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Look for "multi-pole" expansions to help reduce the N^2 interaction cost (perhaps.) Also the PIC (particle-in-cell may help.) There was some work on this done about 50 years ago on the Maniac computer.

I once built a system (for other purposes, network intrusion detection, actually, it wasn't used), that has similar problems to yours. What I did was sample which particle to move. The driving force was the average field of the other particles. I think this is similar to "mean field" approximations. Of course, you get only the "weak behavior" of the system (on the average, not detailed.)

I'll definitely look at your ideas, but I'm hoping (at least for basic n-body systems) to get far better efficiency then that (possibly even down to O(2n)), without losing detail. I still have a fair amount of work to do on it, but so far it looks promising. It may take some extra dimensional calculations, but that'd still be better then N^2. Not sure I can apply that efficiency to anything beyond, but perhaps I'll run into some other ideas.

For my axis problems, I have a feeling it has to do with time compression along the axises, inherent in the orthigonally-produced ('simple') rectangular coordinate system. This would produce a dimensional system not having 2 dimensions, but something slightly greater (or smaller, depending on dimensional curvature on the axises). Have to finish a piece up though to test that theory though. Could be wrong, but it'll be fun to find out.

-The Hajman-