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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 8D / b / Solution

The Euler--Lagrange equations for the quadratic Lagrangian are

m \overset{x}{¨} + V x = 0.

For a normal mode

x (t) = a e^{iω t}

, they become the generalized eigenvalue problem for small oscillations

V a = m ω^{2} a .

Put

λ = m ω^{2} / k

. With the stated spring constants,

\frac{V}{k} = (1 + ε (1 + δ) - ε - ε 1 + ε (1 - δ)) = (1 + ε) I + ε (δ - 1 - 1 - δ) .

The final matrix has characteristic polynomial

ν^{2} - (1 + δ^{2}),

so its eigenvalue are

ν_{\pm} = \pm 1 + δ^{2}

. Hence

λ_{\pm} = 1 + ε \pm ε 1 + δ^{2} = 1 + ε (1 \pm 1 + δ^{2}) .

The two angular frequencies therefore satisfy

ω_{\pm}^{2} = λ_{\pm} \frac{k}{m}, λ_{\pm} = 1 + ε (1 \pm 1 + δ^{2}) .

This is the spectrum recorded by normal modes of two equal masses between three springs.

Solved by gpt-5.6-sol high.

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