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For simplicity, assume that this material is a physical spring made up of
N
particles and
the potential function is represented by the Taylor expansion
0
0
q
j
−
q
j
q
j
−
q
j
q
k
−
q
k
N
N
2
V
∂
V
1
2
∂
V
(
q
)
=
V
0
+
+
+···
.
∂
q
j
∂
q
j
∂
q
k
j
=
1
j
,
k
=
1
(3.16)
If the particles are near equilibrium, the net force acting on each one vanishes,
resulting in
0
=
∂
V
,
=
,
,...,
.
0
j
1
2
N
(3.17)
∂
q
j
The first term in (
3.16
) is a constant and defines the reference potential, which for
convenience is chosen to be zero. The leading-order term in the Taylor expansion of
the potential function is therefore the third one for small displacements and has the
quadratic form
0
q
j
−
q
j
q
k
−
q
k
N
2
V
1
2
∂
V
(
q
)
=
(3.18)
∂
q
j
∂
q
k
j
,
k
=
1
in terms of the oscillator displacements from their equilibrium positions and the
symmetric coupling coefficients.
The Hamiltonian for the
N
particles of our spring is therefore, with the equilibrium
displacements of the individual particles set to zero, or equivalently, with the variables
shifted to the new set
q
=
q
0
,
−
q
p
j
m
+
N
N
1
2
q
)
=
V
jk
q
j
q
k
H
(
p
,
,
(3.19)
j
=
1
k
=
1
and the quantities
V
jk
are the derivatives of the potential with respect to
q
j
and
q
k
at
equilibrium. From Hamilton's equations we arrive at the equations of motion written in
vector form (suppressing the primes),
m
d
2
q
dt
2
+
Vq
=
0
,
(3.20)
where the matrix
V
has elements given by
V
jk
.
The main ideas can be developed by
restricting considerations to a one-dimensional lattice and considering only nearest-
neighbor interactions, like Newton did in his model of the air column. Imposing the
simplification of nearest-neighbor interactions, we have the string of oscillator equations
m
d
2
q
j
dt
2
=
κ
[
q
j
+
1
−
2
q
j
+
q
j
−
1
]
,
j
=
1
,
2
,...,
N
,
(3.21)
where
is the elastic force constant of an individual spring (oscillator). The solution to
this equation is determined by the boundary conditions selected and the initial condi-
tions. A particularly simple and convenient form of the solution for the
n
th oscillator's
complex displacement is
κ
