An idealized $\infty$-body problem: is an infinite and regular configuration of massive objects stable?

Question

Suppose I have an infinite amount of massive point shaped objects, and I arrange the objects by putting one object on each point of $\mathbb{Z}^2$ within $\mathbb{R}^2$. By symmetry, the gravity exerted on every object should add up to zero, so that nothing should happen to the system. Now suppose that I slightly perturb one of the objects. Will the resulting system collapse or will it stabilise?

There are obvious generalisations involving either higher dimensions or other kinds of regular configurations, and I'd be happy to hear more about that too, but for now let's restrict to the simplest situation stated above. Also, feel free to alter the tags, as I wasn't sure where to put this.

Answer 1

The particle will return to its original position if that position is a local minimum of the gravitational potential due to all other particles. However, for particles in 3D under the inverse square law, the potential satisfies Laplace's equation in free space, therefore there are no local minima. So the configuration cannot be stable.

I believe this argument can be extended to particles in 2D by considering them to lie in $\mathbb R^2\times\{0\}$ -- one just has to show that the potential is still not a local minimum when restricted to the plane.

Answer 2

Suppose you slightly move one of the masses, say the one lying at $(0,0)$, along $x$ axis in the positive direction. Then all distances from that mass to any mass with $x>0$ decrease, while all distances from that mass to any mass with $x\le0$ increase. The result is a net force along $x$ in the positive direction, leading to instability of the system.