# Infinitely many planets on a line, with Newtonian gravity

(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 $\:\epsilon\:$)

The acceleration of planet number $n$ except for the planet $0$ will go like $-1/n^3$ because the shift of planet $0$ from zero to $\epsilon$ is equivalent to adding a "dipole" (a pair of positive and negative mass, relatively shifted) at the location $0$ relatively to the balanced (but unstable) uniform chain and this dipole acts with inverse cube, instead of the inverse law.
We see that indeed the planets $+1$ and $-1$ are most affected and fastest to get some acceleration. However, planet $-1$ will move to the left, away from a potential collision. Nevertheless, planet $-2$ is trying to escape from planet $-1$, although by a smaller speed, but that will be enough to guarantee that the $0-1$ collision will be the first one. Other collisions will follow. You may numerically simulate it – the problem isn't integrable even for small $\epsilon$, I think, simply because you're interested in the moments when the distance $\epsilon$ grew to a large number $O(1)$, anyway. 