 Length extremals from a constant quadratic energy

For an autonomous smooth Lagrangian homogeneous of degree two in velocities, the Beltrami identity gives $\dot{q}\cdot L_{\dot{q}}-L=L$ constant on its Euler-Lagrange extremals. On a regular segment with this constant positive, the Euler-Lagrange expression for $\sqrt{L}$ equals that for $L$ divided by $2\sqrt{L}$, since the extra term is proportional to $\dot{L}$. Hence an energy extremal has the same path as a length extremal. Length is invariant under regular reparametrization: in two coordinates, on a graph $v=v(u)$, its density is $\sqrt{E+2F{v}_u+G{v}_u^2}|du|$. If a nonnegative semidefinite energy vanishes along a curve, the curve instead minimizes length at zero; division by $\sqrt{L}$ is not permitted there. For a positive definite metric, regular nonconstant curves have positive energy.