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3.1
Hamilton's equations
The most ubiquitous model in theoretical physics is the harmonic oscillator. The oscil-
lator is the basis of understanding a wide variety of phenomena ranging from water
waves to lattice phonons and all the
-ons
in quantum mechanics. The stature of the
lowly harmonic oscillator derives from the notion that complex phenomena, such as
celestial mechanics, have equations of motion that can be generated by means of the
conserved total energy and a sequence of weak perturbations that modulate the dominant
behavior. When the conditions appropriate for using perturbation theory are realized, the
harmonic oscillator is often a good lowest-order approximation to the dynamics.
Of course, not all linear dynamical webs are Hamiltonian in nature. Some have
dissipation leading to an asymptotic state in which all the energy has been bled out.
Physically the energy extracted from the state of interest has to appear elsewhere, so
dissipation is actually a linear coupling of one web to another, where the second web
is absorbing the energy the first web is losing. In simplest terms the second web must
be much larger than the first in order that it does not change its character by absorbing
the energy. Such a second web is referred to in statistical physics as a heat bath and we
shall spend some time discussing its properties.
Consider the total energy of a dynamical web described by the displacements
q
={
q
1
,
q
2
,...,
q
N
}
and momenta
p
={
p
1
,
p
2
,...,
p
N
}
such that the total energy
E
defines the Hamiltonian
E
In a conservative network the energy is con-
stant so that, indicating the variation in a function
F
by
=
H
(
q
,
p
).
δ
F
, we have for a conservative
network
δ
H
(
q
,
p
)
=
0
.
(3.1)
The variation of the Hamiltonian is zero; however, the Hamiltonian does change due
to variations in the momenta and displacements. What occurs is that the variations in
momenta produce changes in the kinetic energy and variations in displacement pro-
duce changes in the potential energy. The overall constant-energy constraint imposes
the condition that the separate changes in energy cancel one another out, so that energy
is transferred from kinetic to potential and back again without change in the total energy.
The general variation given by (
3.1
) can be replaced by the time derivative, which also
vanishes, giving rise to
∂
N
dH
dt
=
H
dq
k
dt
+
∂
H
dp
k
dt
=
0
,
(3.2)
∂
q
k
∂
p
k
k
=
1
where the Hamiltonian is not an explicit function of time. In the situation where each
of the indexed variables is independent we have each term in the sum being separately
equal to zero, resulting in the relations
dq
k
dt
=
∂
H
p
k
,
(3.3)
∂
dp
k
dt
=−
∂
H
q
k
,
(3.4)
∂
