Note on Legendre transforms.
This note is supplementary to my note on symmetric hyperbolic
form and the existence of entropy.
If two convex functions f and g are Legendre transforms of one
another then their derivatives are inverses of one another and
thus define a bijection between a pair of variables x and y (in
d-space):
y(x) = f'(x),
x(y) = g'(y).
The constant of integration determines the Legendre transform
and is specified by an additional requirement, that
f(x) + g(y) = x.y.
We will justify this requirement below.
Consider the problem of finding g given f.
We would first take the derivative of f and invert it to get g'.
We might then look for an antiderivative of g'.
But in fact we can use integration by parts to find
a general antiderivative of g' in terms of f and g'
(and thus make a general choice of constant of integration).
Let id denote the identity function on d-space.
(So id' = identity matrix.)
Let \int denote the (path-independent)
path integral of a conservative (i.e. gradient) field
from an arbitrary base point to a variable point.
Let "o" denote function composition.
Integration by parts gives us:
g = \int g'
= \int id'.g'
= id.g' - \int id.g''
= id.g' - \int (f' o g').g''
= id.g' - \int (f o g')'
= id.g' - (f o g') (by choice of constant of integration).
Differentiating confirms the result.
Written in terms of the dual variables x and y, this becomes
g(y) = y.g'(y) - (f o g'(y)), i.e.,
g(y) + f(x) = y.x.
Thus the Legendre transform states that if x and y are
bijectively corresponding variables and
g_y = x
then the Legendre transform
f(x) = x.y - g(y)
satisfies
f_x = y.
A generalization of the Legendre tranform (which is used to show
the equivalence of symmetric hyperbolic form with the existence
of an entropy) states that if x and y are bijectively
corresponding variables and
G_y = H(x)
(which highly constrains and might or might not be sufficient to
specify the relationship between x and y), then the "generalized
Legendre transform of G with respect to H" (for the simple
Legendre transform H is the identity function) is F(x),
defined by
F(x) := H(x).y - G(y),
which satisfies
F_x = y.H_x.
(We remark that F(x) := G_y.y - G(y).)