1170 |
|
V(q,Q) + \frac{1}{2}tr[(QQ^T - 1)\Lambda ]. |
1171 |
|
\label{introEquation:RBHamiltonian} |
1172 |
|
\end{equation} |
1173 |
< |
Here, $q$ and $Q$ are the position and rotation matrix for the |
1174 |
< |
rigid-body, $p$ and $P$ are conjugate momenta to $q$ and $Q$ , and |
1175 |
< |
$J$, a diagonal matrix, is defined by |
1173 |
> |
Here, $q$ and $Q$ are the position vector and rotation matrix for |
1174 |
> |
the rigid-body, $p$ and $P$ are conjugate momenta to $q$ and $Q$ , |
1175 |
> |
and $J$, a diagonal matrix, is defined by |
1176 |
|
\[ |
1177 |
|
I_{ii}^{ - 1} = \frac{1}{2}\sum\limits_{i \ne j} {J_{jj}^{ - 1} } |
1178 |
|
\] |
1182 |
|
\begin{equation} |
1183 |
|
Q^T Q = 1, \label{introEquation:orthogonalConstraint} |
1184 |
|
\end{equation} |
1185 |
< |
which is used to ensure rotation matrix's unitarity. Differentiating |
1186 |
< |
Eq.~\ref{introEquation:orthogonalConstraint} and using |
1187 |
< |
Eq.~\ref{introEquation:RBMotionMomentum}, one may obtain, |
1188 |
< |
\begin{equation} |
1189 |
< |
Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0 . \\ |
1190 |
< |
\label{introEquation:RBFirstOrderConstraint} |
1191 |
< |
\end{equation} |
1192 |
< |
Using Equation (\ref{introEquation:motionHamiltonianCoordinate}, |
1185 |
> |
which is used to ensure the rotation matrix's unitarity. Using |
1186 |
> |
Equation (\ref{introEquation:motionHamiltonianCoordinate}, |
1187 |
|
\ref{introEquation:motionHamiltonianMomentum}), one can write down |
1188 |
|
the equations of motion, |
1189 |
|
\begin{eqnarray} |
1192 |
|
\frac{{dQ}}{{dt}} & = & PJ^{ - 1}, \label{introEquation:RBMotionRotation}\\ |
1193 |
|
\frac{{dP}}{{dt}} & = & - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP} |
1194 |
|
\end{eqnarray} |
1195 |
+ |
Differentiating Eq.~\ref{introEquation:orthogonalConstraint} and |
1196 |
+ |
using Eq.~\ref{introEquation:RBMotionMomentum}, one may obtain, |
1197 |
+ |
\begin{equation} |
1198 |
+ |
Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0 . \\ |
1199 |
+ |
\label{introEquation:RBFirstOrderConstraint} |
1200 |
+ |
\end{equation} |
1201 |
|
In general, there are two ways to satisfy the holonomic constraints. |
1202 |
|
We can use a constraint force provided by a Lagrange multiplier on |
1203 |
< |
the normal manifold to keep the motion on constraint space. Or we |
1204 |
< |
can simply evolve the system on the constraint manifold. These two |
1205 |
< |
methods have been proved to be equivalent. The holonomic constraint |
1206 |
< |
and equations of motions define a constraint manifold for rigid |
1207 |
< |
bodies |
1203 |
> |
the normal manifold to keep the motion on the constraint space. Or |
1204 |
> |
we can simply evolve the system on the constraint manifold. These |
1205 |
> |
two methods have been proved to be equivalent. The holonomic |
1206 |
> |
constraint and equations of motions define a constraint manifold for |
1207 |
> |
rigid bodies |
1208 |
|
\[ |
1209 |
|
M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0} |
1210 |
|
\right\}. |
1211 |
|
\] |
1212 |
< |
Unfortunately, this constraint manifold is not the cotangent bundle |
1213 |
< |
$T^* SO(3)$ which can be consider as a symplectic manifold on Lie |
1214 |
< |
rotation group $SO(3)$. However, it turns out that under symplectic |
1215 |
< |
