# Applied 3-sphere model

Hi everyone,

I'm currently implementing some molecular dynamics code. What I have so far is a system of particles interacting over cubic periodic boundaries..so when one particle leaves through one face of the cube, it pop back in through the other face. In reality however, the implementation simply replicates cubes and takes the minimum image. The fact that a boundary is introduced means that there is a cutoff potential...

To make a long story short, I am now implementing a model where the system would be implemented in a 3-sphere so that there would be no boundary considerations for interacting particles. If you consider the 2D analogy of particles sitting in a 2-sphere, they can all force interact with each other, without the need for a cutoff due to the boundary. This would be a better model. I wish to do this without embedding in 4-space.

While I understand the theoretical framework involved, I'm not so sure about how to handle the coordinate systems as an applied problem. I will use a spherical metric to calculate distance and angles, and I don't need the geodesic equation since I am doing a timestep integration to calculate the next position....but in what coordinates?

here's what i was thinking:

a) use x,y,z for coordinates
b) set 3 poles as 3 origins for a coordinate system, that cycle from (-a, a), they will just go round and round always having a coordinate in this range.
c) calculate distance between particles using the metric tensor, and apply force calculations to get acceleration
d) move the particles by newtonian integration,
$$x = x_i + v_x dt + a_x (dt)(dt)$$

so, will keeping a standard coordinate system (but just cycling the coordinates without boundary) but using a spherical spacial metric work? any ideas?