From Goldstein, 3rd edition, section 9.6, page 399 after equation 9.101:

[...] The motion of a system in a time interval $dt$ can be described by an infinitesimal contact transformation generated by the Hamiltonian. The system motion in a finite time interval from $t_0$ to $t$ is represented by a succession of infinitesimal contact transformations which is equivalent to a single finite canonical transformation. [...]

How does the contact transformation differ from the canonical transformation?


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 transformation is 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


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