 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2025

Paper 3

For the metric
$$
du^2+Y(u)^2\,dv^2,
$$
the Christoffel symbols needed in the $v$ equation are
$$
\Gamma^v_{uv}=\Gamma^v_{vu}=\frac{Y'}Y.
$$
Thus a geodesic obeys
$$
\ddot v+2\frac{Y'}Y\dot u\dot v=0.
$$
Multiplication by $Y^2$ turns this into
$$
\frac d{dt}\left(Y(u(t))^2\dot v(t)\right)=0.
$$
Consequently the <Clairaut first integral for a surface of revolution> is
$$
\boxed{Y(u)^2\dot v=\text{constant}}.
$$
It is also the conserved momentum associated with the cyclic coordinate $v$.

Solution

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

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

 Ancestors (12)

 Discussion (0)

 Articles by others on the same topic (0)

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

 Ancestors (12)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)