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In mathematics, it is often desirable to express a functional relationship as a different function, whose argument is the derivative of f , rather than x . If we let p = df / dx be the argument of this new function, then this new function is written and is called the
Legendre transform
of the original function, after
Adrien-Marie Legendre.
The Legendre transform of a function is defined as follows:
The notation max x
indicates the maximization of the expression with respect to the variable x while p is held constant. The
Legendre transform is its own inverse. Like the familiar Fourier transform, the Legendre transform takes a function f
( x
) and produces a function of a different variable p
. However, while the
Fourier transform consists of an integration with a kernel, the
Legendre transform uses maximization as the transformation procedure. The transform is well behaved only if function: f
( x
) is a convex
Diagram illustrating the Legendre transformation of the function f
( x
) . The function is shown in red, and the tangent line at x
0
is shown in blue. The tangent line intersects the vertical axis at (0, − f
*
) and f
* is the value of the Legendre transform f
*
( p
) where
. Note that for any other point on the red curve, a line drawn through that point with the same slope as the blue line will have a y
-intercept above the point (0, − f that f
*
*
), showing
is indeed a maximum.
The Legendre transformation is an application of the duality relationship between points and lines. The functional relationship specified by f
( x
) can be represented equally well as a set of ( x
, y
) points, or as a set of tangent lines specified by their slope and intercept values.
The Legendre transformation can be generalized to the Legendre-Fenchel transformation. It is commonly used in thermodynamics and in the hamiltonian formulation of classical mechanics.
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1 Definitions
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1.1 Another definition
2 Applications
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2.1 Thermodynamics
2.2 Hamilton-Lagrange mechanics
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2.3 An example - variable capacitor
3 Examples
4 Legendre transformation in one dimension
5 Geometric interpretation
6 Legendre transformation in more than one dimension
7 Further properties
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7.1 Scaling properties
7.2 Behavior under translation
7.3 Behavior under inversion
7.4 Behavior under linear transformations

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7.5 Infimal convolution
8 See also
9 References
The definition of the Legendre transform can be made more explicit. To maximize f
* derivative equal to zero:
with respect to x
, we set its
Thus, the expression is maximized when
.
This is a maximum because the second derivative is negative: since f
was assumed convex. Next we invert (2) to obtain x
as a function of p
and plug this into (1) , which gives the more useful form,
This definition gives the conventional procedure for calculating the Legendre transform of f
( x
): find , invert for x and substitute into the expression xp − f ( x ). This definition makes clear the following interpretation: the
Legendre transform produces a new function, in which the independent variable is the derivative of the original function with respect to x
. x is replaced by , which
There is a third definition of the Legendre transform: and are said to be Legendre transforms of each other if their first derivatives are inverse functions of each other:
We can see this by taking derivative of :
Combining this equation with the maximization condition results in the following pair of reciprocal equations:

We see that
Df
and are inverses, as promised. They are unique up to an additive constant which is fixed by the additional requirement that
Although in some cases (e.g. thermodynamic potentials) a non-standard requirement is used:
The standard constraint will be considered in this article unless otherwise noted. The Legendre transformation is its own inverse, and is related to integration by parts.
The strategy behind the use of Legendre transforms is to shift the dependence of a function from one independent variable to another (the derivative of the original function with regard to this independent variable) by taking the difference between the original function and their product. They are used to transform among the various thermodynamic potentials. For example, while the internal energy is an explicit function of the extensive variables , entropy, volume (and chemical composition) the enthalpy, the (non standard) Legendre transform of U with respect to − PV becomes a function of the entropy and the intensive quantity
, pressure, as natural variables, and is useful when the
(external) P is constant. The free energies (Helmholtz and Gibbs), are obtained through further Legendre transforms, by subtracting TS (from U and H respectively), shift dependence from the entropy S to its conjugate intensive variable temperature
T
, and are useful when it is constant.
A Legendre transform is used in classical mechanics to derive the Hamiltonian formulation from the Lagrangian one, and conversely. While the Lagrangian is an explicit function of the positional coordinates q j velocities d q j
and generalized
/d t
(and time), the Hamiltonian shifts the functional dependence to the positions and momenta
,defined as

