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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|Duality]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l0580801.png" /> be a smooth function considered on an open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l0580802.png" /> of a normed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l0580803.png" /> and having the property that the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l0580804.png" /> (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l0580805.png" /> is the [[Fréchet derivative|Fréchet derivative]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l0580806.png" />) maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l0580807.png" /> one-to-one onto a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l0580808.png" />. Then the Legendre transform of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l0580809.png" /> is the function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808010.png" /> defined by the formula
+
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808012.png" /> is a function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808013.png" /> and the determinant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808014.png" /> is non-zero in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808015.png" />, the Legendre transform is given by the formulas
+
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|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|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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1prm)</td></tr></table>
+
$$ \tag{1 }
 +
f ^ { * } ( x  ^ {*} )  = < x  ^ {*} , ( f ^ { \prime } )  ^ {-} 1
 +
( x  ^ {*} ) > - f ( ( f ^ { \prime } )  ^ {-} 1 ( x  ^ {*} ) ) .
 +
$$
 +
 
 +
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
 
here
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808017.png" /></td> </tr></table>
+
$$
 +
\langle  x , y \rangle  = \
 +
\sum _ { i= } 1 ^ { n }
 +
x  ^ {i} y  ^ {i} ,\  f ^ { \prime } ( x)  = \
 +
\left (
 +
 
 +
\frac{\partial  f }{\partial  x  ^ {1} }
 +
\dots
  
The transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808018.png" /> 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).
+
\frac{\partial  f }{\partial  x  ^ {n} }
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808019.png" /> 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:
+
\right ) .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
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).
  
The expression (2) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808021.png" /> replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808022.png" /> was taken (see [[#References|[2]]]) as the basis of the theory of duality of convex functions (see [[Conjugate function|Conjugate function]]).
+
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 [[#References|[2]]]) as the basis of the theory of duality of convex functions (see [[Conjugate function|Conjugate function]]).
  
 
Examples. The Legendre transform of the function
 
Examples. The Legendre transform of the function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808023.png" /></td> </tr></table>
+
$$
 +
f _ {p} ( x)  = \
 +
 
 +
\frac{| x |  ^ {p} }{p}
 +
,\  1 < p < \infty ,
 +
$$
  
 
of one variable is the function
 
of one variable is the function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808024.png" /></td> </tr></table>
+
$$
 +
f _ {p  ^  \prime  } ( y)  = \
  
The Legendre transform of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808025.png" /> in a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808026.png" /> with scalar product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808027.png" /> is the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808028.png" />.
+
\frac{| y | ^ {p  ^  \prime  } }{p  ^  \prime  }
 +
,\ \
  
The Legendre transformation, based on a change of variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808029.png" />, is a special case of a [[Proximity transformation|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808030.png" /> and as the enveloping family of its tangent planes, given by the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808031.png" />, consisting of a linear functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808032.png" /> and an affine tangent function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808033.png" />.
+
\frac{1}{p}
 +
+
 +
\frac{1}{p  ^  \prime  }
 +
= 1 .
 +
$$
  
The Legendre transformation plays an important role in analysis, particularly in convex analysis (see [[#References|[1]]], [[#References|[2]]], [[#References|[4]]]), in the theory of differential equations, in variational calculus (see [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808034.png" /> of a differential equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808035.png" /> reduces it to the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808036.png" /> of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808037.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808039.png" />, which can sometimes be integrated more easily than the original equation. The application of the Legendre transformation to the [[Lagrangian|Lagrangian]] of a problem in classical variational calculus reduces it to the [[Hamilton function|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.
+
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|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 [[#References|[1]]], [[#References|[2]]], [[#References|[4]]]), in the theory of differential equations, in variational calculus (see [[#References|[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|Lagrangian]] of a problem in classical variational calculus reduces it to the [[Hamilton function|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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.M. Fichtenholz,  "Differential und Integralrechnung" , '''1''' , Deutsch. Verlag Wissenschaft.  (1964)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Goursat,  "Cours d'analyse mathématique" , '''1''' , Gauthier-Villars  (1918)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.I. Arnol'd,  "Mathematical methods of classical mechanics" , Springer  (1978)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R.T. Rockafellar,  "Convex analysis" , Princeton Univ. Press  (1970)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  W. Fenchel,  "On conjugate convex functions"  ''Canad. J. Math.'' , '''1'''  (1949)  pp. 73–77</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  C. Carathéodory,  "Variationsrechnung und partielle Differentialgleichungen erster Ordnung" , Teubner  (1956)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.M. Fichtenholz,  "Differential und Integralrechnung" , '''1''' , Deutsch. Verlag Wissenschaft.  (1964)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Goursat,  "Cours d'analyse mathématique" , '''1''' , Gauthier-Villars  (1918)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.I. Arnol'd,  "Mathematical methods of classical mechanics" , Springer  (1978)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R.T. Rockafellar,  "Convex analysis" , Princeton Univ. Press  (1970)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  W. Fenchel,  "On conjugate convex functions"  ''Canad. J. Math.'' , '''1'''  (1949)  pp. 73–77</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  C. Carathéodory,  "Variationsrechnung und partielle Differentialgleichungen erster Ordnung" , Teubner  (1956)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
Line 46: Line 123:
 
