  For the models of $N$-body identical harmonic oscillators interacting through
potentials of homogeneous degree -2, the unitary operator that transforms a
system of time-dependent parameters into that of unit spring constant and unit
mass of different timescale is found. If the interactions can be written in
terms of the differences between positions of two particles, it is also shown
that the Schr\"{o}dinger equation is invariant under a unitary transformation.
These unitary relations can be used not only in finding coherent states from
the given stationary states in a system, but also in finding exact wave
functions of the Hamiltonian systems of time-dependent parameters from those of
time-independent Hamiltonian systems. Both operators are invariant under the
exchange of any pair of particles. The transformations are explicitly applied
for some of the Calogero-Sutherland models to find exact coherent states.
