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Ae
i
(
n
θ
−
ω
t
)
,
q
n
(
t
)
=
(3.22)
which, when substituted into (
3.21
), yields the dispersion relation between the fre-
quency of motion and the spring constant:
2
m
[1
2
ω
=
−
cos
θ
]
.
(3.23)
Of course, the value of the angle
is determined by the boundary conditions. If we
assume that the string of particles is wrapped into a necklace so that the last particle
couples to the first, then
q
N
+
1
=
θ
q
1
and this forces the phase in the dispersion relation
to satisfy the requirement
e
iN
θ
=
1
,
implying that the phase takes on a discrete number of values
l
2
N
,
θ
l
=
l
=
1
,
2
,...,
N
.
(3.24)
The dispersion relation when indexed to the phase values becomes
1
cos
l
2
2
m
N
2
l
ω
=
−
;
(3.25)
other conditions on the phase-dispersion relation can be derived from other boundary
conditions. The general solution (
3.22
) can be written in terms of these phases as
N
A
l
e
i
(
n
θ
l
−
ω
l
t
)
,
q
n
(
t
)
=
(3.26)
l
=
1
where the expansion coefficients are determined by the initial conditions obtained by
the inverse discrete Fourier transform
N
1
N
e
−
in
θ
l
q
n
(
A
l
=
0
).
(3.27)
n
=
1
Note that the solution of each particle displacement is dependent on the initial displace-
ment of all the other particles in the chain due to the coupling. This is the simplest
manifestation of the propagation of influence through a web and establishes a way of
viewing how the elements of a web interact and transmit information from one member
to another. Of course, to make such a claim is a bit facile on our part because we know
in advance what the important variables are for a linear physical network, but we do not
necessarily know that about social or physiologic webs.
In the absence of certainty on how to represent linear web dynamics it is worth
emphasizing the important matter of representation. Consider an even simpler linear
case than the one described above, namely a set of
N uncoupled
oscillators with dis-
tinct masses and elastic constants. The Hamiltonian in the momentum-displacement
canonical representation is
