Degenerate normal modes from permutation symmetry

If the mass and stiffness matrices commute with a permutation representation, each symmetry subspace is invariant under the generalized eigenvalue problem for small oscillations. On an irreducible subspace, Schur lemma forces a commuting operator to be scalar, producing a degenerate normal mode multiplet.

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 withThe two vibrational modes are degenerate by the permutation symmetry.

For three unit masses with unit springs between every pair and one unit spring anchoring each mass to the origin,where $J$ is the all-ones matrix. The collective mode $(1,1,1)$ has angular frequency $1$, while the two-dimensional sum-zero subspace has angular frequency $2$.