An integral transform
 
An integral transform
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808040.png" /></td> </tr></table>
+
$$
 +
f ( n)  = T \{ F ( x) \}  = \int\limits _ { - } 1 ^ { 1 }  P _ {n} ( x) F ( x)  dx
 +
,\  n = 0 , 1 \dots
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808041.png" /> is the Legendre polynomial (cf. [[Legendre polynomials|Legendre polynomials]]) of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808042.png" />. The inversion formula has the form
+
where $  P _ {n} ( x) $
 +
is the Legendre polynomial (cf. [[Legendre polynomials|Legendre polynomials]]) of order $  n $.  
 +
The inversion formula has the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808043.png" /></td> </tr></table>
+
$$
 +
T  ^ {-} 1 \{ f ( n) \}  = F ( x)  = \sum _ { n= } 0 ^  \infty  \left ( n +
 +
\frac{1}{2}
 +
\right ) P _ {n} ( x) f ( n) ,\ \
 +
- 1 < x < 1 ,
 +
$$
  
 
if the series converges. The Legendre transform reduces the differential operation
 
if the series converges. The Legendre transform reduces the differential operation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808044.png" /></td> </tr></table>
+
$$
 +
 
 +
\frac{d}{dx}
 +
( 1 - x  ^ {2} )
 +
\frac{d}{dx}
 +
 
 +
$$
  
 
to an algebraic operation by means of the formula
 
to an algebraic operation by means of the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808045.png" /></td> </tr></table>
+
$$
 +
T \left \{
 +
\frac{d}{dx}
 +
( 1 - x  ^ {2} )
 +
\frac{dF ( x) }{dx}
 +
\right \}  = \
 +
- n ( n+ 1 ) f ( n) ,\  n = 0 , 1 ,\dots .
 +
$$
  
 
For the Legendre transform there is a convolution theorem: If
 
For the Legendre transform there is a convolution theorem: If
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808046.png" /></td> </tr></table>
+
$$
 +
T \{ F _ {i} ( x) \}  = f _ {i} ( n) ,\  i = 1 , 2 ,
 +
$$
  
 
then
 
then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808047.png" /></td> </tr></table>
+
$$
 +
f _ {1} ( n) f _ {2} ( n)  = T \{ h ( x) \} ,
 +
$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808048.png" /></td> </tr></table>
+
$$
 +
h ( x)  =
 +
\frac{1} \pi
 +
{\int\limits \int\limits } _ {E ( x) }
 +
\frac{f _ {1} ( \xi ) f _ {2} (
 +
\xi ) }{\sqrt {1 - x  ^ {2} - \xi  ^ {2} - \eta  ^ {2} + 2 x \xi \eta
 +
} }
 +
  d \xi  d \eta
 +
$$
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808049.png" /> is the interior of the ellipse <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058080/l05808050.png" />. The Legendre transform is a special case of the [[Jacobi transform|Jacobi transform]].
+
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|Jacobi transform]].
  
 
====References====
 
====References====

Revision as of 22:16, 5 June 2020


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

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

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)

Comments

References

[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

How to Cite This Entry:
Legendre transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Legendre_transform&oldid=47610
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article