Contact Transformation

contact transformation

[′kän‚takt ‚tranz·fər′mā·shən]
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Contact Transformation


a transformation of plane curves such that two mutually tangent curves are transformed into two other curves that are also tangent to each other. A contact transformation is defined by the formulas

(*) X = f (x, y, yʹ) Y = ɸ(x, y, yʹ)

where x and y are the coordinates of the variable point of a curve and X and Y are the coordinates of the variable points of the curve’s image. In order that the formula (*) define a contact transformation, = dY/dX must be independent of y” = d2y/dx2. Examples of contact transformations are point transformations, which are defined by the formulas

X = f (x, y) Y = ɸ(x, y)

and the Legendre transformations.

Contact transformations are used in the theory of differential equations and in differential geometry. The general theory of contact transformations was developed by M. S. Lie. Contact transformations of surfaces in space are defined in an analogous fashion.


Goursat, E. Kurs matematicheskogo analiza, 3rd ed., vol. 1. Moscow-Leningrad, 1936. (Translated from French.)
Rashevskii, P. K. Geometricheskaia teoriia uravnenii s chastnymi proizvodnymi. Moscow-Leningrad, 1947. [20–1721—2]
The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
we can verify that this equation cannot be linearized by a point transformation [10] or contact transformation [17].
Ibragimov, "Criteria for fourth-order ordinary differential equations to be linearizable by contact transformations," Communications in Nonlinear Science and Numerical Simulation, vol.
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There are many ways to convert to linear form like Lie point transformations, contact transformations, non-local transformations, and so forth, but for convenience we will use the term "linearization" to signify the Lie point transformations only and qualify the term if we mean any other transformation.
The Hamiltonian portion of the text covers Hamilton's canonical equations, contact transformations, and Hamilton-Jacoby theory.