The **Helmholtz theorem of classical mechanics** reads as follows:

Let

H
(
x
,
p
;
V
)
=
K
(
p
)
+
φ
(
x
;
V
)
be the Hamiltonian of a one-dimensional system, where

K
=
p
2
2
m
is the kinetic energy and

φ
(
x
;
V
)
is a "U-shaped" potential energy profile which depends on a parameter
V
. Let
⟨
⋅
⟩
t
denote the time average. Let

E
=
K
+
φ
,
T
=
2
⟨
K
⟩
t
,
P
=
⟨
−
∂
φ
∂
V
⟩
t
,
S
(
E
,
V
)
=
log
∮
2
m
(
E
−
φ
(
x
,
V
)
)
d
x
.
Then

d
S
=
d
E
+
P
d
V
T
.

The thesis of this theorem of classical mechanics reads exactly as the heat theorem of thermodynamics. This fact shows that thermodynamic-like relations exist between certain mechanical quantities. This in turn allows to define the "thermodynamic state" of a one-dimensional mechanical system. In particular the temperature
T
is given by time average of the kinetic energy, and the entropy
S
by the logarithm of the action (i.e.
∮
d
x
2
m
(
E
−
φ
(
x
,
V
)
)
).

The importance of this theorem has been recognized by Ludwig Boltzmann who saw how to apply it to macroscopic systems (i.e. multidimensional systems), in order to provide a mechanical foundation of equilibrium thermodynamics. This research activity was strictly related to his formulation of the ergodic hypothesis. A multidimensional version of the Helmholtz theorem, based on the ergodic theorem of George David Birkhoff is known as generalized Helmholtz theorem.