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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2024

Paper 309

For the quadratic <geodesic Lagrangian>
$$
L_E=\frac12g_{\mu\nu}(x)\dot x^\mu\dot x^\nu,
$$
the <Euler-Lagrange equation> is
$$
g_{\mu\nu}\ddot x^\nu
+\partial_\rho g_{\mu\nu}\dot x^\rho\dot x^\nu
-\frac12\partial_\mu g_{\alpha\beta}\dot x^\alpha\dot x^\beta=0.
$$
Raising $\mu$ and symmetrizing the coefficient of the two velocities produces exactly
$$
\ddot x^\rho+\Gamma^\rho{}_{\alpha\beta}
\dot x^\alpha\dot x^\beta=0.
$$
The quadratic <action> fixes an <affine parameter>; proper time is an affine parameter for a timelike geodesic.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 309 / 1 / a / iii / Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 309 / 1 / a / iii / Solution

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Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 309 / 1 / a / iii / Solution

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