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
bodyframe 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.
Centerofmass variables are initialized with a twostep process. The
initial centerofmass velocities are determined from the
atom properties VX,VY,VZ:

(11.16) 
These velocities are then used to advance the centerofmass coordinates

(11.17) 
The more stable EulerCayley
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 firstorder approximation to their equation of
motion:

(11.18) 
where
is the fourvector (
,
,
,
),
is the fourvector (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 halfstep advanced angular momentum can be expressed as

(11.22) 
and the first advancement of the centerofmass coordinates
can be written as

(11.23) 
XplorNIH 20240913