Coordinates and Democracy in the N-Body Problem

It is shown that the construction of ``democratic'' subgroups of O3N−3 in the N-body problem is greatly facilitated by the ``proper'' choice of relative vectors in the center of mass frame. The word ``proper'' is taken to mean that the set of (N − 1) relative vectors forms a basis for reduced representations of the corresponding democracy subgroup of SN. The imposition of this requirement easily leads us to the reduction chains O6⊃SU3⊃SO3rot and O9⊃SO31×SO32×SO33⊃SO3rot in the three- and four-body problems, respectively, and to O3N−3⊃SON−1×SO3rot in the N > 4 case.