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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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    $\begingroup$ 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. $\endgroup$
    – Anthony Quas
    Commented Apr 26, 2013 at 7:06
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    $\begingroup$ Of course if you really live in a 1D world, gravitational force presumably doesn't decay at all? $\endgroup$
    – Anthony Quas
    Commented Apr 26, 2013 at 7:07
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    $\begingroup$ 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. $\endgroup$
    – Anthony Quas
    Commented Apr 26, 2013 at 7:10
  • $\begingroup$ 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. youtube.com/watch?v=8C_dnP2fvxk However, dissipative effects such as the inelastic contraction of gas clouds are important, too. $\endgroup$
    – Douglas Zare
    Commented Apr 26, 2013 at 7:42
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    $\begingroup$ Excellent question! Perhaps You should start as in thermodynamics with a large number $N$ of particles located at $-N,\dotsc,-1, \epsilon, 1,\dotsc, N$ and see what happens then. At least in this case the total energy is finite, which is not the case when infinitely many particles are present. $\endgroup$
    – Liviu Nicolaescu
    Commented Apr 26, 2013 at 20:39

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