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

====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====
<table>
  <TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Libermann,   C.-M. Marle,   "Symplectic geometry and analytical mechanics" , Reidel  (1987)  (Translated from French)</TD></TR>
</table>

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