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Normal Modes Lab

Learning goal

See why eigenvalue problems are worth all that machinery. A coupled oscillator system is a mess of coupled ODEs; in the eigenbasis it is \(N\) independent oscillators. The eigenvalues are the frequencies you can hear, and the eigenvectors are the shapes you can see.

The solver builds the mass matrix \(A\) and stiffness matrix \(B\), reduces the generalised problem \(B\mathbf{q} = \omega^2 A\mathbf{q}\) to the symmetric standard form \(A^{-1/2}BA^{-1/2}\), and diagonalises it with Jacobi rotations. Nothing is hard-coded: the frequencies in the table are computed, and the "exact" column is the analytic answer for comparison.

Things to try

  1. The triatomic, μ = 0.75 (CO₂). Three modes:
  2. \(\omega_1 = 0\) — not a bug. It is uniform translation of the whole molecule, a zero mode protected by momentum conservation. Any unconstrained system has one zero mode per symmetry direction, and finding them is a good check that you built \(B\) correctly.
  3. \(\omega_2 = \sqrt{k/m}\) — the antisymmetric stretch. Notice that \(\mu\) does not appear. Look at the mode shape and you will see why: the centre mass does not move at all (the widget confirms its amplitude is zero to \(10^{-16}\)), so its mass cannot possibly matter.
  4. \(\omega_3 = \sqrt{(k/m)(1 + 2/\mu)}\) — the symmetric stretch, the only mode that moves the centre, hence the only one that depends on \(\mu\).

That the algebra "knows" the centre mass is stationary — without anyone telling it — is the moment eigenvalue methods start to feel like more than bookkeeping.

  1. Slide μ. Only \(\omega_3\) moves. Watch it diverge as \(\mu \to 0\) (a massless centre is infinitely easy to shake) and approach \(\sqrt{k/m}\) as \(\mu \to \infty\) (an infinitely heavy centre acts like a wall, so the two outer masses become independent oscillators).

  2. Switch to the uniform chain. The frequencies match $\(\omega_n = 2\sqrt{k/m}\,\sin\!\left(\frac{n\pi}{2(N+1)}\right)\)$ to machine precision. The mode shapes are sampled sine waves — this is a discrete standing wave, and it is the finite-dimensional ancestor of the continuous string. Take \(N\to\infty\) and you get the wave equation; the eigenvectors become \(\sin(n\pi x/L)\) and the eigenvalue problem becomes Fourier series.

  3. Raise N and watch the low modes. For small \(n\), \(\sin(n\pi/2(N+1)) \approx n\pi/2(N+1)\), so \(\omega_n \propto n\) — the linear dispersion of a continuous string. The high modes bend away from linearity: that curvature is the discreteness of the lattice showing through, and it is exactly the phonon dispersion of a 1-D crystal.

  4. Add a heavy defect. Make one mass four times heavier and look at the lowest mode: it concentrates on the defect instead of spreading over the chain. A heavy impurity traps a vibration. This is localisation in miniature, and it is why impurities in a crystal produce localised vibrational states.

  5. Check the orthogonality claim. The modes are orthogonal in the mass-weighted inner product \(\mathbf{q}_i^{\mathsf T}A\,\mathbf{q}_j = 0\), not the plain dot product. This is generalised eigenvalue structure: with unequal masses, the "right" inner product is the one the kinetic energy defines. It is the first place in the course where the inner product is something you choose rather than something given.

Why this is the payoff of Chapter 2

The equations of motion are \(A\ddot{\mathbf{q}} = -B\mathbf{q}\) — coupled, because \(B\) is not diagonal. Change to normal coordinates and every equation becomes \(\ddot{Q}_n = -\omega_n^2 Q_n\): independent simple harmonic oscillators. The general motion is then just a superposition with amplitudes fixed by initial conditions.

This is the entire content of "diagonalise the matrix", and it recurs everywhere: normal modes in mechanics, energy eigenstates in quantum mechanics, principal axes in rigid-body rotation, principal components in statistics.

Eigenvalues & eigenvectors · Diagonalization · Hermitian matrices · Matrices & determinants · Fourier series — the continuum limit · Characteristic equation (eq.)