= Normal modes of a four-bead spring ring
{title2=$\omega^2=0,\ 2k/m,\ 2k/m,\ 4k/m$}
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 $(1,1,1,1)/2$, $(1,0,-1,0)/\sqrt2$, $(0,1,0,-1)/\sqrt2$, 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.
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