Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2018 / S7 / 2 / a

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Solution

Differentiation gives

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

ϕ

and

ψ

are cyclic, so their canonical momenta are conserved. The Lagrangian is autonomous, so the total energy is also conserved. The momentum

p_{θ}

is not generally constant because gravity and the angular kinetic terms depend on

θ

.

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