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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/iii/paper-321/3/b/solution

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Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 321 / 3 / b / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-06

Let

d_{jn} = h (j - n) e_{y}

and

Δ ξ = ξ_{j} - ξ_{n}

, with

ξ_{n} = (x_{n}^{'}, y_{n}^{'})

. Linearizing the Newton's law of universal gravitation uses

δ (\frac{d}{∣ d ∣ ^{3}}) = \frac{1}{∣ d ∣ ^{3}} (I - 3 \frac{d d ^{T}}{∣ d ∣ ^{2}}) Δ ξ .

For an azimuthal separation, the matrix is

diag (1, - 2)

. After canceling the equilibrium forces, the linearized shearing sheet equations are

\overset{x}{¨}_{n}^{'} - 2Ω \overset{y}{˙}_{n}^{'} = 3 Ω^{2} x_{n}^{'} + \frac{G m}{h ^{3}} j \neq = n \sum \frac{x _{j}^{'} - x _{n}^{'}}{∣ j - n ∣ ^{3}},

\overset{y}{¨}_{n}^{'} + 2Ω \overset{x}{˙}_{n}^{'} = - \frac{2 G m}{h ^{3}} j \neq = n \sum \frac{y _{j}^{'} - y _{n}^{'}}{∣ j - n ∣ ^{3}} .

The transverse gravitational response tends to restore a radial displacement, while the longitudinal response reinforces an azimuthal density concentration. Both sums converge for the bounded Fourier modes considered next.

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