| 570 |
|
\dot x = J(x)\nabla _x H \label{introEquation:poissonHamiltonian} |
| 571 |
|
\end{equation} |
| 572 |
|
The most obvious change being that matrix $J$ now depends on $x$. |
| 573 |
– |
The free rigid body is an example of Poisson system (actually a |
| 574 |
– |
Lie-Poisson system) with Hamiltonian function of angular kinetic |
| 575 |
– |
energy. |
| 576 |
– |
\begin{equation} |
| 577 |
– |
J(\pi ) = \left( {\begin{array}{*{20}c} |
| 578 |
– |
0 & {\pi _3 } & { - \pi _2 } \\ |
| 579 |
– |
{ - \pi _3 } & 0 & {\pi _1 } \\ |
| 580 |
– |
{\pi _2 } & { - \pi _1 } & 0 \\ |
| 581 |
– |
\end{array}} \right) |
| 582 |
– |
\end{equation} |
| 583 |
– |
|
| 584 |
– |
\begin{equation} |
| 585 |
– |
H = \frac{1}{2}\left( {\frac{{\pi _1^2 }}{{I_1 }} + \frac{{\pi _2^2 |
| 586 |
– |
}}{{I_2 }} + \frac{{\pi _3^2 }}{{I_3 }}} \right) |
| 587 |
– |
\end{equation} |
| 573 |
|
|
| 574 |
|
\subsection{\label{introSection:exactFlow}Exact Flow} |
| 575 |
|
|
| 934 |
|
method using quaternion representation was developed by Omelyan. |
| 935 |
|
However, both of these methods are iterative and inefficient. In |
| 936 |
|
this section, we will present a symplectic Lie-Poisson integrator |
| 937 |
< |
for rigid body developed by Dullweber and his coworkers\cite{}. |
| 938 |
< |
|
| 954 |
< |
\subsection{\label{introSection:lieAlgebra}Lie Algebra} |
| 937 |
> |
for rigid body developed by Dullweber and his |
| 938 |
> |
coworkers\cite{Dullweber1997} in depth. |
| 939 |
|
|
| 940 |
|
\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Body} |
| 941 |
< |
|
| 941 |
> |
The motion of the rigid body is Hamiltonian with the Hamiltonian |
| 942 |
> |
function |
| 943 |
|
\begin{equation} |
| 944 |
|
H = \frac{1}{2}(p^T m^{ - 1} p) + \frac{1}{2}tr(PJ^{ - 1} P) + |
| 945 |
|
V(q,Q) + \frac{1}{2}tr[(QQ^T - 1)\Lambda ]. |
| 960 |
|
Differentiating \ref{introEquation:orthogonalConstraint} and using |
| 961 |
|
Equation \ref{introEquation:RBMotionMomentum}, one may obtain, |
| 962 |
|
\begin{equation} |
| 963 |
< |
Q^t PJ^{ - 1} + J^{ - 1} P^t Q = 0 . \\ |
| 963 |
> |
Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0 . \\ |
| 964 |
|
\label{introEquation:RBFirstOrderConstraint} |
| 965 |
|
\end{equation} |
| 966 |
|
|
| 972 |
|
\frac{{dq}}{{dt}} = \frac{p}{m} \label{introEquation:RBMotionPosition}\\ |
| 973 |
|
\frac{{dp}}{{dt}} = - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\ |
| 974 |
|
\frac{{dQ}}{{dt}} = PJ^{ - 1} \label{introEquation:RBMotionRotation}\\ |
| 975 |
< |
\frac{{dP}}{{dt}} = - \nabla _q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}\\ |
| 975 |
> |
\frac{{dP}}{{dt}} = - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}\\ |
| 976 |
|
\end{array} |
| 977 |
|
\] |
| 978 |
|
|
| 979 |
+ |
In general, there are two ways to satisfy the holonomic constraints. |
| 980 |
+ |
We can use constraint force provided by lagrange multiplier on the |
| 981 |
+ |
normal manifold to keep the motion on constraint space. Or we can |
| 982 |
+ |
simply evolve the system in constraint manifold. The two method are |
| 983 |
+ |
proved to be equivalent. The holonomic constraint and equations of |
| 984 |
+ |
motions define a constraint manifold for rigid body |
| 985 |
+ |
\[ |
| 986 |
+ |
M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0} |
| 987 |
+ |
\right\}. |
| 988 |
+ |
\] |
| 989 |
|
|
| 990 |
+ |
Unfortunately, this constraint manifold is not the cotangent bundle |
| 991 |
+ |
$T_{\star}SO(3)$. However, it turns out that under symplectic |
| 992 |
+ |
transformation, the cotangent space and the phase space are |
| 993 |
+ |
diffeomorphic. Introducing |
| 994 |
|
\[ |
| 995 |
< |
M = \left\{ {(Q,P):Q^T Q = 1,Q^t PJ^{ - 1} + J^{ - 1} P^t Q = 0} |
