Normal modes of three equal masses with a symmetric gap potential
= Normal modes of three equal masses with a symmetric gap potential
Let three equal masses move on a circle of radius $r$, and let the potential be $U(\alpha)+U(\beta)+U(\gamma)$ with $\alpha+\beta+\gamma=2\pi$. At equal spacing $s_0=2\pi/3$, angular displacements have a zero-frequency rigid-rotation mode $(1,1,1)$ and a two-dimensional sum-zero eigenspace with
$$
\omega^2=\frac{3U''(s_0)}{mr^2}.
$$
The two vibrational modes are degenerate by the permutation symmetry.