# Legendre Transformation

Also found in: Wikipedia.

## 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 It might be outdated or ideologically biased.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.

References in periodicals archive ?
Equation (1) is similar to the Legendre transformation between Helmholtz free energy F and internal energy U or between Gibbs free energy G and enthalpy H, and therefore these energy components of quasi-localized energy-momentum complexes [mathematical expression not reproducible] could correspond to thermodynamic potentials.
In conclusion, for the SdS black hole solution, the establishment of Legendre transformation in (27) exhibits that [E.sub.E][|.sub.M] and [absolute value of ([E.sub.M][|.sub.M])] would play the role of thermodynamic potential.
Accordingly the yield surfaces of the dual efforts satisfy the associated plastic flow rule and follow the degenerate special case of Legendre transformation.
The Legendre transformation C : [L.sup.n] [right arrow] [H.sup.n] transforms the fundamental geometrical object fields on [L.sup.n] into the fundamental geometrical object fields on [H.sup.n].
The dual theory via Legendre transformation, leads to the geometrical study of the Hamiltonian mechanical systems [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where T* is energy, K(x,p) is the fundamental function of a given Cartan space and H(x,p) is a regular Hamiltonian on the cotangent bundle T*M.
The Hamiltonian density H is derived from the Lagrangian density by the following well known Legendre transformation
The passage from the Hamiltonian H = H(u, [u.sub.1]) to the Lagrangian L = L(u, [u.sub.2]) ought to be realized through the equation (Legendre transformation)1
The following Hamiltonian can be associated with the particle, through a classical Legendre transformation:
Given a [k.sup.th] order Lagrangian [Mathematical Expression Omitted], the Legendre transformation F is a map from [Mathematical Expression Omitted] to ([q.sup.i], [p.sup.i]) defined by
where the first one is a celebrated Legendre transformation. Replacing the dots by the deformation gradient F of the finite strain theory in expression (1), we have the standard theory of thermoelasticity, which is complemented by Fourier's law of heat conduction.
From the Lagrangian formalism, we can pass to the Hamiltonian one through the Legendre transformation
There are also two new appendices, on exact differential equations and on generation of auxiliary functions as Legendre transformations. There is also a CD containing thermodynamic properties of 50 commonly used compounds.

Site: Follow: Share:
Open / Close