(I apologize if this question is too theoretical for this site.)
This is related to the answer here, although I came up with it independently of that. $\:$ Suppose we
have a unit mass planet at each integer point in 1-d space. $\:$ As described in that answer, the sum
of the forces acting on any particular planet is absolutely convergent. $\;\;$ Suppose we move planet_0
to point $\epsilon$, where $\: 0< \epsilon< \frac12 \:$. $\;\;$ For similar reasons, those sums will still be absolutely convergent.
Now we let Newtonian gravity apply. $\:$ What will happen?
If it's unclear what an answer might look like, you could consider the following more specific questions:
planet_0 will start out moving right, and all of the other planets will start out moving to the left.
Will there be a positive amount of time before any of them turn around?
(As opposed to, for example, each planet_n for $\: n\neq 0 \:$ turning around at time 1/|n|.)
Will there be a positive amount of time before any collisions occur?
"Obviously" (at least, I hope I'm right), planet_0 will collide with planet_1. $\:$ Will that be the first collision?
How long will it be before there are any collisions? $\:\:$ (perhaps just an approximation for small $\:\epsilon\:$)