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(I previously asked essentially this on physics.stackexchange, but was actually
hoping for answers with something closer to a proof than what I got there.)

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:

Will there be a positive amount of time before any collisions occur?
(As opposed to, for example, a collision at time $\frac1n$ for each positive integer $n$.)

"Obviously" (at least, I hope I'm right), planet_0 will collide with planet_1. $\:$ Will that be the first collision?

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?

How long will it be before there are any collisions? $\:\:$ (perhaps just an approximation for small $\:\epsilon\:$)

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I read recently about a very similar problem that appeared in a 1949 letter from Ulam to von Neumann. (In that case the particles started at points of $\mathbb Z$ with each node being occupied with probability 1/2). He showed(?) that something analogous to the universe happens: nearby groups of particles come together; and then those "solar systems" form galaxies etc. – Anthony Quas Apr 26 '13 at 7:06
Of course if you really live in a 1D world, gravitational force presumably doesn't decay at all? – Anthony Quas Apr 26 '13 at 7:07
I think that as long as each planet is at most $\delta$ from its nearest integer, the total force on each planet is $O(\delta)$. This can be used to prove rigorously that there's a positive $\tau>0$ before any collision can occur. – Anthony Quas Apr 26 '13 at 7:10
Yes, particles clump together, typically forming smaller systems first. This is studied extensively in cosmology, both analytically and numerically. Gravitational effects are easy to model, and affect dark matter. However, dissipative effects such as the inelastic contraction of gas clouds are important, too. – Douglas Zare Apr 26 '13 at 7:42
Are the planets point masses, or some radius $r \lt \frac{1}{2}$? – David Roberts Apr 26 '13 at 7:56

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