On N-body potentials

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Päällikkö
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I suppose I could've equally well posted this to the math forum, but here goes...

One often sees the two-body potential f(r1, r2) being reduced to to f(r12). Can this be done in a more general case (can n-body potentials be reduced to just the distances between all the particles, e.g. f(r1, r2) -> f(r12, r13, r23))? I assume that the point particles don't have any internal structure. Do I need to add further constraints? I'm mainly aiming for representations of angular and dihedral (torsional) potentials.

From the looks of it, I'd guess it works at least with 3 point particles, due to the SSS triangle congruence.

Any ideas?
 

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