= Length extremals from a constant quadratic energy
{title2=$L=\dot q^Tg(q)\dot q,\quad\ell=\sqrt L$}
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 equations>[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 $\sqrt{E+2Fv_u+Gv_u^2}\,|du|$. 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.
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