Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 336 / 2 / a / i

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- Solution i

Solution

At leading order,

u = R cos θ, u_{t} = - R sin θ,

so

f = - u (u_{t})^{2} = - R^{3} cos θ sin^{2} θ .

The required period averages are

⟨ cos θ sin^{3} θ ⟩ = 0, ⟨ cos^{2} θ sin^{2} θ ⟩ = \frac{1}{8} .

Hence

R^{'} = 0, R ϕ^{'} = \frac{R ^{3}}{8} .

The initial data give

R (0) = 1

and

ϕ (0) = 0

, so

R (T) = 1, ϕ (T) = \frac{T}{8} .

Thus the perturbation produces a slow frequency shift but no leading amplitude decay:

u (t) \sim cos (t + \frac{ϵ t}{8}) .

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