# Legendre Transformation

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## Legendre transformation

[lə′zhän·drə ‚tranz·fər′mā·shən] (fluid mechanics)

The basis for a version of the hodograph method for compressible flow in which a replacement is made not only of the independent variables but also of the dependent variables, that is, of the velocity potential and the stream function.

(mathematics)

A mathematical procedure in which one replaces a function of several variables with a new function which depends on partial derivatives of the original function with respect to some of the original independent variables. Also known as Legendre contact transformation.

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 It might be outdated or ideologically biased.## Legendre Transformation

a transformation given by

*X = y′(x), Y(X) = xy′(x) - y(x) Y′(X) = x*

It follows from these formulas that, conversely,

*x = y′(x), y(x) = XY′(X) - Y(X), y′(x) = X*

Thus, the transformation is self-dual. The Legendre transformation converts the first-order differential equation

(1) *F(x, y, y′) =* 0

into the equation

(2) *F(Y′, XY′ - Y, x) =* 0

which can sometimes be integrated more easily than the initial equation. If we know a solution of (2), we can obtain a solution of (1). The Legendre transformation is also used in the study of differential equations of hydrodynamics. It was named after A. Legendre, who first investigated it in 1789.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.