Legendre Transform

Legendre Transform

Definition 1

(Legendre Transformation).

Let

f

:

ℝ

n

⟶

ℝ

be a

C

1

function

and consider the

transformation

x

=

(

x

1

,

⋯

,

x

n

)

⟶

y

=

(

y

1

,

⋯

,

y

n

)

=

(

∂

1

⁡

f

⁢

(

x

)

,

x

2

,

⋯

,

x

n

)

. Provided it is
possible to invert

1

1

The Inverse Mapping Theorem and its

implications

must be used here; in order to be possible to invert
for

x

, the

Jacobian

must be different from zero. The Jacobian
being

∂

2

⁡

f

⁢

(

x

)

∂

⁡

x

2

in this case indicates
that

∂

2

⁡

f

⁢

(

x

)

∂

⁡

x

2

≠

0

, which means that

f

⁢

(

x

)

must be

strictly concave

or strictly convex; this seems clear
graphically

for

x

,

x

=

φ

⁢

(

y

)

, we define the

Legendre
Transform

of

f

,

ℒ

⁢

f

, as the function

ℒ

⁢

f

=

g

:

ℝ

n

⟶

ℝ

y

⟶

g

⁢

(

y

)

=

φ

⁢

(

y

)

⋅

y

-

f

⁢

(

φ

⁢

(

y

)

)

(here ’

⋅

’ denotes the usual

scalar product

on

ℝ

n

).

ℒ

is called the Legendre Transformation.

Remark 1

.

As

x

=

φ

⁢

(

y

)

, the

defining relation

is often written
as

g

⁢

(

y

)

=

x

⋅

y

-

f

⁢

(

x

)

, without explicitly indicating that

x

must be a function of

y

Remark 2

.

Note that, in inverting for

x

,

x

=

φ

⁢

(

y

)

, we are
making

y

=

(

y

1

,

⋯

,

y

n

)

the

independent

variables

.
This is more an issue related to the Inverse Mapping Theorem, but it
is well worth to state it explicitly.

Remark 3

.

From the

definition

we see that the Legendre
Transformation allows us to pass from a function

f

of

(

x

1

,

⋯

⁢

x

n

)

to a function in which we have substituted the first

coordinate

by the

derivative

of

∂

1

⁡

f

. We will deal here
with the case in which just one coordinate is changed but proceeding
by

induction

it is easy to prove the following facts for any number
of variables.

The rationale behind the Legendre transformation is the
following. Let’s begin by considering the unidimensional case.
Suppose we have the function

x

→

f

⁢

(

x

)

. We could be interested in
expressing the values of

f

as function of the derivative

m

=

f

x

⁢

(

x

)

instead of as function of

x

itself without losing any
information about

f

(some examples of this situation will be given
below). At first glance one could think of just inverting the

relation

m

=

f

x

⁢

(

x

)

for

x

to write

f

⁢

(

x

)

=

f

⁢

(

f

x

-

1

⁢

(

m

)

)

≡

g

⁢

(

m

)

. However, this would result in a loss of information because
there would be

infinite

functions

f

which will give rise to the
same

g

; namely the family of translated functions

f

⁢

(

x

-

a

)

for any

a

∈

ℝ

will result in the same

g

. This can be easily
visualized in the figure.

Figure 1:

Translated
versions of the same function have the same relation

f

⁢

(

f

x

-

1

⁢

(

m

)

)

This is because we can not entirely determine a curve by knowing its
slope at every point. The key point is that we can, nevertheless,
determine a curve by knowing its slope

and

its origin
ordinate at every point.

Take a point P on the curve with abscise

x

-see figure
2-.

Figure 2:

Meaning of
Legendre Transformation

Call the origin ordinate of its

tangent

ψ

and its slope

m

which is given by

m

=

f

⁢

(

x

)

-

ψ

x

-

0

Then

ψ

=

f

-

x

⋅

m

. So, intuitively we see that

the Legendre transform is
nothing but the origin ordinate of the slope of

f

at x

. It is

obvious

-at least graphically- that we can recover

f

knowing

ψ

⁢

(

m

)

. We now prove it rigourously.

