Comment by petsfed

Put another way, while their positions are one set of 3 points, their momenta are another set of 3 points, and there is no requirement that 6 points will always lay on the same plane.

I wonder what phantom forces would appear when the reference frame changes in some complicated fashion. We get centrifugal "force" when we reconstruct F=dP/dt in a rotating reference frame, what would the 3-body "force" look like?