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In-Depth Information
generalized coordinates are defined by the vector
q
= (
q
1
, …,
q
n
)
T
of
displacements from equilibrium. Typically the three Cartesian coordinates of
each node are considered separately, giving
q
a total of 3
N
components (notable
exceptions include the GNM, in which
q
has
N
components, and highly
symmetric systems, such as viral capsids, for which symmetry can be exploited
to reduce the dimensionality of
q
). Near the equilibrium structure, the potential
energy can be expanded as a power series in
q
as
2
∂
V
1
∂
V
∑
∑
⋯
.
V
( )
q
=
V
(0)
+
q
+
q q
+
(7.1)
i
i
j
i
∂
q
2
ij
∂ ∂
q
q
i
i
j
0
0
The first term of the above expression is a constant that may be set to zero, and
the second term is identically zero at a potential minimum. To second order, the
potential is a sum of pairwise potentials
∂
2
1
V
∑
V
( )
q
=
q q
(7.2)
i
j
2
ij
∂ ∂
q
q
i
j
0
1
2
=
∑
ij
q U q
(7.3)
i
ij
j
1
2
q Uq
T
=
,
(7.4)
where
U
is the matrix of second derivatives of the potential with respect to the
generalized coordinates. It should be noted that
U
is symmetric and nonnegative
definite.
Equation of motion.
The kinetic energy can similarly be written in compact
form as
1
2
q Mq
T
T
=
ɺ ɺ
,
(7.
5)
where the elements of the diagonal matrix
M
are the masses of the nodes and
q
ɺ
is
the time derivative of
q
. The equations of motion of the system are
Mq + Uq
ɺɺ
= 0
.
(7.6)
Here the double dot denotes the second derivative with respect to time.
Analytical solution
. We solve Eq. (7.6) by transforming to mass-weighted
coordinates,
r
=
M
1/2
q
,
K
=
M
-1/2
UM
-1/2
, which yield
ɺɺ
r = Kr
-
,
(7.7)