| 997 |
< |
\right\} . |
| 995 |
> |
\tilde Q = Q,\tilde P = \frac{1}{2}\left( {P - QP^T Q} \right), |
| 996 |
|
\] |
| 997 |
+ |
the mechanical system subject to a holonomic constraint manifold $M$ |
| 998 |
+ |
can be re-formulated as a Hamiltonian system on the cotangent space |
| 999 |
+ |
\[ |
| 1000 |
+ |
T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \tilde Q = |
| 1001 |
+ |
1,\tilde Q^T \tilde PJ^{ - 1} + J^{ - 1} P^T \tilde Q = 0} \right\} |
| 1002 |
+ |
\] |
| 1003 |
|
|
| 1004 |
< |
\subsection{\label{introSection:DLMMotionEquation}The Euler Equations of Rigid Body Motion} |
| 1004 |
> |
For a body fixed vector $X_i$ with respect to the center of mass of |
| 1005 |
> |
the rigid body, its corresponding lab fixed vector $X_0^{lab}$ is |
| 1006 |
> |
given as |
| 1007 |
> |
\begin{equation} |
| 1008 |
> |
X_i^{lab} = Q X_i + q. |
| 1009 |
> |
\end{equation} |
| 1010 |
> |
Therefore, potential energy $V(q,Q)$ is defined by |
| 1011 |
> |
\[ |
| 1012 |
> |
V(q,Q) = V(Q X_0 + q). |
| 1013 |
> |
\] |
| 1014 |
> |
Hence, the force and torque are given by |
| 1015 |
> |
\[ |
| 1016 |
> |
\nabla _q V(q,Q) = F(q,Q) = \sum\limits_i {F_i (q,Q)}, |
| 1017 |
> |
\] |
| 1018 |
> |
and |
| 1019 |
> |
\[ |
| 1020 |
> |
\nabla _Q V(q,Q) = F(q,Q)X_i^t |
| 1021 |
> |
\] |
| 1022 |
> |
respectively. |
| 1023 |
|
|
| 1024 |
< |
\subsection{\label{introSection:symplecticDiscretizationRB}Symplectic Discretization of Euler Equations} |
| 1024 |
> |
As a common choice to describe the rotation dynamics of the rigid |
| 1025 |
> |
body, angular momentum on body frame $\Pi = Q^t P$ is introduced to |
| 1026 |
> |
rewrite the equations of motion, |
| 1027 |
> |
\begin{equation} |
| 1028 |
> |
\begin{array}{l} |
| 1029 |
> |
\mathop \Pi \limits^ \bullet = J^{ - 1} \Pi ^T \Pi + Q^T \sum\limits_i {F_i (q,Q)X_i^T } - \Lambda \\ |
| 1030 |
> |
\mathop Q\limits^{{\rm{ }} \bullet } = Q\Pi {\rm{ }}J^{ - 1} \\ |
| 1031 |
> |
\end{array} |
| 1032 |
> |
\label{introEqaution:RBMotionPI} |
| 1033 |
> |
\end{equation} |
| 1034 |
> |
, as well as holonomic constraints, |
| 1035 |
> |
\[ |
| 1036 |
> |
\begin{array}{l} |
| 1037 |
> |
\Pi J^{ - 1} + J^{ - 1} \Pi ^t = 0 \\ |
| 1038 |
> |
Q^T Q = 1 \\ |
| 1039 |
> |
\end{array} |
| 1040 |
> |
\] |
| 1041 |
|
|
| 1042 |
+ |
For a vector $v(v_1 ,v_2 ,v_3 ) \in R^3$ and a matrix $\hat v \in |
| 1043 |
+ |
so(3)^ \star$, the hat-map isomorphism, |
| 1044 |
+ |
\begin{equation} |
| 1045 |
+ |
v(v_1 ,v_2 ,v_3 ) \Leftrightarrow \hat v = \left( |
| 1046 |
+ |
{\begin{array}{*{20}c} |
| 1047 |
+ |
0 & { - v_3 } & {v_2 } \\ |
| 1048 |
+ |
{v_3 } & 0 & { - v_1 } \\ |
| 1049 |
+ |
{ - v_2 } & {v_1 } & 0 \\ |
| 1050 |
+ |
\end{array}} \right), |
| 1051 |
+ |
\label{introEquation:hatmapIsomorphism} |
| 1052 |
+ |
\end{equation} |
| 1053 |
+ |
will let us associate the matrix products with traditional vector |
| 1054 |
+ |
operations |
| 1055 |
+ |
\[ |
| 1056 |
+ |
\hat vu = v \times u |
| 1057 |
+ |
\] |
| 1058 |
|
|
| 1059 |
< |
\section{\label{introSection:langevinDynamics}Langevin Dynamics} |
| 1059 |
> |
Using \ref{introEqaution:RBMotionPI}, one can construct a skew |
| 1060 |
> |
matrix, |
| 1061 |
> |
\begin{equation} |
| 1062 |
> |
(\mathop \Pi \limits^ \bullet - \mathop \Pi \limits^ \bullet ^T |
| 1063 |
> |
){\rm{ }} = {\rm{ }}(\Pi - \Pi ^T ){\rm{ }}(J^{ - 1} \Pi + \Pi J^{ |
| 1064 |
> |
- 1} ) + \sum\limits_i {[Q^T F_i (r,Q)X_i^T - X_i F_i (r,Q)^T Q]} - |
| 1065 |
> |
(\Lambda - \Lambda ^T ) . \label{introEquation:skewMatrixPI} |
| 1066 |
> |
\end{equation} |
| 1067 |
> |
Since $\Lambda$ is symmetric, the last term of Equation |
| 1068 |
> |
\ref{introEquation:skewMatrixPI} is zero, which implies the Lagrange |
| 1069 |
