  We consider M clusters of interacting particles, whose in-group interactions
are arbitrary, and inter-group interactions are approximated by oscillator
potentials. We show that there are masses and frequencies that decouple the
in-group and inter-group degrees of freedom, which reduces the initial problem
to M independent problems that describe each of the relative in-group systems.
The dynamics of the M center-of-mass coordinates is described by the
analytically solvable problem of M coupled harmonic oscillators. This paper
derives and discusses these decoupling conditions. Furthermore, to illustrate
our findings, we consider a charged impurity interacting with a ring of ions.
We argue that the impurity can be used to probe the center-of-mass dynamics of
the ions.
