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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025480/c0254801.png" /> of a contact manifold (that is, a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025480/c0254802.png" /> endowed with a [[Contact structure|contact structure]] with a form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025480/c0254803.png" />), such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025480/c0254804.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025480/c0254805.png" /> is a non-zero function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025480/c0254806.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025480/c0254807.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025480/c0254808.png" /> is called a strict contact transformation. A vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025480/c0254809.png" /> on a contact manifold is called a contact (or strict contact) infinitesimal transformation if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025480/c02548010.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025480/c02548011.png" />), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025480/c02548012.png" /> is the [[Lie derivative|Lie derivative]] along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025480/c02548013.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025480/c02548014.png" /> onto another contact element. A contact element at that time was defined as a point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025480/c02548015.png" /> 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|Pfaffian equation]]). In this setting a contact transformation is a mapping

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025480/c02548016.png" /></td> </tr></table>

such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025480/c02548017.png" /> for some nowhere-vanishing function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025480/c02548018.png" />.

====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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