Legendre transformation

In mathematics, two differentiable functions ''f'' and ''g'' are said to be Legendre transforms of each other if their first derivatives are inverse functions of each other: :

Df = \left( Dg \right)^{-1}

''f'' and ''g'' are then said to be related by a Legendre transformation. Legendre transformations are named after Adrien-Marie Legendre. They are unique up to an additive constant which is usually fixed by the additional requirement that :

f(x) + g(y) = \left\langle x,y\right\rangle

holds. A Legendre transformation is its own inverse, and is related to integration by parts. ==Applications== Legendre transformations are used in thermodynamics to transform between the different thermodynamic potentials, and in classical mechanics to derive Hamiltonian mechanics from Lagrangian mechanics, as well as the other way around. ==Examples== The exponential function e^''x'' has ''x'' ln ''x'' − ''x'' as a Legendre transform since the respective first derivatives e^''x'' and ln ''x'' are inverse to each other. This example shows that the respective domain (mathematics)s of a function and its Legendre transform need not agree. Similarly, the quadratic form :

u(x) = \frac{1}{2} \, x^t \, A \, x

with ''A'' an invertible matrix ''n''-by-''n''-Matrix (mathematics) has :

v(y) = \frac{1}{2} \, y^t \, A^{-1} \, y

as a Legendre transform. ==Legendre transformation in one dimension== In one dimension, a Legendre transform to a function ''f'' : R → R with an invertible first derivative may be found using the formula :

g(y) = y \, x - f(x), \, x = f^{\prime-1}(y)

This can be seen by integrating both sides of the defining condition restricted to one-dimension :

f^\prime(x) = g^{\prime-1}(x)

from x₀ to x₁, making use of the Fundamental theorem of calculus on the left hand side and Substitution rule :

y = g^{\prime-1}(x)

on the right hand side to find :

f(x_1) - f(x_0) = \int_{y_0}^{y_1} y \, g^{\prime\prime}(y) \, dy

with ''g''′(''y''₀) = ''x''₀, ''g''′(''y''₁) = ''x''₁. Using integration by parts the last integral simplifies to :

y_1 \, g^\prime(y_1) - y_0 \, g^\prime(y_0) - \int_{y_0}^{y_1} g^\prime(y) \, dy
= y_1 \, x_1 - y_0 \, x_0 - g(y_1) + g(y_0)

Therefore, :

f(x_1) + g(y_1) - y_1 \, x_1 = f(x_0) + g(y_0) - y_0 \, x_0

Since the left hand side of this equation does only depend on x₁ and the right hand side only on x₀, they have to evaluate to the same constant. :

f(x) + g(y) - y \, x = C,\, x = g^\prime(y) = f^{\prime-1}(y)

Solving for ''g'' and choosing ''C'' to be zero results in the above-mentioned formula. ==Geometric interpretation== For a strictly convex function the Legendre-transformation can be interpreted as the mapping between the graph of a function of the function and the family of tangents of the graph. (The tangents are well-defined at all but at most countable points since a convex function is derivative at all but at most countably many points.) The equation of a line with slope ''m'' and y-intercept ''b'' is given by :

y = mx + b

For this line to be tangent to the graph of a function ''f'' at the point (''x''₀, ''f''(''x''₀)) requires :

f\left(x_0\right) = m x_0 + b

and :

m = f^{\prime}\left(x_0\right)

''f''′ is strictly monotone as the derivative of a strictly convex function, and the second equation can be solved for ''x''₀, allowing to eliminate ''x''₀ from the first giving the ''y''-intercept ''b'' of the tangent as a function of its slope ''m'': :

b = f\left(f^{\prime-1}\left(m\right)\right) - m \cdot f^{\prime-1}\left(m\right)

==Legendre transformation in more than one dimension== For a differentiable real-valued function on an open subset ''U'' of Rⁿ the Legendre conjugate of the pair (''U'', ''f'') is defined to be the pair (''V'', ''g''), where ''V'' is the image of ''U'' under the gradient mapping D''f'', and ''g'' is the function on ''V'' given by the formula :

g(y) = \left\langle y, x \right\rangle - f\left(x\right), \,
x = \left(Df\right)^{-1}(y)

where :

\left\langle u,v\right\rangle = \sum_{k=1}^{n}u_{k} \cdot v_{k}

is the scalar product on Rⁿ. Alternatively, if ''X'' is a real vector space and ''Y'' is its dual space, then for each point ''x'' of ''X'' and ''y'' of ''Y'', there is a natural identification of the cotangent spaces T^*''X''_''x'' with ''Y'' and T^*''Y''_''y'' with ''X''. If ''f'' is a real differentiable function over ''X'', then ∇''f'' is a section of the cotangent bundle T^*''X'' and as such, we can construct a map from ''X'' to ''Y''. Similarly, if ''g'' is a real differentiable function over ''Y'', ∇''g'' defines a map from ''Y'' to ''X''. If both maps happen to be inverses of each other, we say we have a Legendre transform. ==Convex conjugates== ===Definition=== For a function :

f:\mathbb{R}^n\rightarrow\mathbb{R}\cup\{+\infty\}

taking values on the extended real number line the Legendre transformation can be generalized to the Legendre-Fenchel transformation or convex conjugate of ''f'' by :

f^\star:\mathbb{R}^n\rightarrow\mathbb{R}\cup\{+\infty\}

f^{\star}\left(p\right) =
 \sup\left\{\left\langle x,p\right\rangle - f\left(x\right) :
   x \in \mathbb{R}^n \right\} =
 - \inf\left\{f\left(x\right) - \left\langle x,p\right\rangle :
   x \in \mathbb{R}^n \right\}

===Examples of convex conjugates=== The convex conjugate of an affine function :

f(x) = \left\langle a,x \right\rangle - b,\,
a \in \mathbb{R}^n, b \in \mathbb{R}

is :

f^\star\left(x^\star\right)
= \left \{ \begin{matrix} b,      & x^\star = a
                       \\ \infty, & x^\star \ne a
           \end{matrix}
  \right.

