Lie point symmetry of an ordinary differential equation

A Lie point symmetry is a local one-parameter transformation of the independent and dependent variables that maps solution graphs of a differential equation to solution graphs.

For $V=\xi (x,u){\partial }_x+\eta (x,u){\partial }_u$, its prolongation to derivatives through order $n$ isIt generates a symmetry of $\Delta =0$ exactly when ${\mathrm{pr}}^{(n)}V(\Delta )$ vanishes on the equation manifold.

For $V=\xi (x,u){\partial }_x+\eta (x,u){\partial }_u$, the first prolongation is

The $n$th jet space for a scalar function $u(x)$ has local coordinatesIt records the value of a function and its derivatives through order $n$ at one point, allowing an order-$n$ ordinary differential equation to be treated as a hypersurface and a point transformation to act through its prolongation of a vector field.

An ordinary differential equation $\Delta (x,u,u^{\prime },\dots ,u^{(n)})=0$ defines a solution manifold in jet space. The prolonged graph of every classical solution lies in this manifold.

For $u^{\prime }=F(x,u)$, a point-symmetry generator $V=\xi {\partial }_x+\eta {\partial }_u$ must satisfyIt follows by applying the first prolongation to $u^{\prime }-F=0$ and restricting to its solution manifold.

The infinitesimal generator $V=u{\partial }_x-x{\partial }_u$ of rotations of the $(x,u)$ plane has second prolongationConsequently 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.

On jet coordinates,It differentiates a differential function along prolonged solution graphs.

For a thrice differentiable function with $u^{\prime }\ne 0$, the Schwarzian derivative isIt vanishes exactly for fractional linear functions $u(x)=(ax+b)/(cx+d)$ on intervals where the denominator does not vanish. In a two-dimensional conformal field theory, it is the inhomogeneous term in the finite conformal transformation of a stress-energy tensor with nonzero central charge.

For $u_j=u^{(j)}$, the $n$th prolongations of the translation, dilation, and special projective generators acting on the independent variable are

For $V_1={\partial }_x$, $V_2=x{\partial }_x$, and $V_3=x^2{\partial }_x$, direct application of their third prolongations givesThus the Schwarzian is invariant under translation and has weight minus two under dilation; the third identity is its infinitesimal projective transformation law.

The vector fields $(cx+d){\partial }_x-2cu{\partial }_u$ generateTheir second prolongations multiply the differential equation by $-4c$, so they are infinitesimal Lie symmetries.