transformation, the cotangent space and the phase space are |
1216 |
< |
diffeomorphic. By introducing |
1212 |
> |
Unfortunately, this constraint manifold is not $T^* SO(3)$ which is |
1213 |
> |
a symplectic manifold on Lie rotation group $SO(3)$. However, it |
1214 |
> |
turns out that under symplectic transformation, the cotangent space |
1215 |
> |
and the phase space are diffeomorphic. By introducing |
1216 |
|
\[ |
1217 |
|
\tilde Q = Q,\tilde P = \frac{1}{2}\left( {P - QP^T Q} \right), |
1218 |
|
\] |
1280 |
|
motion. This unique property eliminates the requirement of |
1281 |
|
iterations which can not be avoided in other methods\cite{Kol1997, |
1282 |
|
Omelyan1998}. Applying the hat-map isomorphism, we obtain the |
1283 |
< |
equation of motion for angular momentum on body frame |
1283 |
> |
equation of motion for angular momentum in the body frame |
1284 |
|
\begin{equation} |
1285 |
|
\dot \pi = \pi \times I^{ - 1} \pi + \sum\limits_i {\left( {Q^T |
1286 |
|
F_i (r,Q)} \right) \times X_i }. |
1293 |
|
\] |
1294 |
|
|
1295 |
|
\subsection{\label{introSection:SymplecticFreeRB}Symplectic |
1296 |
< |
Lie-Poisson Integrator for Free Rigid Body} |
1296 |
> |
Lie-Poisson Integrator for Free Rigid Bodies} |
1297 |
|
|
1298 |
|
If there are no external forces exerted on the rigid body, the only |
1299 |
|
contribution to the rotational motion is from the kinetic energy |
1345 |
|
\end{array}} \right),\theta _1 = \frac{{\pi _1 }}{{I_1 }}\Delta t. |
1346 |
|
\] |
1347 |
|
To reduce the cost of computing expensive functions in $e^{\Delta |
1348 |
< |
tR_1 }$, we can use Cayley transformation to obtain a single-aixs |
1349 |
< |
propagator, |
1350 |
< |
\[ |
1351 |
< |
e^{\Delta tR_1 } \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1 |
1352 |
< |
). |
1353 |
< |
\] |
1354 |
< |
The propagator maps for $T_2^r$ and $T_3^r$ can be found in the same |
1348 |
> |
tR_1 }$, we can use the Cayley transformation to obtain a |
1349 |
> |
single-aixs propagator, |
1350 |
> |
\begin{eqnarray*} |
1351 |
> |
e^{\Delta tR_1 } & \approx & (1 - \Delta tR_1 )^{ - 1} (1 + \Delta |
1352 |
> |
tR_1 ) \\ |
1353 |
> |
% |
1354 |
> |
& \approx & \left( \begin{array}{ccc} |
1355 |
> |
1 & 0 & 0 \\ |
1356 |
> |
0 & \frac{1-\theta^2 / 4}{1 + \theta^2 / 4} & -\frac{\theta}{1+ |
1357 |
> |
\theta^2 / 4} \\ |
1358 |
> |
0 & \frac{\theta}{1+ \theta^2 / 4} & \frac{1-\theta^2 / 4}{1 + |
1359 |
> |
\theta^2 / 4} |
1360 |
> |
\end{array} |
1361 |
> |
\right). |
1362 |
> |
\end{eqnarray*} |
1363 |
> |
The propagators for $T_2^r$ and $T_3^r$ can be found in the same |
1364 |
|
manner. In order to construct a second-order symplectic method, we |
1365 |
|
split the angular kinetic Hamiltonian function into five terms |
1366 |
|
\[ |
1386 |
|
function $G$ is zero, $F$ is a \emph{Casimir}, which is the |
1387 |
|
conserved quantity in Poisson system. We can easily verify that the |
1388 |
|
norm of the angular momentum, $\parallel \pi |
1389 |
< |
\parallel$, is a \emph{Casimir}. Let$ F(\pi ) = S(\frac{{\parallel |
1389 |
> |
\parallel$, is a \emph{Casimir}\cite{McLachlan1993}. Let$ F(\pi ) = S(\frac{{\parallel |
1390 |
|
\pi \parallel ^2 }}{2})$ for an arbitrary function $ S:R \to R$ , |
1391 |
|
then by the chain rule |
1392 |
|
\[ |
1533 |
|
differential equations,the Laplace transform is the appropriate tool |
