 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ib/paper-3/6b/solution

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

\ddot{θ} - sin θ cos θ \dot{ϕ}^{2} = 0, \frac{d}{d t} (sin^{2} θ \dot{ϕ}) = 0.

The coordinate

ϕ

l = sin^{2} θ \dot{ϕ}

is conserved. Substitution into the first equation gives

\ddot{θ} = \frac{l ^{2} c o s θ}{s i n ^{3} θ} .

Multiplying by

2 \dot{θ}

shows that

\frac{d}{d t} (\dot{θ}^{2} + \frac{l ^{2}}{s i n ^{2} θ}) = 0.

Thus

h

is conserved. Since

l^{2} / sin^{2} θ = sin^{2} θ \dot{ϕ}^{2}

, $h$ is the squared speed on the unit sphere, equivalently twice the kinetic energy.

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