  We point out a map between the dynamics of a non-relativistic system of $N$
particles in one dimension interacting via the pair-wise potentials $U_I(q) =
(\nu^2/4R^2)\sin^2(q/2R)$ and the one of the particles with the pair potential
$U_{II}(q) = \nu^2/q^2$ and the external potential $U_{ext} = \omega^2 q^2/2$.
The natural relation between the frequency $\omega$ and the radius $R$ is:
$\omega R = 1$.
