So, I have now generalized the 2-d Universe a little more, and here are two particles interacting with each other within the surface of a sphere.
grumpy about the words I use - it works :)).
Starting point is that we are on a sphere, and so it makes sense to use spherical polar coordinates. Now, one painful thing is that different people define which angle is ϑ and which is φ, so I will be following the convention shown in the top figure on Wikipedia. Remember, however, we are working in the surface of the sphere, and so we have no radial (r) coordinate.
If we have two infinitesimal displacements in our coordinate (and assuming a unit sphere for convenience), then the separation between the displaced coordinates is given by;
Christoffel symbols, which tell us the equations of motion are given by
where we are using the Einstein summation convention, and the greek characters are just our coordinates that we are using. The v-terms are velocities in our coordinate directions. If we plug all the terms in, calculating the Christoffel symbols, the equations of motion become;
So, that all works. Particles happily travel through the spherical surface on effectively free-fall paths, covering great circles. But now we need to add the interaction between the particles, and for that we need to calculate a few things. Firstly, there is the distance between particles in the surface of the sphere! That's relatively straight-forward. Then we need to work out the vector components of the force, that is somewhat harder. Once we have these, we then use the modified form of the geodesic equation, which includes the influences of forces, that is