A system of coupled oscillators looks tangled: push one mass and the other answers. But the tangle is a disguise. Diagonalize the stiffness matrix and the motion splits into independent pure tones — the symmetric (in-phase) mode at ω=√(k/m) and the antisymmetric (out-of-phase) mode at ω=√((k+2kc)/m). Every possible motion is a sum of these. Rendered, not quoted.
source J.-L. Lagrange, Mécanique analytique (Paris, 1788) — small-oscillation theory, simultaneous diagonalization of the kinetic and potential quadratic forms. No stable canonical URL for the 1788 treatise; see Mécanique analytique. AMBER — primary text cited by author/title/year.
Two equal masses (m=1), each tied to a wall by spring k, coupled to each other by spring kc. Natural units k=1, kc=0.5.
m·ẍ₁ = −k·x₁ − kc(x₁−x₂)
m·ẍ₂ = −k·x₂ − kc(x₂−x₁)
Stiffness K = [[k+kc, −kc], [−kc, k+kc]], mass M = I. Eigenproblem Kv = ω²Mv.
Coupled becomes independent. Diagonalizing K turns tangled masses into a set of decoupled oscillators, each with a fixed eigenfrequency.
Quantize one such mode and it is the-quantum-harmonic-oscillator. Chain infinitely many and the mode shapes become the-standing-wave on a string. The same diagonalization runs underneath all three.
Re-runs the full eigen-solve live and confirms the two frequencies and their ordering. If window 6 tampers the coupling, the ordering breaks and this badge flips red.
k = 1 kc = 0.5 m = 1
initial x₀ = (0.70, 0.10), v₀ = (0, 0)
A general nudge — not a pure mode — fed in to be decomposed.
Live eigen-solve of K and analytic mode evolution. The masses below move by the exact superposition x(t)=Σ Qi(t)·vi.
ωsym = √(k/m) = 1.000000
ωanti = √((k+2kc)/m) = 1.414214
modes = DOF = 2 · eigenvectors M-orthogonal
"A coupled system has one blurry shared frequency."
→ It has exactly DOF distinct eigenfrequencies; the beat you hear is their interference.
"The mode shapes depend on the amplitude."
→ In the small-oscillation limit the eigenvectors are amplitude-independent — fixed ratios (1,1) and (1,−1).
"Energy leaks between modes."
→ Once diagonalized, each mode energy is separately conserved; the modes never talk.
Flip the coupling sign in K (off-diagonal +kc instead of −kc). The antisymmetric frequency drops below the symmetric one — a physically wrong ordering. The witness (7) catches it.