Each inertia tensor is diagonalized by a rotational transformation
to the body coordinate system:
|
(11.14) |
The transformation matrix
is used to initialize rotational
variables such that
=
. Values for the
body-frame coordinates
of group elements are obtained by
|
(11.15) |
The net force and torque acting on each body are determined by
summing the force and torque acting on each of its constituent atoms.
Center-of-mass variables are initialized with a two-step process. The
initial center-of-mass velocities are determined from the
atom properties VX,VY,VZ:
|
(11.16) |
These velocities are then used to advance the center-of-mass coordinates
|
(11.17) |
The more stable Euler-Cayley
parameters (also referred to as quaternions)
are used as rotational variables instead of the Euler angles ,
, (cf. Goldstein (1980)).
They are defined in
Eq. 2.1.
The quaternions
are initialized using a first-order approximation to their equation of
motion:
|
(11.18) |
where
is the four-vector (
,
,
,
),
is the four-vector (0,
,
,
), and
is the matrix that gives their time evolution:
|
(11.19) |
Thus one obtains
|
(11.20) |
The initial angular velocity
follows directly from the
initial angular momentum, which is determined by
|
(11.21) |
where
is the momentum of the
atom of the rigid body.
The initial half-step advanced angular momentum can be expressed as
|
(11.22) |
and the first advancement of the center-of-mass coordinates
can be written as
|
(11.23) |
Xplor-NIH 2024-09-13