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.
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.