> |
multiplier $\Lambda$ is absent from the equations of motion. This |
| 1070 |
> |
unique property eliminate the requirement of iterations which can |
| 1071 |
> |
not be avoided in other methods\cite{}. |
| 1072 |
|
|
| 1073 |
< |
\subsection{\label{introSection:LDIntroduction}Introduction and application of Langevin Dynamics} |
| 1073 |
> |
Applying hat-map isomorphism, we obtain the equation of motion for |
| 1074 |
> |
angular momentum on body frame |
| 1075 |
> |
\begin{equation} |
| 1076 |
> |
\dot \pi = \pi \times I^{ - 1} \pi + \sum\limits_i {\left( {Q^T |
| 1077 |
> |
F_i (r,Q)} \right) \times X_i }. |
| 1078 |
> |
\label{introEquation:bodyAngularMotion} |
| 1079 |
> |
\end{equation} |
| 1080 |
> |
In the same manner, the equation of motion for rotation matrix is |
| 1081 |
> |
given by |
| 1082 |
> |
\[ |
| 1083 |
> |
\dot Q = Qskew(I^{ - 1} \pi ) |
| 1084 |
> |
\] |
| 1085 |
> |
|
| 1086 |
> |
\subsection{\label{introSection:SymplecticFreeRB}Symplectic |
| 1087 |
> |
Lie-Poisson Integrator for Free Rigid Body} |
| 1088 |
> |
|
| 1089 |
> |
If there is not external forces exerted on the rigid body, the only |
| 1090 |
> |
contribution to the rotational is from the kinetic potential (the |
| 1091 |
> |
first term of \ref{ introEquation:bodyAngularMotion}). The free |
| 1092 |
> |
rigid body is an example of Lie-Poisson system with Hamiltonian |
| 1093 |
> |
function |
| 1094 |
> |
\begin{equation} |
| 1095 |
> |
T^r (\pi ) = T_1 ^r (\pi _1 ) + T_2^r (\pi _2 ) + T_3^r (\pi _3 ) |
| 1096 |
> |
\label{introEquation:rotationalKineticRB} |
| 1097 |
> |
\end{equation} |
| 1098 |
> |
where $T_i^r (\pi _i ) = \frac{{\pi _i ^2 }}{{2I_i }}$ and |
| 1099 |
> |
Lie-Poisson structure matrix, |
| 1100 |
> |
\begin{equation} |
| 1101 |
> |
J(\pi ) = \left( {\begin{array}{*{20}c} |
| 1102 |
> |
0 & {\pi _3 } & { - \pi _2 } \\ |
| 1103 |
> |
{ - \pi _3 } & 0 & {\pi _1 } \\ |
| 1104 |
> |
{\pi _2 } & { - \pi _1 } & 0 \\ |
| 1105 |
> |
\end{array}} \right) |
| 1106 |
> |
\end{equation} |
| 1107 |
> |
Thus, the dynamics of free rigid body is governed by |
| 1108 |
> |
\begin{equation} |
| 1109 |
> |
\frac{d}{{dt}}\pi = J(\pi )\nabla _\pi T^r (\pi ) |
| 1110 |
> |
\end{equation} |
| 1111 |
|
|
| 1112 |
+ |
One may notice that each $T_i^r$ in Equation |
| 1113 |
+ |
\ref{introEquation:rotationalKineticRB} can be solved exactly. For |
| 1114 |
+ |
instance, the equations of motion due to $T_1^r$ are given by |
| 1115 |
+ |
\begin{equation} |
| 1116 |
+ |
\frac{d}{{dt}}\pi = R_1 \pi ,\frac{d}{{dt}}Q = QR_1 |
| 1117 |
+ |
\label{introEqaution:RBMotionSingleTerm} |
| 1118 |
+ |
\end{equation} |
| 1119 |
+ |
where |
| 1120 |
+ |
\[ R_1 = \left( {\begin{array}{*{20}c} |
| 1121 |
+ |
0 & 0 & 0 \\ |
| 1122 |
+ |
0 & 0 & {\pi _1 } \\ |
| 1123 |
+ |
0 & { - \pi _1 } & 0 \\ |
| 1124 |
+ |
\end{array}} \right). |
| 1125 |
+ |
\] |
| 1126 |
+ |
The solutions of Equation \ref{introEqaution:RBMotionSingleTerm} is |
| 1127 |
+ |
\[ |
| 1128 |
+ |
\pi (\Delta t) = e^{\Delta tR_1 } \pi (0),Q(\Delta t) = |
| 1129 |
+ |
Q(0)e^{\Delta tR_1 } |
| 1130 |
+ |
\] |
| 1131 |
+ |
with |
| 1132 |
+ |
\[ |
| 1133 |
+ |
e^{\Delta tR_1 } = \left( {\begin{array}{*{20}c} |
| 1134 |
+ |
0 & 0 & 0 \\ |
| 1135 |
+ |
0 & {\cos \theta _1 } & {\sin \theta _1 } \\ |
| 1136 |
+ |
0 & { - \sin \theta _1 } & {\cos \theta _1 } \\ |
| 1137 |
+ |
\end{array}} \right),\theta _1 = \frac{{\pi _1 }}{{I_1 }}\Delta t. |
| 1138 |
+ |
\] |
| 1139 |
+ |
To reduce the cost of computing expensive functions in e^{\Delta |
| 1140 |
+ |
tR_1 }, we can use Cayley transformation, |
| 1141 |
+ |
\[ |
| 1142 |
+ |
e^{\Delta tR_1 } \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1 |
| 1143 |
+ |
) |
| 1144 |
+ |
