Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 309 / 3 / b / i

Table of contents
- Solution i

Solution

Write

I_{a} = \overset{ˉ}{I}_{aa}

and use the right-handed rotation

L (t) = cos Ω t sin Ω t 0 - sin Ω t cos Ω t 0 001 .

In the integral defining the second mass moment tensor, change variables to

y = L^{- 1} x

. The determinant is one and

x_{i} = L_{ik} y_{k}

I_{ij} (t) = \int ρ (y) L_{ik} y_{k} L_{j l} y_{l} d^{3} y = L_{ik} L_{j l} \overset{ˉ}{I}_{k l}, I (t) = L (t) \overset{ˉ}{I} L (t)^{T} .

C = (I_{1} + I_{2}) /2

and

Δ = I_{1} - I_{2}

. Multiplication gives

I_{11} (t) I_{22} (t) I_{12} (t) = I_{21} (t) I_{33} (t) = C + \frac{Δ}{2} cos (2Ω t), = C - \frac{Δ}{2} cos (2Ω t), = \frac{Δ}{2} sin (2Ω t), = I_{3}, I_{13} (t) = I_{23} (t) = 0.

In particular,

I_{kk} = I_{1} + I_{2} + I_{3}

is constant. These expressions also show how the rotating anisotropy enters the mass quadrupole moment.

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