 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/ii/paper-1/9c/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

q_{i} + ε η_{i}

with

η_{i} (t_{1}) = η_{i} (t_{2}) = 0

δ S = \int_{t_{1}}^{t_{2}} (L_{q_{i}} - \frac{d}{d t} L_{\overset{q}{˙}_{i}}) η_{i} d t .

d L_{\overset{q}{˙}_{i}} / d t = L_{q_{i}}

L_{q_{j}} = 0

p_{j} = L_{\overset{q}{˙}_{j}}

is constant.

For the sphere,

φ

p_{φ} = m a^{2} sin^{2} θ \overset{φ}{˙}, H = \frac{m a ^{2}}{2} (\dot{θ}^{2} + sin^{2} θ \overset{φ}{˙}^{2}) + V (θ) .

The Hamiltonian is the mechanical energy. Along solutions,

\dot{H} = - \partial L / \partial t = 0

, because the Lagrangian has no explicit time dependence.

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