\] |
| 1145 |
+ |
|
| 1146 |
+ |
The flow maps for $T_2^r$ and $T_2^r$ can be found in the same |
| 1147 |
+ |
manner. |
| 1148 |
+ |
|
| 1149 |
+ |
In order to construct a second-order symplectic method, we split the |
| 1150 |
+ |
angular kinetic Hamiltonian function can into five terms |
| 1151 |
+ |
\[ |
| 1152 |
+ |
T^r (\pi ) = \frac{1}{2}T_1 ^r (\pi _1 ) + \frac{1}{2}T_2^r (\pi _2 |
| 1153 |
+ |
) + T_3^r (\pi _3 ) + \frac{1}{2}T_2^r (\pi _2 ) + \frac{1}{2}T_1 ^r |
| 1154 |
+ |
(\pi _1 ) |
| 1155 |
+ |
\]. |
| 1156 |
+ |
Concatenating flows corresponding to these five terms, we can obtain |
| 1157 |
+ |
an symplectic integrator, |
| 1158 |
+ |
\[ |
| 1159 |
+ |
\varphi _{\Delta t,T^r } = \varphi _{\Delta t/2,\pi _1 } \circ |
| 1160 |
+ |
\varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t,\pi _3 } |
| 1161 |
+ |
\circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi |
| 1162 |
+ |
_1 }. |
| 1163 |
+ |
\] |
| 1164 |
+ |
|
| 1165 |
+ |
The non-canonical Lie-Poisson bracket ${F, G}$ of two function |
| 1166 |
+ |
$F(\pi )$ and $G(\pi )$ is defined by |
| 1167 |
+ |
\[ |
| 1168 |
+ |
\{ F,G\} (\pi ) = [\nabla _\pi F(\pi )]^T J(\pi )\nabla _\pi G(\pi |
| 1169 |
+ |
) |
| 1170 |
+ |
\] |
| 1171 |
+ |
If the Poisson bracket of a function $F$ with an arbitrary smooth |
| 1172 |
+ |
function $G$ is zero, $F$ is a \emph{Casimir}, which is the |
| 1173 |
+ |
conserved quantity in Poisson system. We can easily verify that the |
| 1174 |
+ |
norm of the angular momentum, $\parallel \pi |
| 1175 |
+ |
\parallel$, is a \emph{Casimir}. Let$ F(\pi ) = S(\frac{{\parallel |
| 1176 |
+ |
\pi \parallel ^2 }}{2})$ for an arbitrary function $ S:R \to R$ , |
| 1177 |
+ |
then by the chain rule |
| 1178 |
+ |
\[ |
| 1179 |
+ |
\nabla _\pi F(\pi ) = S'(\frac{{\parallel \pi \parallel ^2 |
| 1180 |
+ |
}}{2})\pi |
| 1181 |
+ |
\] |
| 1182 |
+ |
Thus $ [\nabla _\pi F(\pi )]^T J(\pi ) = - S'(\frac{{\parallel \pi |
| 1183 |
+ |
\parallel ^2 }}{2})\pi \times \pi = 0 $. This explicit |
| 1184 |
+ |
Lie-Poisson integrator is found to be extremely efficient and stable |
| 1185 |
+ |
which can be explained by the fact the small angle approximation is |
| 1186 |
+ |
used and the norm of the angular momentum is conserved. |
| 1187 |
+ |
|
| 1188 |
+ |
\subsection{\label{introSection:RBHamiltonianSplitting} Hamiltonian |
| 1189 |
+ |
Splitting for Rigid Body} |
| 1190 |
+ |
|
| 1191 |
+ |
The Hamiltonian of rigid body can be separated in terms of kinetic |
| 1192 |
+ |
energy and potential energy, |
| 1193 |
+ |
\[ |
| 1194 |
+ |
H = T(p,\pi ) + V(q,Q) |
| 1195 |
+ |
\] |
| 1196 |
+ |
The equations of motion corresponding to potential energy and |
| 1197 |
+ |
kinetic energy are listed in the below table, |
| 1198 |
+ |
\begin{center} |
| 1199 |
+ |
\begin{tabular}{|l|l|} |
| 1200 |
+ |
\hline |
| 1201 |
+ |
% after \\: \hline or \cline{col1-col2} \cline{col3-col4} ... |
| 1202 |
+ |
Potential & Kinetic \\ |
| 1203 |
+ |
$\frac{{dq}}{{dt}} = \frac{p}{m}$ & $\frac{d}{{dt}}q = p$ \\ |
| 1204 |
+ |
$\frac{d}{{dt}}p = - \frac{{\partial V}}{{\partial q}}$ & $ \frac{d}{{dt}}p = 0$ \\ |
| 1205 |
+ |
$\frac{d}{{dt}}Q = 0$ & $ \frac{d}{{dt}}Q = Qskew(I^{ - 1} j)$ \\ |
| 1206 |
+ |
$ \frac{d}{{dt}}\pi = \sum\limits_i {\left( {Q^T F_i (r,Q)} \right) \times X_i }$ & $\frac{d}{{dt}}\pi = \pi \times I^{ - 1} \pi$\\ |
| 1207 |
+ |
\hline |
| 1208 |
+ |
\end{tabular} |
| 1209 |
+ |
\end{center} |
| 1210 |
+ |
A second-order symplectic method is now obtained by the composition |
| 1211 |
+ |
of the flow maps, |
| 1212 |
+ |
\[ |
| 1213 |
+ |
\varphi _{\Delta t} = \varphi _{\Delta t/2,V} \circ \varphi |
| 1214 |
+ |
_{\Delta t,T} \circ \varphi _{\Delta t/2,V}. |
| 1215 |
+ |
