Unparametrized geodesic equation
= Unparametrized geodesic equation
{title2=$\nabla_vv=fv$}
A regular curve with tangent $v=dx/du$ is an unparametrized <geodesic> of an <affine connection> if $\nabla_vv=f(u)v$. Choose a new parameter $s$ with $ds/du=r(u)>0$ and $r'/r=f$. Then $w=dx/ds=v/r$ obeys $\nabla_ww=r^{-2}(\nabla_vv-(r'/r)v)=0$. Conversely a reparametrized affine geodesic has acceleration proportional to its tangent. This criterion concerns the geometric curve, not a fixed parametrization.