Legendre Transform
Definition 1 (Legendre Transformation).
Let be a function and consider the
transformation . Provided it is
possible to invert 11The Inverse Mapping Theorem and its
implications must be used here; in order to be possible to invert
for , the Jacobian
must be different from zero. The Jacobian
being in this case indicates
that , which means that
must be strictly concave or strictly convex; this seems clear
graphically for , , we define the Legendre
Transform of , , as the function
(here ’’ denotes the usual scalar product on ).
is called the Legendre Transformation.
Remark 1.
As , the defining relation is often written as , without explicitly indicating that must be a function of
Remark 2.
Note that, in inverting for , , we are making 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 of to a function in which we have substituted the first
coordinate by the derivative
of . 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 . We could be interested in
expressing the values of as function of the derivative
instead of as function of itself without losing any
information about (some examples of this situation will be given
below). At first glance one could think of just inverting the
relation for to write . However, this would result in a loss of information because
there would be infinite
functions which will give rise to the
same ; namely the family of translated functions for any
will result in the same . This can be easily
visualized in the figure.
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 -see figure 2-.
Call the origin ordinate of its tangent and its slope
which is given by
Then . So, intuitively we see that the Legendre transform is nothing but the origin ordinate of the slope of at x. It is obvious -at least graphically- that we can recover knowing . 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,
or .
Proof.
Evaluate the function at point to get
this is
Now, it is easy to show that . So, according to the definition, this is the Legendre transform of induced by the transformation , .
∎
Example 1.
In thermodynamics, a thermodynamic system is completely described by knowing its fundamental equation in energetic form: where is the energy, is entropy and 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 , temperature. So, we would like to being able to swap for which is an easily measurable quantity. We just take the Legendre transform of induced by the transformation :
which is called the Helmholtz Potential and hence is a function of the independent variables Analogously, as it happens that , pressure, we can swap and and consider the Legendre Transformation of induced by the transformation :
which is called Enthaply and hence is a function of the independent variables .
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, which is a function of generalized coordinates , generalized velocities 22The customary notation for generalized velocities is ; however this notation is somehow obscure because it is prone to establish a functional relation between and as variables of L. As variables of L they are just points in and time : ). The generalized moments are defined as 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 and . Thus we consider the Legendre transform of induced by the transformation :
which is called the Hamiltonian. As pointed out in Remark 2, and are independent variables, as we have inverted the mentioned transformation
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 |