Second prolongation of the rotation generator for plane graphs
= Second prolongation of the rotation generator for plane graphs
The infinitesimal generator $V=u\partial_x-x\partial_u$ of rotations of the $(x,u)$ plane has second prolongation
$$
\operatorname{pr}^{(2)}V
=u\partial_x-x\partial_u
-(1+u_x^2)\partial_{u_x}
-3u_xu_{xx}\partial_{u_{xx}}.
$$
Consequently it preserves both $u_{xx}=0$ and $u_{xx}=(1+u_x^2)^{3/2}$, the equations for straight lines and consistently oriented unit-circle arcs.