  In the context of classical or quantum many-body problems involving identical
bodies, a linear change of coordinates can be constructed with the properties
that it includes the center-of-mass as one of the new coordinates and preserves
the inherent permutation symmetry of both the Hamiltonian and the admissible
states. This has advantages over the usual system of Jacobi coordinates in the
study of many-body problems for which permutation symmetry of the bodies plays
an important role. This paper contains the details of the construction of this
system and the proof that these properties uniquely determine it, up to trivial
modifications. Examples of applications to both classical and quantum problems
are explored, including a generalization to problems involving groups of
different species of bodies.
