# How do contact transformations differ from canonical transformations?

In the 2nd (but not the 3rd!) edition of Goldstein, *Classical Mechanics,* the word contact transformation appears in its index, and there is a 13 line long footnote on p. 382, which (among other things) states

[...] In much of the physics literature the term

contact transformationis used as fully synonomous to canonical transformation, [...]

Concerning canonical transformation, see also this related Phys.SE post.

Contact transformations were discovered by Sophus Lie in the 19th century. Within this context an infinitesimal homogeneous (time independent) contact transformation: $$ \delta q^i = \frac{\partial H}{\partial p_i}\delta t,\qquad \delta p_i = - \frac{\partial H}{\partial q^i}\delta t $$ is a coordinate transformation that leaves the system of equations: $$ \Delta = \begin{vmatrix} dp_1 ,\dots,dp_n\\ p_1,\dots,p_n\\ dq^1 ,\dots,dq^n \end{vmatrix} =0,\qquad \sum_ip_idq^i =0 $$

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 $dim(2n)$ symplectic manifolds and contact transformations on $dim(2n+1)$ 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, $$ \theta = pdq, \qquad \Theta = pdq-Hdt $$ respectively, that becomes invariant under their respective transformations.

[1] The infinitesimal contact transformations of mechanics. Sophus Lie. 1889. Translated by D. H. Delphenich.

[2] https://arxiv.org/pdf/1604.08266.pdf, Contact Hamiltonian Mechanics, Alessandro Bravettia, Hans Cruzb, Diego Tapias, 2016