Re: Gravity on a torus

Gerard Westendorp wrote:
I was playing with a 2D model with point particles and gravitational
attraction. To avoid particles going off to infinity, I thought I might
turn the 2D space into a topological torus: connect x=L = x=0 and y=L to
y=0.

But now a problem occurs: What is the distance between 2 points?

Because of the periodic boundary, you get for the distance (s):

s^2 = (x2-x1+i*L)^2 + (y2-y1+j*L)^2

You could set i = j = 0, but that would destroy translation symmetry, it
would make the coordinates at 0 and L physically different from other
points.

So there are a multitude of distances, a grid of them described by the
integers (i,j).

But that does seem a bit weird. Any comments on this?

Think I get it now.

The particles simply see each other particle, plus all the infinite
number of images of of them. In 2D (ie gravity force varies as 1/r) this
would be a problem, because the series 1/x + 1/(x+L) + 1/(x+2L) + ...
doesn't converge. But with the inverse square law it is OK.

Gerard

.

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