Each of the two formulations has its own applicability, both in the theoretical foundations of the subject, and in practice, depending on the ease of calculation for a particular problem. The coordinates are not necessarily rectilinear, but can also be angles, etc. An optimum choice takes advantage of the actual physical symmetries.
As another example from physics, consider a parallel-plate capacitor whose plates can approach or recede from one another, exchanging work with external mechanical forces which maintain the plate separation — analogous to a gas in a cylinder with a piston. We want the attractive force f
between the plates as a function of the variable separation x
. (The two vectors point in opposite directions.) If the charges
on the plates remain constant as they move, the force is the negative gradient of the electrostatic energy
However, if the voltage
between the plates
V
is maintained constant by connection to a battery, which is a reservoir for charge at constant potential difference, the force now becomes the negative gradient of the Legendre transform
The two functions happen to be negatives only because of the linearity of the capacitance. Of course, for given charge, voltage and distance, the static force must be the same by either calculation since the plates cannot "know" what might be held constant as they move.
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 domains of a function and its Legendre transform need not agree.
Similarly, the quadratic form with
A
a symmetric invertible n
-byn
-matrix has as a Legendre transform.
In one dimension, a Legendre transform to a function f :
R
→
R
with an invertible first derivative may be found using the formula
This can be seen by integrating both sides of the defining condition restricted to one-dimension

from x
0
to x
1
, making use of the fundamental theorem of calculus on the left hand side and substituting on the right hand side to find with f* ′ ( y
0
) = x
0
, f* ′ ( y
1
) = x
1
. Using integration by parts the last integral simplifies to
Therefore,
Since the left hand side of this equation does only depend on x
1
and the right hand side only on x
0
, they have to evaluate to the same constant.
Solving for f* and choosing C to be zero results in the above-mentioned formula.
For a strictly convex function the Legendre-transformation can be interpreted as a mapping between the graph of the function and the family of tangents of the graph. (For a function of one variable, the tangents are well-defined at all but at most countably many points since a convex function is differentiable at all but at most countably many points.)
The equation of a line with slope m and y-intercept b is given by
.
For this line to be tangent to the graph of a function f at the point ( x
0
, f ( x
0
)) requires and 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 :
,

Here f* denotes the Legendre transform of f .
The family of tangents of the graph of f
is therefore (parameterized by m
) given by or, written implicitly, by the solutions of the equation
The graph of the original function can be reconstructed from this family of lines as the envelope of this family by demanding
Eliminating m from these two equations gives
Identifying y with f ( x ) and recognizing the right side of the preceding equation as the Legendre transform of f * we find
For a differentiable real-valued function on an open subset U of
R n
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 where is the scalar product on R n .
Alternatively, if X is a real vector space and Y is its dual vector space, then for each point x of X and y of Y , there is a natural identification of the cotangent spaces T* over X , then ∇ f
X x
with Y and T* Y
is a section of the cotangent bundle T* X y
with X . If f is a real differentiable function
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.
In the following the Legendre transform of a function f is denoted as f *.

The Legendre transformation has the following scaling properties:
It follows that if a function is homogeneous of degree r
then its image under the Legendre transformation is a homogeneous function of degree s , where 1/ r + 1/ s = 1.
Let A be a linear transformation from R n
to R m
. For any convex function f on R n
, one has where A * is the adjoint operator of A defined by
A closed convex function f
is symmetric with respect to a given set
G
of orthogonal linear transformations, if and only if f * is symmetric with respect to G .
The infimal convolution
of two functions f
and g
is defined as
Let f
1
, …, f m
be proper convex functions on R n
. Then

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Projective duality
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Arnol'd, Vladimir Igorevich (1989).
Mathematical Methods of Classical Mechanics (second edition)
.
Springer. ISBN 0-387-96890-3.
Rockafellar, Ralph Tyrell (1996). Convex Analysis . Princeton University Press. ISBN 0-691-01586-4.
Retrieved from "http://en.wikipedia.org/wiki/Legendre_transformation"
Categories: Transforms | Duality theories
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