A transformation of the jet bundle $J^n(E)$ is called contact if it preserves the Cartan distribution.
Example
Given $\varphi:E \to E$ (usually called point transformation), the prolongation $\varphi^{(n)}$ 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 $\varphi^{(n)}$ for certain $\varphi:E \to E$ if and only if it preserves the vertical distribution $V$.
Here the vertical distribution is $V_q:=\mbox{Ker}d\pi_q$ for $q\in J^n (E)$ and being
$$ \pi: J^n(E) \to E. $$See [Doubrov 2016].
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Author of the notes: Antonio J. Pan-Collantes
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