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 Incoming links: Quartic equation for one component of torque-free angular velocity

Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 14B / Solution Created 2026-09-24 Updated 2026-10-03

A principal body frame is an orthonormal basis

e_{a} (t)

fixed in the rotating body and aligned with its principal axes. If

R (t)

is the rotation matrix whose columns are these vectors, differentiating

R^{T} R = I

shows that

\dot{R} R^{T}

is skew-symmetric. Every skew-symmetric linear map in three dimensions is cross product with a unique vector

ω

, so

\dot{e}_{a} = ω \times e_{a} .

For any vector

A = A_{a} e_{a}

,

(\frac{d A}{d t})_{m s p a ce} = \dot{A}_{a} e_{a} + ω \times A .

Applying this to the conserved angular momentum

L = I_{1} ω_{1} e_{1} + I_{2} ω_{2} e_{2} + I_{3} ω_{3} e_{3}

gives the Euler equations for a torque-free rigid body:

I_{1} \overset{ω}{˙}_{1} = (I_{2} - I_{3}) ω_{2} ω_{3}, I_{2} \overset{ω}{˙}_{2} = (I_{3} - I_{1}) ω_{3} ω_{1}, I_{3} \overset{ω}{˙}_{3} = (I_{1} - I_{2}) ω_{1} ω_{2} .

Multiplying these equations respectively by

ω_{i}

and summing proves conservation of kinetic energy; multiplying by

I_{i} ω_{i}

and summing proves conservation of squared angular momentum. Thus the torque-free rigid-body invariants are

2 E = 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} .

Solving these two linear equations for the first two squared components gives

ω_{1}^{2} = \frac{L ^{2} - 2 E I _{2} + I _{3} ( I _{2} - I _{3} ) ω _{3}^{2}}{I _{1} ( I _{1} - I _{2} )},

ω_{2}^{2} = \frac{L ^{2} - 2 E I _{1} + I _{3} ( I _{1} - I _{3} ) ω _{3}^{2}}{I _{2} ( I _{2} - I _{1} )} .

Using

I_{3} \overset{ω}{˙}_{3} = (I_{1} - I_{2}) ω_{1} ω_{2}

yields the quartic equation for one component of torque-free angular velocity:

\overset{ω}{˙}_{3}^{2} = f (ω_{3}) = - \frac{[ L ^{2} - 2 E I _{2} + I _{3} ( I _{2} - I _{3} ) ω _{3}^{2} ] [ L ^{2} - 2 E I _{1} + I _{3} ( I _{1} - I _{3} ) ω _{3}^{2} ]}{I _{1} I _{2} I _{3}^{2}} .

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