1534 |
|
to solve this problem. The basic idea is to transform the difficult |
1535 |
|
differential equations into simple algebra problems which can be |
1536 |
< |
solved easily. Then, by applying the inverse Laplace transform, also |
1537 |
< |
known as the Bromwich integral, we can retrieve the solutions of the |
1538 |
< |
original problems. Let $f(t)$ be a function defined on $ [0,\infty ) |
1539 |
< |
$, the Laplace transform of $f(t)$ is a new function defined as |
1536 |
> |
solved easily. Then, by applying the inverse Laplace transform, we |
1537 |
> |
can retrieve the solutions of the original problems. Let $f(t)$ be a |
1538 |
> |
function defined on $ [0,\infty ) $, the Laplace transform of $f(t)$ |
1539 |
> |
is a new function defined as |
1540 |
|
\[ |
1541 |
|
L(f(t)) \equiv F(p) = \int_0^\infty {f(t)e^{ - pt} dt} |
1542 |
|
\] |
1554 |
|
p^2 L(x_\alpha ) - px_\alpha (0) - \dot x_\alpha (0) & = & - \omega _\alpha ^2 L(x_\alpha ) + \frac{{g_\alpha }}{{\omega _\alpha }}L(x), \\ |
1555 |
|
L(x_\alpha ) & = & \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }}. \\ |
1556 |
|
\end{eqnarray*} |
1557 |
< |
By the same way, the system coordinates become |
1557 |
> |
In the same way, the system coordinates become |
1558 |
|
\begin{eqnarray*} |
1559 |
|
mL(\ddot x) & = & |
1560 |
|
- \sum\limits_{\alpha = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}\frac{p}{{p^2 + \omega _\alpha ^2 }}pL(x) - \frac{p}{{p^2 + \omega _\alpha ^2 }}g_\alpha x_\alpha (0) - \frac{1}{{p^2 + \omega _\alpha ^2 }}g_\alpha \dot x_\alpha (0)} \right\}} \\ |
1578 |
|
& & + \sum\limits_{\alpha = 1}^N {\left\{ {\left[ {g_\alpha |
1579 |
|
x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha \omega _\alpha }}} |
1580 |
|
\right]\cos (\omega _\alpha t) + \frac{{g_\alpha \dot x_\alpha |
1581 |
< |
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t)} \right\}} |
1582 |
< |
\end{eqnarray*} |
1583 |
< |
\begin{eqnarray*} |
1584 |
< |
m\ddot x & = & - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t |
1585 |
< |
{\sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
1586 |
< |
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha |
1581 |
> |
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t)} \right\}}\\ |
1582 |
> |
% |
1583 |
> |
& = & - |
1584 |
> |
\frac{{\partial W(x)}}{{\partial x}} - \int_0^t {\sum\limits_{\alpha |
1585 |
> |
= 1}^N {\left( { - \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha |
1586 |
> |
^2 }}} \right)\cos (\omega _\alpha |
1587 |
|
t)\dot x(t - \tau )d} \tau } \\ |
1588 |
|
& & + \sum\limits_{\alpha = 1}^N {\left\{ {\left[ {g_\alpha |
1589 |
|
x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha \omega _\alpha }}} |
1661 |
|
which is known as the Langevin equation. The static friction |
1662 |
|
coefficient $\xi _0$ can either be calculated from spectral density |
1663 |
|
or be determined by Stokes' law for regular shaped particles. A |
1664 |
< |
briefly review on calculating friction tensor for arbitrary shaped |
1664 |
> |
brief review on calculating friction tensors for arbitrary shaped |
1665 |
|
particles is given in Sec.~\ref{introSection:frictionTensor}. |
1666 |
|
|
1667 |
|
\subsubsection{\label{introSection:secondFluctuationDissipation}\textbf{The Second Fluctuation Dissipation Theorem}} |