The convex conjugate of the absolute value function :

f(x) = \left| x \right|

is :

f^\star\left(x^\star\right)
= \left \{ \begin{matrix} 0,      & \left|x\right| \le 1
                       \\ \infty, & \left|x\right| > 1
           \end{matrix}
  \right.

The convex conjugate of the exponential function is :

\exp^\star\left(x^\star\right)
= \left \{ \begin{matrix} x \ln x - x, & x > 0
                       \\ 0          , & x = 0
                       \\ \infty     , & x < 0
           \end{matrix}
  \right.

===Properties of convex conjugation=== Convex-conjugation is Order theory: if ''f'' ≤ ''g'' then ''f''^* ≥ ''g''^*. The convex conjugate of a closed convex function is again a closed convex function. The convex conjugate of a polyhedral convex function (a convex function with polyhedral epigraph) is again a polyhedral convex function. For any proper convex function ''f'' and its convex conjugate ''f''^* Fenchel's inequality (also known as the Fenchel-Young inequality) holds: :

\left\langle p,x \right\rangle \le f(x) + f^\star(p)

The convex conjugate of a function is always lower semi-continuous. The biconjugate ''f''^** (the convex conjugate of the convex conjugate) is also the closed convex hull, i.e. the largest lower semi-continuous convex function smaller than ''f''. Therefore, ''f'' = ''f''^** iff ''f'' is convex and lower semi-continuous. ==Further properties== ===Scaling properties=== The Legendre transformation has the following scaling properties: :

f(x) = a \cdot g(x)
\Rightarrow
f^\star(p) = a \cdot g^\star\left(\frac{p}{a}\right)

f(x) = g(a \cdot x)
\Rightarrow
f^\star(p) = g^\star\left(\frac{p}{a}\right)

It follows that if a function is homogeneous function then its image under the Legendre transformation is a homogeneous function of degree ''s'', where 1/''r'' + 1/''s'' = 1. ===Behavior under translation=== :

f(x) = g(x) + b
\Rightarrow
f^\star(p) = g^\star(p) - b

f(x) = g(x + y)
\Rightarrow
f^\star(p) = g^\star(p) - p \cdot y

===Behavior under inversion=== :

f(x) = g^{-1}(x)
\Rightarrow
f^\star(p) = - p \cdot g^\star\left(\frac{1}{p}\right)

===Behavior under linear transformations=== Let ''A'' be a linear transformation from Rⁿ to R^m. For any convex function ''f'' on Rⁿ, one has :

\left(A f\right)^\star = f^\star A^\star

where ''A''^* is the adjoint operator of ''A'' defined by :

\left \langle Ax, y^\star \right \rangle = \left \langle x, A^\star y^\star \right \rangle

A closed convex function ''f'' is symmetric with respect to a given set ''G'' of orthogonal matrixs, :

f\left(A x\right) = f(x), \; \forall x, \; \forall A \in G

if and only if ''f''^* is symmetric with respect to ''G''. ===Infimal convolution=== The infimal convolution of two functions ''f'' and ''g'' is defined as :

\left(f \star_\inf  g\right)(x) = \inf \left \{ f(x-y) + g(y) \, | \, y \in \mathbb{R}^n \right \}

Let ''f''₁, …, ''f''_m be proper convex functions on Rⁿ. Then :

\left( f_1 \star_\inf \cdots \star_\inf f_m \right)^\star = f_1^\star + \cdots + f_m^\star

== References == * Rockafellar, Ralph Tyrell, ''Convex Analysis'', Princeton University Press (1996). ISBN 0691015864 * Vladimir Igorevich Arnol'd, ''Mathematical Methods of Classical Mechanics'', second edition, Springer (1989). ISBN 0387968903 transforms

Legendre transformation

The abrupt way this article begins with no context-setting and no clear definition should grate upon the sensibilities of thoughtful people. Those who know this topic, please de-stubbify. User:Michael Hardy 22:13, 4 Feb 2004 (UTC) == Huh? == This is absolutely useless as explanation of Legendre transformation. It's not even a definition, let alone explanation. :I'm inclined to agree. I don't know the subject, but I know enough to know that whoever wrote this did not write clearly. Physicists often seem opposed to writing clearly about mathematics; they prefer a touchy-feely style. Could the author of the above identify "themself"? Thanks. User:Michael Hardy 22:14, 10 Sep 2004 (UTC) ::Mathematicians often write like Bourbaki and write incomprehensible abstract stuff at a far higher level of abstraction and generality than needed for practical applications and are ever ready to criticize physicists for being "nonrigorous" and "intuitive". ---- Should the section on convex conjugates be moved to a separate page? -- User:Tobias Bergemann 10:54, 2004 Nov 4 (UTC)

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