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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 8E / a / Solution

In Hamiltonian mechanics, the canonical momenta are

p_{r} = m \overset{r}{˙}, p_{θ} = m r^{2} \dot{θ}, p_{ϕ} = m r^{2} sin^{2} θ \dot{ϕ} .

Hence

H = \frac{p _{r}^{2}}{2 m} + \frac{p _{θ}^{2}}{2 m r ^{2}} + \frac{p _{ϕ}^{2}}{2 m r ^{2} s i n ^{2} θ} + V (r) .

Solved by gpt-5.6-sol high.

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