Three equal masses lie on a line

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Mechanics & Dynamical Systems · Oscillations and normal modes

Invariance, Ch Representations, Schur's lemma and characters

Problem

Three equal masses lie on a line, each attached to a fixed wall by a spring and to their neighbour by another spring. Writing the equations of motion as x¨=−Ωx\ddot{\mathbf{x}} = -\Omega\mathbf{x} (with mass absorbed into the spring constants, so Ω\Omega has units of s−2\mathrm{s^{-2}}), the dynamical matrix is Ω=(3−20−25−20−23) s−2.\Omega = \begin{pmatrix} 3 & -2 & 0 \\ -2 & 5 & -2 \\ 0 & -2 & 3 \end{pmatrix}\,\mathrm{s^{-2}}. The arrangement has a mirror symmetry that swaps masses 11 and 33 and leaves mass 22 fixed, represented by R=(001010100).R = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix}. Ω\Omega and RR commute (each spring's effect on a mass and its mirror image is identical). Someone argues: "since Ω\Omega and RR commute, they must have the same eigenvalues; RR's eigenvalues are +1,+1,−1+1, +1, -1, so the squared normal-mode frequencies must be 1,1,−1 s−21, 1, -1\ \mathrm{s^{-2}}." ((a)) Find the true eigenvalues of Ω\Omega (the squared normal-mode frequencies) and their eigenvectors. ((b)) Explain what commuting matrices actually share, and say why the quoted argument is wrong.

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