\] |
| 1216 |
+ |
Moreover, \varphi _{\Delta t/2,V} can be divided into two sub-flows |
| 1217 |
+ |
which corresponding to force and torque respectively, |
| 1218 |
+ |
\[ |
| 1219 |
+ |
\varphi _{\Delta t/2,V} = \varphi _{\Delta t/2,F} \circ \varphi |
| 1220 |
+ |
_{\Delta t/2,\tau }. |
| 1221 |
+ |
\] |
| 1222 |
+ |
Since the associated operators of $\varphi _{\Delta t/2,F} $ and |
| 1223 |
+ |
$\circ \varphi _{\Delta t/2,\tau }$ are commuted, the composition |
| 1224 |
+ |
order inside \varphi _{\Delta t/2,V} does not matter. |
| 1225 |
+ |
|
| 1226 |
+ |
Furthermore, kinetic potential can be separated to translational |
| 1227 |
+ |
kinetic term, $T^t (p)$, and rotational kinetic term, $T^r (\pi )$, |
| 1228 |
+ |
\begin{equation} |
| 1229 |
+ |
T(p,\pi ) =T^t (p) + T^r (\pi ). |
| 1230 |
+ |
\end{equation} |
| 1231 |
+ |
where $ T^t (p) = \frac{1}{2}p^T m^{ - 1} p $ and $T^r (\pi )$ is |
| 1232 |
+ |
defined by \ref{introEquation:rotationalKineticRB}. Therefore, the |
| 1233 |
+ |
corresponding flow maps are given by |
| 1234 |
+ |
\[ |
| 1235 |
+ |
\varphi _{\Delta t,T} = \varphi _{\Delta t,T^t } \circ \varphi |
| 1236 |
+ |
_{\Delta t,T^r }. |
| 1237 |
+ |
\] |
| 1238 |
+ |
Finally, we obtain the overall symplectic flow maps for free moving |
| 1239 |
+ |
rigid body |
| 1240 |
+ |
\begin{equation} |
| 1241 |
+ |
\begin{array}{c} |
| 1242 |
+ |
\varphi _{\Delta t} = \varphi _{\Delta t/2,F} \circ \varphi _{\Delta t/2,\tau } \\ |
| 1243 |
+ |
\circ \varphi _{\Delta t,T^t } \circ \varphi _{\Delta t/2,\pi _1 } \circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t,\pi _3 } \circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi _1 } \\ |
| 1244 |
+ |
\circ \varphi _{\Delta t/2,\tau } \circ \varphi _{\Delta t/2,F} .\\ |
| 1245 |
+ |
\end{array} |
| 1246 |
+ |
\label{introEquation:overallRBFlowMaps} |
| 1247 |
+ |
\end{equation} |
| 1248 |
+ |
|
| 1249 |
+ |
\section{\label{introSection:langevinDynamics}Langevin Dynamics} |
| 1250 |
+ |
As an alternative to newtonian dynamics, Langevin dynamics, which |
| 1251 |
+ |
mimics a simple heat bath with stochastic and dissipative forces, |
| 1252 |
+ |
has been applied in a variety of studies. This section will review |
| 1253 |
+ |
the theory of Langevin dynamics simulation. A brief derivation of |
| 1254 |
+ |
generalized Langevin Dynamics will be given first. Follow that, we |
| 1255 |
+ |
will discuss the physical meaning of the terms appearing in the |
| 1256 |
+ |
equation as well as the calculation of friction tensor from |
| 1257 |
+ |
hydrodynamics theory. |
| 1258 |
+ |
|
| 1259 |
|
\subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics} |
| 1260 |
|
|
| 1261 |
|
\begin{equation} |
| 1448 |
|
\label{introEquation:secondFluctuationDissipation} |
| 1449 |
|
\end{equation} |
| 1450 |
|
|
| 1201 |
– |
\section{\label{introSection:hydroynamics}Hydrodynamics} |
| 1202 |
– |
|
| 1451 |
|
\subsection{\label{introSection:frictionTensor} Friction Tensor} |
| 1452 |
< |
\subsection{\label{introSection:analyticalApproach}Analytical |
| 1453 |
< |
Approach} |
| 1452 |
> |
Theoretically, the friction kernel can be determined using velocity |
| 1453 |
> |
autocorrelation function. However, this approach become impractical |
| 1454 |
> |
when the system become more and more complicate. Instead, various |
| 1455 |
> |
approaches based on hydrodynamics have been developed to calculate |
| 1456 |
> |
the friction coefficients. The friction effect is isotropic in |
| 1457 |
> |
Equation, \zeta can be taken as a scalar. In general, friction |
| 1458 |
> |
tensor \Xi is a $6\times 6$ matrix given by |
| 1459 |
> |
\[ |
| 1460 |
> |
\Xi = \left( {\begin{array}{*{20}c} |
| 1461 |
> |
