B.3 Calculus of Variations: But Briefly
617
T and P , which are referred to in thermodynamics as the ‘natural variables’ for
G. We could, of course, arrive at precisely the same new thermodynamic function
via the Legendre transform G ≡ H − T S of the enthalpy H . Moreover, we
could also have transformed both extensive independent thermodynamic variables
simultaneously into intensive independent thermodynamic variables by carrying out
a double Legendre transform, G ≡ U − T S + P V , of the internal energy U ,
corresponding to g = f − ux − vy, so that the resultant total differential becomes
dg = −xdu − ydv.
It is perhaps worthwhile to take a closer look at the Legendre transform to obtain
the thermodynamic enthalpy H from the internal energy U . To do this, we note that
from the combined first and second law expression dU = T dS − P dV , we have the
relations
T ≡
∂U
∂S
V
,
P ≡ −
∂U
∂V
S
,
so that the defining relation for the enthalpy, namely,
H (P , S) = U(V , S) + P V ,
indeed corresponds to our general definition of the Legendre transform
t i = y − y
x
upon identifying H (P , S) with t i , U(V , S) with y, −P with y , and V with x. This
expression also makes it clear that the enthalpy correlates with the internal energy
treated as a function of the pressure P and entropy S, rather than as a function of
the volume V and entropy S. Similar considerations show that the Helmholtz energy
correlates with the internal energy as a function of the temperature T and volume
V , while the Gibbs energy can be viewed as the internal energy treated as a function
of temperature T and pressure P .
B.3 Calculus of Variations: But Briefly
We shall examine the problem of finding a path y = y(x) between two values x 1
and x 2 of x which has the property that the integral of some function f (y, y , x)
is an extremum. We use the notation y ≡
dy
dx
in the following. Further, we shall
restrict ourselves to dealing only with a one-dimensional version of the calculus of
variations.
We begin by desiring to find the particular path y = y(x) that makes the line
integral
I ≡
x 2
x 1
f (y, y
, x) dx
(B.3.1)
between the points (x 1 , y 1 ) and (x 2 , y 2 ) either a maximum or a minimum.
617
T and P , which are referred to in thermodynamics as the ‘natural variables’ for
G. We could, of course, arrive at precisely the same new thermodynamic function
via the Legendre transform G ≡ H − T S of the enthalpy H . Moreover, we
could also have transformed both extensive independent thermodynamic variables
simultaneously into intensive independent thermodynamic variables by carrying out
a double Legendre transform, G ≡ U − T S + P V , of the internal energy U ,
corresponding to g = f − ux − vy, so that the resultant total differential becomes
dg = −xdu − ydv.
It is perhaps worthwhile to take a closer look at the Legendre transform to obtain
the thermodynamic enthalpy H from the internal energy U . To do this, we note that
from the combined first and second law expression dU = T dS − P dV , we have the
relations
T ≡
∂U
∂S
V
,
P ≡ −
∂U
∂V
S
,
so that the defining relation for the enthalpy, namely,
H (P , S) = U(V , S) + P V ,
indeed corresponds to our general definition of the Legendre transform
t i = y − y
x
upon identifying H (P , S) with t i , U(V , S) with y, −P with y , and V with x. This
expression also makes it clear that the enthalpy correlates with the internal energy
treated as a function of the pressure P and entropy S, rather than as a function of
the volume V and entropy S. Similar considerations show that the Helmholtz energy
correlates with the internal energy as a function of the temperature T and volume
V , while the Gibbs energy can be viewed as the internal energy treated as a function
of temperature T and pressure P .
B.3 Calculus of Variations: But Briefly
We shall examine the problem of finding a path y = y(x) between two values x 1
and x 2 of x which has the property that the integral of some function f (y, y , x)
is an extremum. We use the notation y ≡
dy
dx
in the following. Further, we shall
restrict ourselves to dealing only with a one-dimensional version of the calculus of
variations.
We begin by desiring to find the particular path y = y(x) that makes the line
integral
I ≡
x 2
x 1
f (y, y
, x) dx
(B.3.1)
between the points (x 1 , y 1 ) and (x 2 , y 2 ) either a maximum or a minimum.