Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 15E / a

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- Solution a

Solution

The kinetic energy and squared angular-momentum magnitude are

T = \frac{1}{2} (I_{1} ω_{1}^{2} + I_{2} ω_{2}^{2} + I_{3} ω_{3}^{2}), L^{2} = I_{1}^{2} ω_{1}^{2} + I_{2}^{2} ω_{2}^{2} + I_{3}^{2} ω_{3}^{2} .

Differentiate and substitute Euler's equations. In

\dot{T}

, the coefficient of

ω_{1} ω_{2} ω_{3}

is

(I_{2} - I_{3}) + (I_{3} - I_{1}) + (I_{1} - I_{2}) = 0.

The analogous weighted sum in

d L^{2} / d t

also cancels, so both quantities are conserved.

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