The https://www-thphys.physics.ox.ac.uk/people/AndreiStarinets/TUTORIALS_NOTES_GR/gr-cosmology-exam-paper-2011-solutions.pdf#page=11[official notes, PDF p. 11, handwritten p. 4] give the <Lagrangian> for the <geodesic equations>.

Solution

Past exam of the general relativity and cosmology and condensed matter physics paper of Part B of the physics course of the University of Oxford / 2011 / Section V / 2 / Unlettered 1 / Solution

 Solution (source code)