 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ii/paper-4/8e/b/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

ω_{1}, ω_{2}, ω_{3}

and add. The cubic terms cancel, giving conservation of the rotational kinetic energy

E = \frac{1}{2} (I_{1} ω_{1}^{2} + I_{2} ω_{2}^{2} + I_{3} ω_{3}^{2}) .

Instead multiply the equations by

I_{1} ω_{1}, I_{2} ω_{2}, I_{3} ω_{3}

and add. The cubic terms again cancel, giving

L^{2} = I_{1}^{2} ω_{1}^{2} + I_{2}^{2} ω_{2}^{2} + I_{3}^{2} ω_{3}^{2},

the squared magnitude of the conserved angular momentum. For an asymmetric body these two quadratic forms are not proportional, so the torque-free rigid-body invariants are independent.

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