 Normal modes of a four-bead spring ring

Four equal beads of mass $m$ are connected cyclically by identical springs of stiffness $k$ following arcs of a frictionless circular guide. In signed arc displacements $q_j$, the quadratic Lagrangian is $L=(m/2)\sum {\dot{q}}_j^2-(k/2)\sum (q_{j+1}-q_j{)}^2$. The stiffness is $k$ times the four-cycle graph Laplacian. Its eigenvectors can be chosen as $(1,1,1,1)/2$, $(1,0,-1,0)/\sqrt{2}$, $(0,1,0,-1)/\sqrt{2}$, and $(1,-1,1,-1)/2$, with squared frequencies in the title. Their dot products with $q$ are decoupled normal mode coordinates. The zero mode is a rigid rotation; the two middle modes span a degenerate eigenspace; the highest mode alternates adjacent arc displacements. Changing the guide radius multiplies angular inertia and angular spring energy by the same factor, leaving these frequencies unchanged.