Rotationally symmetric surface metric

An intrinsic two-dimensional metric locally written $d{s}^2=d{r}^2+f(r{)}^{2}d{\varphi }^2$ with positive $f$. Rotation of the angular coordinate is an isometry. Such a metric need not have a global realization as a surface of revolution in three-dimensional Euclidean space.

For the affinely parametrized geodesic Lagrangian $\mathcal{L}=({\dot{r}}^2+f(r{)}^2{\dot{\varphi }}^2)/2$, the conserved cyclic momentum is $J=f(r{)}^2\dot{\varphi }$. Its Euler-Lagrange equations are $\ddot{r}-f{f}^{\prime }{\dot{\varphi }}^2=0$ and $\ddot{\varphi }+2f^{\prime }\dot{r}\dot{\varphi }/f=0$.