# Legendre Transformation

(redirected from Fenchel duality)

## 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.

## 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.

References in periodicals archive ?
By the Fenchel duality [43] [Chap 3, Chap 10], it follows that
As mentioned above, the functional equation (33) can be transformed using Fenchel duality into a smooth functional with a box constraint, which can be solved effectively using projected gradient-type methods.

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