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.

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 ?
Accordingly the yield surfaces of the dual efforts satisfy the associated plastic flow rule and follow the degenerate special case of Legendre transformation.
Indeed, the geometrical theory of the Hamilton spaces can be constructed by using the theory of Lagrange spaces and the duality, via Legendre transformation which is suggested by Theoretical Mechanics and is presented in [section]6.
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
As is well known, an operation of the Legendre transformation (6) on a Lagrangian density that is linear in time derivatives yields an expression that is independent of time derivatives.
Tulczyjiew and P Urbanski: A slow and careful Legendre transformation for singular Lagrangians, arXiv:math-ph/9909029.
In fact, on the basis of this momentum a Hamiltonian can also be associated with the particle through a classical Legendre transformation.
th] order Lagrangian [Mathematical Expression Omitted], the Legendre transformation F is a map from [Mathematical Expression Omitted] to ([q.
Substituting (25) into the Legendre transformation (10) gives
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.