Initialization

Each inertia tensor is diagonalized by a rotational transformation to the body coordinate system:

$${\bf I^P_{J}=A^{-1}_{J}I_JA_J}$$ (11.14)

The transformation matrix ${\bf A_J}$ is used to initialize rotational variables such that ${\bf A_J}$= ${\bf (A^{-1}_{J})^t}$. Values for the body-frame coordinates $r^B_{i}$ of group elements are obtained by

$$r^B_{i}={\bf A^t_{J}}(r_i-R_J)$$ (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:

$$V_J(t=0)={1\over M_J}\sum_{i\in J} m_iv_i$$ (11.16)

These velocities are then used to advance the center-of-mass coordinates

$$R_J(dt)=R_J(0)+V_J(0)dt+{1\over M_J}f_j(0)(dt)^2$$ (11.17)

The more stable Euler-Cayley parameters (also referred to as quaternions) are used as rotational variables instead of the Euler angles $\theta_1$, $\theta_2$, $\theta_3$ (cf. Goldstein (1980)). They are defined in Eq. 2.1. The quaternions are initialized using a first-order approximation to their equation of motion:

$$\dot{q}={\bf Q}\omega'$$ (11.18)

where $\dot{q}$ is the four-vector ($\dot{q}_0$,$\dot{q}_1$, $\dot{q}_2$,$\dot{q}_3$), $\omega'$ is the four-vector (0,$\omega_x$,$\omega_y$,$\omega_z$), and ${\bf Q}$ is the matrix that gives their time evolution:

$${\bf Q}={1\over 2}\left(\begin{array}{rrrr} q_0 & -q_1 & -q... ...3 & q_0 & -q_1 \\ q_3 & -q_2 & q_1 & q_0 \end{array} \right)$$ (11.19)

Thus one obtains

$$q({1\over 2}dt)=q(0)+{1\over 2}{\bf Q}(0)\omega'(0)dt$$ (11.20)

The initial angular velocity $\omega$ follows directly from the initial angular momentum, which is determined by

$$J_J(0)=\sum_{i\in J} (r_i-R_J)\times p_i$$ (11.21)

where $p_i$ is the momentum of the ${\rm i^{th}}$ atom of the rigid body. The initial half-step advanced angular momentum can be expressed as

$$J_J({1\over 2}dt)=J_J(0)+{1\over 2}N(0)$$ (11.22)

and the first advancement of the center-of-mass coordinates can be written as

$$R_J(dt)=R_J(0)+V_J(0)dt+{1\over 2}f_J(0){(dt)^2 \over M_J}$$ (11.23)