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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2025

Paper 309

Affinely parametrized <geodesic>[geodesics] are the critical curves of the <geodesic Lagrangian>
$$
L=\frac12\left[-w\dot t^2+\frac{\dot r^2}{w}
+r^2\dot\theta^2+r^2\sin^2\theta\,\dot\phi^2\right].
$$
For $w=1+r^2$ and $w'=2r$, the nonzero <Christoffel symbol>[Christoffel symbols], up to symmetry in the lower indices, are
$$
\Gamma^t{}_{tr}=\frac{w'}{2w}=\frac r w,
\quad
\Gamma^r{}_{tt}=\frac{ww'}2=rw,
\quad
\Gamma^r{}_{rr}=-\frac{w'}{2w}=-\frac r w,
$$
$$
\Gamma^r{}_{\theta\theta}=-rw,
\quad
\Gamma^r{}_{\phi\phi}=-rw\sin^2\theta,
\quad
\Gamma^\theta{}_{r\theta}=\Gamma^\phi{}_{r\phi}=\frac1r,
$$
$$
\Gamma^\theta{}_{\phi\phi}=-\sin\theta\cos\theta,
\qquad
\Gamma^\phi{}_{\theta\phi}=\cot\theta.
$$
They follow either from the Euler-Lagrange equations or directly from the <Levi-Civita connection> formula.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 309 / 1 / b / Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 309 / 1 / b / Solution

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Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 309 / 1 / b / Solution

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