Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 316 / 3 / vi

Table of contents
- Solution vi

Solution

For

t ≫ τ

, the free term has disappeared. Exactly at

A = g_{1}

, the resonant contribution tends to

z_{1} (t) = i τ ν_{1} e^{i (g_{1} t + β_{1})},

with finite eccentricity

∣ ν_{1} ∣ τ

and a quarter-cycle phase shift. Without damping, exact resonance instead gives

z_{1} (t) = e^{i A t} (C + i ν_{1} t e^{i β_{1}}),

whose secular resonance amplitude grows linearly in the ideal linear theory.

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