Contact transformations were discovered by Sophus Lie in the 19th CENTURY. Within this context an infinitesimal homogeneous (time independent) contact transformation:

δqi=∂H∂piδt,δpi=−∂H∂qiδtδqi=∂H∂piδt,δpi=−∂H∂qiδt

is a coordinate transformation that leaves the system of EQUATIONS:

Δ=∣∣∣∣dp1,…,dpnp1,…,pndq1,…,dqn∣∣∣∣=0,∑ipidqi=0Δ=|dp1,…,dpnp1,…,pndq1,…,dqn|=0,∑ipidqi=0

invariant [1]. In this context we can interchange contact with canonical ACCORDING to Qmechanic's answer.

In the context of differential geometry, we make a distinction between symplectic transformations on dim(2n)dim(2n) symplectic manifolds and contact transformations on dim(2n+1)dim(2n+1) contact manifolds. This extends the time independent formulation into an extended phase space (time dependent). [2] We must now take care on how we use the phrase contact.

In both symplectic and contact frameworks, we can DEFINE a canonical STRUCTURE,

θ=pdq,Θ=pdq−Hdtθ=pdq,Θ=pdq−Hdt

respectively, that becomes invariant under their respective transformations.