= Solution
In local coordinates $u=(u^1,u^2)$ with metric $g_{ij}=\partial_i\sigma\cdot\partial_j\sigma$, the energy is
$$
E(\gamma)=\frac12\int_a^b g_{ij}(u)\dot u^i\dot u^j\,dt.
$$
The Euler–Lagrange equations are equivalently
$$
\ddot u^k+\Gamma^k_{ij}\dot u^i\dot u^j=0,
$$
where $\Gamma^k_{ij}=\tfrac12g^{k\ell}(\partial_i g_{j\ell}+\partial_jg_{i\ell}-\partial_\ell g_{ij})$.
For the plane-section curve $\eta$, constant speed gives $\eta''\perp\eta'$. Both $\eta''$ and the surface normal lie in $P$, while $P$ is spanned by $\eta'$ and that normal along the intersection. Hence $\eta''$ is normal to the surface. Its tangential <acceleration> vanishes, which is the <geodesic equation>.
Back to article page
