Contact transformation

A transformation of the jet bundle $J^n(E)$ is called contact if it preserves the Cartan distribution.

Example
Given $\phi :E\to E$ (usually called point transformation), the prolongation ${\phi }^{(n)}$ is a contact transformation. The "converse" of this example is studied by the Bäcklund theorem (see the end of Cartan distribution#Symmetries of the Cartan distribution).

Lemma
An arbitrary contact transformation has the form ${\phi }^{(n)}$ for certain $\phi :E\to E$ if and only if it preserves the vertical distribution $V$.
Here the vertical distribution is $V_q:=\text{Ker}d{\pi }_q$ for $q\in J^n(E)$ and being

See [Doubrov 2016].