Contact Transformation

contact transformation

[′kän‚takt ‚tranz·fər′mā·shən]
(mathematics)

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.

REFERENCES

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]
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.
The following are Accenture & Cisco Business Group's first three solutions: Customer Contact Transformation Solutions, which help companies to win and retain customers with an improved customer service experience; Unified Communications and Collaboration Solutions, which include services that are tailored to companies' environments to enable new ways of doing business and help create a competitive advantage; and Infrastructure Transformation Solutions, which combine the capabilities of Accenture's business-process experts and Cisco networking experts to help companies optimise their network and data centre infrastructures, the company said.
"We formed the Accenture & Cisco Business Group to provide a seamless services team that can help clients improve their performance." <p>The initial offerings focus on customer contacts, unified communications, and collaboration and infrastructure transformation.<p>The Oracle-Cisco Customer Contact Transformation Solution promises to bring together customer relationship management, analytics, workflow performance, and hiring software applications with collaboration products.
"When the goal is to simply automate more functions, essentially adding more paths for the customer to take in self-service channels, companies risk further frustrating customers by making the experience more complex and lost in the maze of self-service roadblocks," says Robert Wollan, global managing partner of Accenture's Customer Contact Transformation Practice (CRM).
Function theory on symplectic manifolds is fruitfully interacting with geometry and algebra of groups of symplectic and contact transformations, which form another objective of this proposal.
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.