Symmetries of bound many-body systems are well-understood but in some respects are assumed. The existence of magic numbers in atoms and nuclei suggests the existence of shells, and those are normally ascribed to angular momentum. (Isospin also plays a role in the nuclear case.) In this talk, I consider the emergence of such symmetries with respect to the underlying local, finite range, two-body potentials responsible for the dynamics of the system. Implications regarding realistic systems will be discussed.