| 40 |
|
The basic equations of motion being integrated are derived from the |
| 41 |
|
Hamiltonian for conservative systems containing rigid bodies, |
| 42 |
|
\begin{equation} |
| 43 |
< |
H = \sum_{i} \frac{{\bf v}_i^T m_i {\bf v}_i}{2} + \sum_i |
| 44 |
< |
\frac{{\bf j}_i^T \cdot {\bf j}_i}{2 \overleftrightarrow{\mathsf{I}}_i} + |
| 43 |
> |
H = \sum_{i} \left( \frac{1}{2} m_i {\bf v}_i^T \cdot {\bf v}_i + |
| 44 |
> |
\frac{1}{2} {\bf j}_i^T \cdot \overleftrightarrow{\mathsf{I}}_i^{-1} \cdot |
| 45 |
> |
{\bf j}_i \right) + |
| 46 |
|
V\left(\left\{{\bf r}\right\}, \left\{\mathsf{A}\right\}\right) |
| 47 |
|
\end{equation} |
| 48 |
|
where ${\bf r}_i$ and ${\bf v}_i$ are the cartesian position vector |
| 62 |
|
where ${\bf f}$ is the instantaneous force on the center of mass |
| 63 |
|
of the particle, |
| 64 |
|
\begin{equation} |
| 65 |
< |
{\bf f} = \frac{\partial}{\partial |
| 65 |
> |
{\bf f} = - \frac{\partial}{\partial |
| 66 |
|
{\bf r}} V(\left\{{\bf r}(t)\right\}, \left\{\mathsf{A}(t)\right\}). |
| 67 |
|
\end{equation} |
| 68 |
|
|
| 68 |
– |
|
| 69 |
|
The equations of motion for the orientational degrees of freedom are |
| 70 |
|
\begin{eqnarray} |
| 71 |
< |
\dot{\mathsf{A}} & = & \mathsf{A} |
| 71 |
> |
\dot{\mathsf{A}} & = & \mathsf{A} \cdot |
| 72 |
|
\mbox{ skew}\left(\overleftrightarrow{I}^{-1} \cdot {\bf j}\right) \\ |
| 73 |
|
\dot{{\bf j}} & = & {\bf j} \times \left( \overleftrightarrow{I}^{-1} |
| 74 |
< |
\cdot {\bf j} \right) - \mbox{ rot}\left(\mathsf{A}^{T} \frac{\partial |
| 74 |
> |
\cdot {\bf j} \right) - \mbox{ rot}\left(\mathsf{A}^{T} \cdot \frac{\partial |
| 75 |
|
V}{\partial \mathsf{A}} \right) |
| 76 |
|
\end{eqnarray} |
| 77 |
|
In these equations of motion, the $\mbox{skew}$ matrix of a vector |
| 94 |
|
\mbox{rot}\left(\mathsf{A}\right) := \mbox{ skew}^{-1}\left(\mathsf{A} |
| 95 |
|
- \mathsf{A}^{T} \right) |
| 96 |
|
\end{equation} |
| 97 |
< |
|
| 97 |
> |
Written this way, the $\mbox{rot}$ operation creates a set of |
| 98 |
> |
conjugate angle coordinates to the body-fixed angular momenta |
| 99 |
> |
represented by ${\bf j}$. This equation of motion for angular momenta |
| 100 |
> |
is equivalent to the more familiar body-fixed form: |
| 101 |
> |
\begin{eqnarray} |
| 102 |
> |
\dot{j_{x}} & = & \tau^b_x(t) + |
| 103 |
> |
\left(\overleftrightarrow{\mathsf{I}}_{yy} - \overleftrightarrow{\mathsf{I}}_{zz} \right) j_y j_z \\ |
| 104 |
> |
\dot{j_{y}} & = & \tau^b_y(t) + |
| 105 |
> |
\left(\overleftrightarrow{\mathsf{I}}_{zz} - \overleftrightarrow{\mathsf{I}}_{xx} \right) j_z j_x \\ |
| 106 |
> |
\dot{j_{z}} & = & \tau^b_z(t) + |
| 107 |
> |
\left(\overleftrightarrow{\mathsf{I}}_{xx} - \overleftrightarrow{\mathsf{I}}_{yy} \right) j_x j_y |
| 108 |
> |
\end{eqnarray} |
| 109 |
> |
which utilize the body-fixed torques, ${\bf \tau}^b$. Torques are |
| 110 |
> |
most easily derived in the space-fixed frame, |
| 111 |
> |
\begin{equation} |
| 112 |
> |
{\bf \tau}^b(t) = \mathsf{A}(t) \cdot {\bf \tau}^s(t) |
| 113 |
> |
\end{equation} |
| 114 |
> |
where |
| 115 |
> |
\begin{equation} |
| 116 |
> |
{\bf \tau}^s(t) = - \hat{\bf u}(t) \times \left( \frac{\partial} |
| 117 |
> |
{\partial \hat{\bf u}} V\left(\left\{ {\bf r}(t) \right\}, \left\{ |
| 118 |
> |
\mathsf{A}(t) \right\}\right) \right) |
| 119 |
> |
\end{equation} |
| 120 |
> |
where $\hat{\bf u}$ is a unit vector pointing along the principal axis |
| 121 |
> |
of the particle in the space-fixed frame. |
| 122 |
|
|
| 123 |
+ |
The DLM method uses a Trotter factorization of the orientational |
| 124 |
+ |
propagator. This has three effects: |
| 125 |
+ |
\begin{enumerate} |
| 126 |
+ |
\item the integrator is area preserving in phase space (i.e. it is |
| 127 |
+ |
{\it symplectic}), |
| 128 |
+ |
\item the integrator is time-{\it reversible}, making it suitable for Hybrid |
| 129 |
+ |
Monte Carlo applications, and |
| 130 |
+ |
\item the error for a single time step is of order $O\left(h^3\right)$ |
| 131 |
+ |
for timesteps of length $h$. |
| 132 |
+ |
\end{enumerate} |
| 133 |
|
|
| 134 |
|
In the integration method, the |
| 135 |
|
orientational propagation involves a sequence of matrix evaluations to |