{\Xi _{}^{tt} } & {\Xi _{}^{rt} } \\ |
| 1462 |
> |
{\Xi _{}^{tr} } & {\Xi _{}^{rr} } \\ |
| 1463 |
> |
\end{array}} \right). |
| 1464 |
> |
\] |
| 1465 |
> |
Here, $ {\Xi^{tt} }$ and $ {\Xi^{rr} }$ are translational friction |
| 1466 |
> |
tensor and rotational resistance (friction) tensor respectively, |
| 1467 |
> |
while ${\Xi^{tr} }$ is translation-rotation coupling tensor and $ |
| 1468 |
> |
{\Xi^{rt} }$ is rotation-translation coupling tensor. When a |
| 1469 |
> |
particle moves in a fluid, it may experience friction force or |
| 1470 |
> |
torque along the opposite direction of the velocity or angular |
| 1471 |
> |
velocity, |
| 1472 |
> |
\[ |
| 1473 |
> |
\left( \begin{array}{l} |
| 1474 |
> |
F_R \\ |
| 1475 |
> |
\tau _R \\ |
| 1476 |
> |
\end{array} \right) = - \left( {\begin{array}{*{20}c} |
| 1477 |
> |
{\Xi ^{tt} } & {\Xi ^{rt} } \\ |
| 1478 |
> |
{\Xi ^{tr} } & {\Xi ^{rr} } \\ |
| 1479 |
> |
\end{array}} \right)\left( \begin{array}{l} |
| 1480 |
> |
v \\ |
| 1481 |
> |
w \\ |
| 1482 |
> |
\end{array} \right) |
| 1483 |
> |
\] |
| 1484 |
> |
where $F_r$ is the friction force and $\tau _R$ is the friction |
| 1485 |
> |
toque. |
| 1486 |
|
|
| 1487 |
< |
\subsection{\label{introSection:approximationApproach}Approximation |
| 1208 |
< |
Approach} |
| 1487 |
> |
\subsubsection{\label{introSection:resistanceTensorRegular}The Resistance Tensor for Regular Shape} |
| 1488 |
|
|
| 1489 |
< |
\subsection{\label{introSection:centersRigidBody}Centers of Rigid |
| 1490 |
< |
Body} |
| 1489 |
> |
For a spherical particle, the translational and rotational friction |
| 1490 |
> |
constant can be calculated from Stoke's law, |
| 1491 |
> |
\[ |
| 1492 |
> |
\Xi ^{tt} = \left( {\begin{array}{*{20}c} |
| 1493 |
> |
{6\pi \eta R} & 0 & 0 \\ |
| 1494 |
> |
0 & {6\pi \eta R} & 0 \\ |
| 1495 |
> |
0 & 0 & {6\pi \eta R} \\ |
| 1496 |
> |
\end{array}} \right) |
| 1497 |
> |
\] |
| 1498 |
> |
and |
| 1499 |
> |
\[ |
| 1500 |
> |
\Xi ^{rr} = \left( {\begin{array}{*{20}c} |
| 1501 |
> |
{8\pi \eta R^3 } & 0 & 0 \\ |
| 1502 |
> |
0 & {8\pi \eta R^3 } & 0 \\ |
| 1503 |
> |
0 & 0 & {8\pi \eta R^3 } \\ |
| 1504 |
> |
\end{array}} \right) |
| 1505 |
> |
\] |
| 1506 |
> |
where $\eta$ is the viscosity of the solvent and $R$ is the |
| 1507 |
> |
hydrodynamics radius. |
| 1508 |
|
|
| 1509 |
< |
\section{\label{introSection:correlationFunctions}Correlation Functions} |
| 1509 |
> |
Other non-spherical shape, such as cylinder and ellipsoid |
| 1510 |
> |
\textit{etc}, are widely used as reference for developing new |
| 1511 |
> |
hydrodynamics theory, because their properties can be calculated |
| 1512 |
> |
exactly. In 1936, Perrin extended Stokes's law to general ellipsoid, |
| 1513 |
> |
also called a triaxial ellipsoid, which is given in Cartesian |
| 1514 |
> |
coordinates by |
| 1515 |
> |
\[ |
| 1516 |
> |
\frac{{x^2 }}{{a^2 }} + \frac{{y^2 }}{{b^2 }} + \frac{{z^2 }}{{c^2 |
| 1517 |
> |
}} = 1 |
| 1518 |
> |
\] |
| 1519 |
> |
where the semi-axes are of lengths $a$, $b$, and $c$. Unfortunately, |
| 1520 |
> |
due to the complexity of the elliptic integral, only the ellipsoid |
| 1521 |
> |
with the restriction of two axes having to be equal, \textit{i.e.} |
| 1522 |
> |
prolate($ a \ge b = c$) and oblate ($ a < b = c $), can be solved |
| 1523 |
> |
exactly. Introducing an elliptic integral parameter $S$ for prolate, |
| 1524 |
> |
\[ |
| 1525 |
> |
S = \frac{2}{{\sqrt {a^2 - b^2 } }}\ln \frac{{a + \sqrt {a^2 - b^2 |
| 1526 |
> |
} }}{b}, |
| 1527 |
> |
\] |
| 1528 |
> |
and oblate, |
| 1529 |
> |
\[ |
| 1530 |
> |
S = \frac{2}{{\sqrt {b^2 - a^2 } }}arctg\frac{{\sqrt {b^2 - a^2 } |
| 1531 |
> |
}}{a} |
| 1532 |
> |
\], |
| 1533 |
> |
one can write down the translational and rotational resistance |
