Contact transformation

A transformation of curves in the plane under which tangent curves are taken into tangent curves. A contact transformation of surfaces in space is defined similarly. A simple example of a contact transformation is the Legendre transformation (cf. Legendre transform).

More generally, a contact transformation is a diffeomorphism $f$ of a contact manifold (that is, a manifold $M^{2n+1}$ endowed with a contact structure with a form $\eta$, such that $f^\star \eta = \sigma \eta$, where $\sigma$ is a non-zero function on $M^{2n+1}$. When $\sigma = 1$, $f$ is called a strict contact transformation. A vector field $X$ on a contact manifold is called a contact (or strict contact) infinitesimal transformation if $L_X \eta = \tilde{\sigma} \eta$ (or $L_X \eta=0$), where $L_X$ is the Lie derivative along $X$. Contact transformations are sometimes called tangency transformations.

References

[1]	P.K. Rashevskii, "Geometric theory of partial differential equations", Moscow-Leningrad (1947) (In Russian)
[2]	L.P. Eisenhart, "Riemannian geometry" , Princeton Univ. Press (1949)
[3]	S.E. Cohn-Vossen, "Some problems of differential geometry in the large", Moscow (1959) (In Russian)
[4]	C. Godbillon, "Géométrie différentielle et mécanique analytique" , Hermann (1969)
[5]	Itogi Nauk. Algebra. Topol. Geom. 1967 (1969) pp. 127–188

Comments

Historically the term contact transformation was first used for "transformations" which map a contact element in $\RR^3$ onto another contact element. A contact element at that time was defined as a point in $\RR^3$ together with a plane passing through that point. The general theory of contact transformations was introduced by S. Lie in his studies of the reduction of Pfaffian forms (cf. Pfaffian equation). In this setting a contact transformation is a mapping

$$ (z, x_1, \ldots, x_n, p_1, \ldots, p_n) \mapsto (z', x_1', \ldots, x_n', p'_1, \ldots, p'_n) $$

such that $dz' - \sum p'_i\, dx'_i$ for some nowhere-vanishing function $f$.

References

[a1]	P. Libermann, C.-M. Marle, "Symplectic geometry and analytical mechanics" , Reidel (1987) (Translated from French)