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

With $u=t-z$, the quadratic <geodesic Lagrangian> is
$$
L=\frac12(-\dot t^2+\dot x^2+\dot y^2+\dot z^2)
+xyA(u)\dot u^2.
$$
Its transverse <Euler-Lagrange equations> are
$$
\boxed{\ddot x=yA(u)\dot u^2,
\qquad \ddot y=xA(u)\dot u^2}.
$$
The longitudinal equations are
$$
\boxed{\ddot t=\ddot z
=2A(u)(\dot x,y+x\dot y)\dot u+xyA'(u)\dot u^2},
$$
after using the difference of those same equations. In particular,
$$
\ddot u=\ddot t-\ddot z=0,
\qquad
\boxed{\dot t-\dot z=\dot u=\text{constant}}.
$$
This conserved quantity also follows from the <Killing vector field> $\partial_t+\partial_z$ and the <geodesic conserved quantity from a Killing vector>.

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

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.

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

For a metric $g_{ij}$, the Lagrangian $L=g_{ij}\dot x^i\dot x^j/2$ has the affinely parametrized geodesic equations as its Euler-Lagrange equations. Its value is half the constant squared norm of the tangent.


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