Legendre Transformation
Shina Tan
Legendre transformation is useful in both mechanics and thermodynamics.
Given any smooth function
f
(
x
) deﬁned in the domain
x
1
< x < x
2
, if
f
00
(
x
)
>
0 for all
x
in this domain, so that
f
(
x
) is
strictly convex
1
, we can deﬁne
g
(
y
) =
x
df
dx

f,
(1)
where
y
=
dx
.
(2)
The function
g
(
y
) is called the
Legendre transform
of
f
(
x
).
For example, the Legendre transform of
L
(
v
) =
1
2
mv
2
(Lagrangian of a free particle in
one dimension) is
H
(
p
) =
p
2
2
m
(Hamiltonian of a free particle in one dimension).
Because
f
00
(
x
) =
y
0
(
x
)
>
0,
y
is a monotonically increasing function of
x
, so the function
y
(
x
) can be inverted to yield
x
=
x
(
y
). The existence of the inverse of
y
(
x
) is necessary,
since we need to ﬁrst compute
xf
0
(
x
)

f
(
x
), and then express
x
in terms of
y
.
The function
g
(
y
) is deﬁned in the domain
y
1
< y < y
2
, where
y
1
= lim
x
→
x
1
dx
,
y
2
= lim
x
→
x
2
dx
.
(3)
Theorem 1:
g
(
y
)
is also a convex function and, moreover,
g
00
(
y
) = 1
/f
00
(
x
)
.
Proof:
dg
dx
=
d
dx
[
xf
0
(
x
)

f
(
x
)] =
x
0
f
0
(
x
)+
xf
00
(
x
)

f
0
(
x
) =
f
0
(
x
)+
xf
00
(
x
)

f
0
(
x
) =
xf
00
(
x
).
So
dg
dy
=
dg/dx
dy/dx
=
xf
00
(
x
)
0
(
x
)
/dx
=
x
. Thus
dg
dy
=
x,
(4)
and
g
00
(
y
) =
d
dy
dg
dy
=
d
dy
x
=
dx
dy
.
(5)
On the other hand,
f
00
(
x
) =
d
dx
dx
=
d
dx
y
=
dy
dx
.
(6)
Thus
g
00
(
y
) is the
reciprocal
of
f
00
(
x
), and is positive. So
g
(
y
) is also a convex function.
Theorem 2: Any smooth and strictly convex function’s Legendre transform’s
Legendre transform is that function itself.
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 Spring '11
 Tan
 Thermodynamics, mechanics

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