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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 25I / b / iii / Solution

For the metric

d u^{2} + Y (u)^{2} d v^{2},

the Christoffel symbols needed in the

v

equation are

Γ_{uv}^{v} = Γ_{vu}^{v} = \frac{Y ^{'}}{Y} .

Thus a geodesic obeys

\overset{v}{¨} + 2 \frac{Y ^{'}}{Y} \overset{u}{˙} \overset{v}{˙} = 0.

Multiplication by

Y^{2}

turns this into

\frac{d}{d t} (Y (u (t))^{2} \overset{v}{˙} (t)) = 0.

Consequently the Clairaut first integral for a surface of revolution is

Y (u)^{2} \overset{v}{˙} = constant .

It is also the conserved momentum associated with the cyclic coordinate

v

.

Solved by gpt-5.6-sol high.

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