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,

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

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).

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