Legendre transform

A transformation in mathematical analysis that establishes a duality between objects in dual spaces (in parallel with projective duality in analytic geometry and polar duality in convex geometry, cf. Duality). Let $ f : A \rightarrow \mathbf R $ be a smooth function considered on an open set $ A $ of a normed space $ X $ and having the property that the mapping $ x \rightarrow f ^ { \prime } ( x) $( here $ f ^ { \prime } ( x) $ is the Fréchet derivative of $ f $) maps $ A $ one-to-one onto a set $ B \subset X ^ {*} $. Then the Legendre transform of $ f $ is the function on $ B $ defined by the formula

$$ \tag{1 } f ^ { * } ( x ^ {*} ) = \langle x ^ {*} , ( f ^ { \prime } ) ^ {-1} ( x ^ {*} ) \rangle - f ( ( f ^ { \prime } ) ^ {-1} ( x ^ {*} ) ) . $$

If $ f $ is a function on $ \mathbf R ^ {n} $ and the determinant $ \mathop{\rm det} ( {\partial ^ {2} f } / {\partial x ^ {i} \partial x ^ {j} } ) $ is non-zero in $ A $, the Legendre transform is given by the formulas

$$ \tag{1'} f ^ { \prime } ( x) = y ,\ \ f ^ { * } ( y) = \langle x , f ^ { \prime } ( x) \rangle - f ( x) ; $$

here

$$ \langle x , y \rangle = \ \sum _ {i=1} ^ { n } x ^ {i} y ^ {i} ,\ f ^ { \prime } ( x) = \ \left ( \frac{\partial f }{\partial x ^ {1} } \dots \frac{\partial f }{\partial x ^ {n} } \right ) . $$

The transformation $ x \rightarrow f ^ { * } ( f ^ { \prime } ( x) ) $ goes back to G. Leibniz; in its general form it was defined by A.M. Legendre (1789), but it was considered earlier by L. Euler (1776).

If $ f $ is a finite-dimensional function that is smooth, strictly convex, and increases at infinity faster than a linear function, the Legendre transform can be defined thus:

$$ \tag{2 } f ^ { * } ( x ^ {*} ) = \ \max _ {x \in \mathbf R ^ {n} } \ ( \langle x ^ {*} , x \rangle - f ( x) ) . $$

The expression (2) with $ \max $ replaced by $ \sup $ was taken (see [2]) as the basis of the theory of duality of convex functions (see Conjugate function).

Examples. The Legendre transform of the function

$$ f _ {p} ( x) = \ \frac{| x | ^ {p} }{p} ,\ 1 < p < \infty , $$

of one variable is the function

$$ f _ {p ^ \prime } ( y) = \ \frac{| y | ^ {p ^ \prime } }{p ^ \prime } ,\ \ \frac{1}{p} + \frac{1}{p ^ \prime } = 1 . $$

The Legendre transform of the function $ ( x , x ) /2 $ in a Hilbert space $ X $ with scalar product $ ( \cdot , \cdot ) $ is the function $ ( y , y ) /2 $.

The Legendre transformation, based on a change of variables $ x \rightarrow y = f ^ { \prime } ( x) $, is a special case of a proximity transformation; the essence of the Legendre transformation lies in the possibility of a dual description of a surface in space — as a set of points $ ( x , f ( x) ) $ and as the enveloping family of its tangent planes, given by the pair $ ( x ^ {*} , \langle x ^ {*} , \cdot \rangle - f ^ { * } ( x ^ {*} ) ) $, consisting of a linear functional $ x ^ {*} $ and an affine tangent function $ x \rightarrow \langle x ^ {*} , x \rangle - f ^ { * } ( x ^ {*} ) $.

The Legendre transformation plays an important role in analysis, particularly in convex analysis (see [1], [2], [4]), in the theory of differential equations, in variational calculus (see [6]), and in classical mechanics, thermodynamics, the theory of elasticity and other branches of mathematical physics. Thus, the application of the Legendre transformation to the solution $ y $ of a differential equation $ F ( x , y , y ^ \prime ) = 0 $ reduces it to the solution $ Y $ of the equation $ F ( Y ^ \prime , XY ^ \prime - Y , X ) = 0 $, where $ X = y ^ \prime ( x) $, $ Y ( X) = y ^ {*} ( X) $, which can sometimes be integrated more easily than the original equation. The application of the Legendre transformation to the Lagrangian of a problem in classical variational calculus reduces it to the Hamilton function. Here, the system of Euler equations (in variational calculus) and the Lagrange equation (in classical mechanics) go over to an equivalent system of canonical equations. In thermodynamics the Legendre transformation brings about a transition from some state functions to others, for example from the specific volume and entropy to the temperature and pressure.

References

[1]	G.M. Fichtenholz, "Differential und Integralrechnung" , 1 , Deutsch. Verlag Wissenschaft. (1964)
[2]	E. Goursat, "Cours d'analyse mathématique" , 1 , Gauthier-Villars (1918)
[3]	V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) (Translated from Russian)
[4]	R.T. Rockafellar, "Convex analysis" , Princeton Univ. Press (1970)
[5]	W. Fenchel, "On conjugate convex functions" Canad. J. Math. , 1 (1949) pp. 73–77
[6]	C. Carathéodory, "Variationsrechnung und partielle Differentialgleichungen erster Ordnung" , Teubner (1956)
[a1]	R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1962) (Translated from German)

An integral transform

$$ f ( n) = T \{ F ( x) \} = \int\limits _ {-1}^ { 1 } P _ {n} ( x) F ( x) dx ,\ n = 0 , 1 \dots $$

where $ P _ {n} ( x) $ is the Legendre polynomial (cf. Legendre polynomials) of order $ n $. The inversion formula has the form

$$ T ^ {-1} \{ f ( n) \} = F ( x) = \sum _{n=0}^ \infty \left ( n + \frac{1}{2} \right ) P _ {n} ( x) f ( n) ,\ \ - 1 < x < 1 , $$

if the series converges. The Legendre transform reduces the differential operation

$$ \frac{d}{dx} ( 1 - x ^ {2} ) \frac{d}{dx} $$

to an algebraic operation by means of the formula

$$ T \left \{ \frac{d}{dx} ( 1 - x ^ {2} ) \frac{dF ( x) }{dx} \right \} = \ - n ( n+ 1 ) f ( n) ,\ n = 0 , 1 ,\dots . $$

For the Legendre transform there is a convolution theorem: If

$$ T \{ F _ {i} ( x) \} = f _ {i} ( n) ,\ i = 1 , 2 , $$

then

$$ f _ {1} ( n) f _ {2} ( n) = T \{ h ( x) \} , $$

where

$$ h ( x) = \frac{1} \pi {\int\limits \int\limits } _ {E ( x) } \frac{f _ {1} ( \xi ) f _ {2} ( \xi ) }{\sqrt {1 - x ^ {2} - \xi ^ {2} - \eta ^ {2} + 2 x \xi \eta } } d \xi d \eta $$

and $ E ( x) $ is the interior of the ellipse $ \xi ^ {2} + \eta ^ {2} - 2x \xi \eta = 1 - x ^ {2} $. The Legendre transform is a special case of the Jacobi transform.

References

[1]	C.J. Tranter, "Legendre transforms" Quart. J. Math. , 1 (1950) pp. 1–8
[2]	Yu.A. Brychkov, A.P. Prudnikov, "Operational calculus" Progress in Math. , 1 (1968) pp. 1–74 Itogi Nauk. Mat. Anal. 1966 (1967) pp. 7–82

Yu.A. BrychkovA.P. Prudnikov