beyond d units from one of the other masses.
3. Any of the bodies escapes to infinity relative to one of the others.




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If necessary to make the problem well-defined one could stipulate that the initial positions are choosen to avoid ever allowing an exact collision of the point particles (also wonder if that is decidable).


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More generally is their some result letting one embed arbitrary computations into a system of bodies acting only under gravity?



Decidability of 3 body problem
The paper [Undecidability in $\mathbb{R}^n$: Riddled Basins,
the KAM Tori, and the Stability of the Solar System](http://philsci-archive.pitt.edu/13175/1/parker2003.pdf) by Matthew W. Parker (*Philosophy of Science* **70** (April 2003), 359–382) comes close to answering your question. A classical problem in the same spirit as your question is the [stability of the solar system](https://en.wikipedia.org/wiki/Stability_of_the_Solar_System). As Parker notes, there are quite a few informal claims in the literature that this type of problem is uncomputable, but for the most part, they gloss over the crucial question of how to define computability in the context of real numbers (sometimes referred to as *[real computation](https://en.wikipedia.org/wiki/Real_computation)* or *[computable analysis](https://en.wikipedia.org/wiki/Computable_analysis)*).


Parker offers his own approach to real computation and analyzes the question accordingly. However, as far as I know, neither Parker nor anyone else has analyzed the setup you suggested, which is to restrict the set of initial conditions to the rational numbers, and ask if the subset of (say) "stable configurations" is computable in the traditional sense.