| 1534 |
> |
tensors |
| 1535 |
> |
\[ |
| 1536 |
> |
\begin{array}{l} |
| 1537 |
> |
\Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}} \\ |
| 1538 |
> |
\Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + 2a}} \\ |
| 1539 |
> |
\end{array}, |
| 1540 |
> |
\] |
| 1541 |
> |
and |
| 1542 |
> |
\[ |
| 1543 |
> |
\begin{array}{l} |
| 1544 |
> |
\Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}} \\ |
| 1545 |
> |
\Xi _b^{rr} = \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}} \\ |
| 1546 |
> |
\end{array}. |
| 1547 |
> |
\] |
| 1548 |
> |
|
| 1549 |
> |
\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}The Resistance Tensor for Arbitrary Shape} |
| 1550 |
> |
|
| 1551 |
> |
Unlike spherical and other regular shaped molecules, there is not |
| 1552 |
> |
analytical solution for friction tensor of any arbitrary shaped |
| 1553 |
> |
rigid molecules. The ellipsoid of revolution model and general |
| 1554 |
> |
triaxial ellipsoid model have been used to approximate the |
| 1555 |
> |
hydrodynamic properties of rigid bodies. However, since the mapping |
| 1556 |
> |
from all possible ellipsoidal space, $r$-space, to all possible |
| 1557 |
> |
combination of rotational diffusion coefficients, $D$-space is not |
| 1558 |
> |
unique\cite{Wegener79} as well as the intrinsic coupling between |
| 1559 |
> |
translational and rotational motion of rigid body\cite{}, general |
| 1560 |
> |
ellipsoid is not always suitable for modeling arbitrarily shaped |
| 1561 |
> |
rigid molecule. A number of studies have been devoted to determine |
| 1562 |
> |
the friction tensor for irregularly shaped rigid bodies using more |
| 1563 |
> |
advanced method\cite{} where the molecule of interest was modeled by |
| 1564 |
> |
combinations of spheres(beads)\cite{} and the hydrodynamics |
| 1565 |
> |
properties of the molecule can be calculated using the hydrodynamic |
| 1566 |
> |
interaction tensor. Let us consider a rigid assembly of $N$ beads |
| 1567 |
> |
immersed in a continuous medium. Due to hydrodynamics interaction, |
| 1568 |
> |
the ``net'' velocity of $i$th bead, $v'_i$ is different than its |
| 1569 |
> |
unperturbed velocity $v_i$, |
| 1570 |
> |
\[ |
| 1571 |
> |
v'_i = v_i - \sum\limits_{j \ne i} {T_{ij} F_j } |
| 1572 |
> |
\] |
| 1573 |
> |
where $F_i$ is the frictional force, and $T_{ij}$ is the |
| 1574 |
> |
hydrodynamic interaction tensor. The friction force of $i$th bead is |
| 1575 |
> |
proportional to its ``net'' velocity |
| 1576 |
> |
\begin{equation} |
| 1577 |
> |
F_i = \zeta _i v_i - \zeta _i \sum\limits_{j \ne i} {T_{ij} F_j }. |
| 1578 |
> |
\label{introEquation:tensorExpression} |
| 1579 |
> |
\end{equation} |
| 1580 |
> |
This equation is the basis for deriving the hydrodynamic tensor. In |
| 1581 |
> |
1930, Oseen and Burgers gave a simple solution to Equation |
| 1582 |
> |
\ref{introEquation:tensorExpression} |
| 1583 |
> |
\begin{equation} |
| 1584 |
> |
T_{ij} = \frac{1}{{8\pi \eta r_{ij} }}\left( {I + \frac{{R_{ij} |
| 1585 |
> |
R_{ij}^T }}{{R_{ij}^2 }}} \right). |
| 1586 |
> |
\label{introEquation:oseenTensor} |
| 1587 |
> |
\end{equation} |
| 1588 |
> |
Here $R_{ij}$ is the distance vector between bead $i$ and bead $j$. |
| 1589 |
> |
A second order expression for element of different size was |
| 1590 |
> |
introduced by Rotne and Prager\cite{} and improved by Garc\'{i}a de |
| 1591 |
> |
la Torre and Bloomfield, |
| 1592 |
> |
\begin{equation} |
| 1593 |
> |
T_{ij} = \frac{1}{{8\pi \eta R_{ij} }}\left[ {\left( {I + |
| 1594 |
> |
\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right) + R\frac{{\sigma |
| 1595 |
> |
_i^2 + \sigma _j^2 }}{{r_{ij}^2 }}\left( {\frac{I}{3} - |
