Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 2 / 7D / a

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The two cyclic Euler angles give conserved canonical momenta

L_{3} = p_{ψ} = I_{3} (\dot{ψ} + \dot{ϕ} cos θ), L_{z} = p_{ϕ} = I_{1} \dot{ϕ} sin^{2} θ + L_{3} cos θ .

(1)

Here

L_{3}

is the body-axis angular momentum and

L_{z}

the vertical angular momentum. Thus $ω_{3} = L_{3} / I_{3}$ is constant. Since the Lagrangian has no explicit time dependence, the energy is also conserved:

E = \frac{I _{1}}{2} \dot{θ}^{2} + \frac{I _{1}}{2} \dot{ϕ}^{2} sin^{2} θ + \frac{L _{3}^{2}}{2 I _{3}} + M g l cos θ .

(2)

These are the three independent conserved quantities furnished by the continuous symmetries of the heavy symmetric top.

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Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 2 / 7D / a

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