Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 4 / 24H (Differential Geometry)

A geodesic with affine parameter is a curve whose velocity is parallel along itself, ${\nabla }_{\dot{\gamma }}\dot{\gamma }=0$. Its energy on a fixed parameter interval is $E(\gamma )=\frac{1}{2}\int ⟨\dot{\gamma },\dot{\gamma }⟩dt$. For a smooth fixed-endpoint variation with variation field $V$, metric compatibility and the torsion-free connection giveThe endpoint term vanishes. Thus a geodesic is a critical point; conversely, vanishing for every interior variation field implies ${\nabla }_t\dot{\gamma }=0$. Affinely parametrized geodesics are precisely the fixed-endpoint energy critical points.

Write $s=U(u)+V(v)>0$. The geodesic energy density $E=\frac{1}{2}s({\dot{u}}^2+{\dot{v}}^2)$ is constant, because an affine geodesic has constant speed. Its Euler-Lagrange equations give, with $p=s\dot{u}$,The requested expression isDifferentiate, using $\dot{u}=p/s$ and $\dot{E}=0$:Therefore $K$ is independent of the affine parameter, the Liouville metric geodesic integral. The assumption of affine parametrization is the standard geodesic convention here.