Theorem 1

(Invertibility and duality of Legendre Transformation).

The Legendre Transformation is

invertible

and the

Inverse

Legendre
Transformation is the Legendre Transformation itself, that is,

L

⁢

L

-

1

⁢

f

=

f

or

L

=

L

-

1

.

Proof.

Evaluate the function

g

at point

y

=

φ

-

1

⁢

(

x

)

to get

g

⁢

(

φ

-

1

⁢

(

x

)

)

=

x

⋅

φ

-

1

⁢

(

x

)

-

f

⁢

(

x

)

this is

f

⁢

(

x

)

=

x

⋅

φ

-

1

⁢

(

x

)

-

g

⁢

(

φ

-

1

⁢

(

x

)

)

Now, it is easy to show that

x

=

(

x

1

,

⋯

,

x

n

)

=

(

∂

1

⁡

g

⁢

(

y

)

,

y

2

,

⋯

,

y

n

)

. So, according to the definition, this is the Legendre
transform of

g

induced by the transformation

y

⟶

x

,

y

=

φ

-

1

⁢

(

x

)

.

∎

Example 1

.

In thermodynamics, a thermodynamic system is completely described by
knowing its

fundamental

equation

in energetic form

:

𝒰

=

U

⁢

(

S

,

V

)

where

𝒰

is the energy,

S

is entropy and

V

is volume. This relation, although of great theoretical value, has a
major drawback, namely that entropy is not a

measurable

quantity.
However, it happens that

∂

⁡

U

∂

⁡

S

=

T

,
temperature. So, we would like to being able to swap

S

for

T

which is an easily measurable quantity. We just take the Legendre
transform

F

of

U

induced by the transformation

(

S

,

V

)

⟶

(

T

,

V

)

=

(

∂

⁡

U

∂

⁡

S

,

V

)

:

F

=

U

-

T

⁢

S

which is called the

Helmholtz

Potential

and hence
is a function of the independent variables

T

,

V

Analogously, as it
happens that

∂

⁡

U

∂

⁡

V

=

-

P

, pressure, we can
swap

V

and

P

and consider the Legendre Transformation

H

of

U

induced by the transformation

(

S

,

V

)

⟶

(

S

,

P

)

:

H

=

U

+

P

⁢

V

which is
called

Enthaply

and hence is a function of the independent
variables

S

,

P

.

Example 2

.

The Lagrangian

formalism

in Mechanics allows to completely determine
the evolution of a general mechanical system by knowledge of the so
called Lagrangian,

ℒ

which is a function of generalized
coordinates

q

, generalized velocities

2

2

The customary
notation for generalized velocities is

q

˙

; however this
notation is somehow obscure because it is prone to establish a
functional relation between

q

and

q

˙

as variables of L. As
variables of L they are just points in

ℝ

n

v

and time

t

:

ℒ

=

L

(

q

,

v

,

t

). The generalized moments are defined as

p

=

∂

⁡

L

∂

⁡

v

and they play the role of usual
linear momentum. Generalized moments are conserved in time under
certain circumstances, so we would like to swap the role of

v

and

p

. Thus we consider the Legendre transform

H

of

L

induced by
the transformation

(

q

,

v

,

t

)

⟶

(

q

,

p

,

t

)

=

(

q

,

∂

⁡

L

∂

⁡

v

,

t

)

:

H

=

L

-

p

⋅

v

which is called the

Hamiltonian

. As pointed out in Remark 2,

q

,

p

and

t

are
independent variables, as we have inverted the mentioned
transformation

(

q

,

v

,

t

)

⟶

(

q

,

p

,

t

)

Title

Legendre Transform

Canonical

name

LegendreTransform

Date of creation

2013-03-22 17:45:58

Last modified on

2013-03-22 17:45:58

Owner

fernsanz (8869)

Last modified by

fernsanz (8869)

Numerical id

11

Author

fernsanz (8869)

Entry type

Definition

Classification

msc 14R99

Classification

msc 26B10

Related topic

InverseFunctionTheorem

Defines

Legendre Transform

Defines

Legendre transformation

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