| 1596 |
> |
\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right)} \right]. |
| 1597 |
> |
\label{introEquation:RPTensorNonOverlapped} |
| 1598 |
> |
\end{equation} |
| 1599 |
> |
Both of the Equation \ref{introEquation:oseenTensor} and Equation |
| 1600 |
> |
\ref{introEquation:RPTensorNonOverlapped} have an assumption $R_{ij} |
| 1601 |
> |
\ge \sigma _i + \sigma _j$. An alternative expression for |
| 1602 |
> |
overlapping beads with the same radius, $\sigma$, is given by |
| 1603 |
> |
\begin{equation} |
| 1604 |
> |
T_{ij} = \frac{1}{{6\pi \eta R_{ij} }}\left[ {\left( {1 - |
| 1605 |
> |
\frac{2}{{32}}\frac{{R_{ij} }}{\sigma }} \right)I + |
| 1606 |
> |
\frac{2}{{32}}\frac{{R_{ij} R_{ij}^T }}{{R_{ij} \sigma }}} \right] |
| 1607 |
> |
\label{introEquation:RPTensorOverlapped} |
| 1608 |
> |
\end{equation} |
| 1609 |
> |
|
| 1610 |
> |
To calculate the resistance tensor at an arbitrary origin $O$, we |
| 1611 |
> |
construct a $3N \times 3N$ matrix consisting of $N \times N$ |
| 1612 |
> |
$B_{ij}$ blocks |
| 1613 |
> |
\begin{equation} |
| 1614 |
> |
B = \left( {\begin{array}{*{20}c} |
| 1615 |
> |
{B_{11} } & \ldots & {B_{1N} } \\ |
| 1616 |
> |
\vdots & \ddots & \vdots \\ |
| 1617 |
> |
{B_{N1} } & \cdots & {B_{NN} } \\ |
| 1618 |
> |
\end{array}} \right), |
| 1619 |
> |
\end{equation} |
| 1620 |
> |
where $B_{ij}$ is given by |
| 1621 |
> |
\[ |
| 1622 |
> |
B_{ij} = \delta _{ij} \frac{I}{{6\pi \eta R}} + (1 - \delta _{ij} |
| 1623 |
> |
)T_{ij} |
| 1624 |
> |
\] |
| 1625 |
> |
where \delta _{ij} is Kronecker delta function. Inverting matrix |
| 1626 |
> |
$B$, we obtain |
| 1627 |
> |
|
| 1628 |
> |
\[ |
| 1629 |
> |
C = B^{ - 1} = \left( {\begin{array}{*{20}c} |
| 1630 |
> |
{C_{11} } & \ldots & {C_{1N} } \\ |
| 1631 |
> |
\vdots & \ddots & \vdots \\ |
| 1632 |
> |
{C_{N1} } & \cdots & {C_{NN} } \\ |
| 1633 |
> |
\end{array}} \right) |
| 1634 |
> |
\] |
| 1635 |
> |
, which can be partitioned into $N \times N$ $3 \times 3$ block |
| 1636 |
> |
$C_{ij}$. With the help of $C_{ij}$ and skew matrix $U_i$ |
| 1637 |
> |
\[ |
| 1638 |
> |
U_i = \left( {\begin{array}{*{20}c} |
| 1639 |
> |
0 & { - z_i } & {y_i } \\ |
| 1640 |
> |
{z_i } & 0 & { - x_i } \\ |
| 1641 |
> |
{ - y_i } & {x_i } & 0 \\ |
| 1642 |
> |
\end{array}} \right) |
| 1643 |
> |
\] |
| 1644 |
> |
where $x_i$, $y_i$, $z_i$ are the components of the vector joining |
| 1645 |
> |
bead $i$ and origin $O$. Hence, the elements of resistance tensor at |
| 1646 |
> |
arbitrary origin $O$ can be written as |
| 1647 |
> |
\begin{equation} |
| 1648 |
> |
\begin{array}{l} |
| 1649 |
> |
\Xi _{}^{tt} = \sum\limits_i {\sum\limits_j {C_{ij} } } , \\ |
| 1650 |
> |
\Xi _{}^{tr} = \Xi _{}^{rt} = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\ |
| 1651 |
> |
\Xi _{}^{rr} = - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j \\ |
| 1652 |
> |
\end{array} |
| 1653 |
> |
\label{introEquation:ResistanceTensorArbitraryOrigin} |
| 1654 |
> |
\end{equation} |
| 1655 |
> |
|
| 1656 |
> |
The resistance tensor depends on the origin to which they refer. The |
| 1657 |
> |
proper location for applying friction force is the center of |
| 1658 |
> |
resistance (reaction), at which the trace of rotational resistance |
| 1659 |
> |
tensor, $ \Xi ^{rr}$ reaches minimum. Mathematically, the center of |
| 1660 |
> |
resistance is defined as an unique point of the rigid body at which |
| 1661 |
> |
the translation-rotation coupling tensor are symmetric, |
| 1662 |
> |
\begin{equation} |
| 1663 |
> |
\Xi^{tr} = \left( {\Xi^{tr} } \right)^T |
| 1664 |
> |
\label{introEquation:definitionCR} |
| 1665 |
> |
\end{equation} |
| 1666 |
> |
Form Equation \ref{introEquation:ResistanceTensorArbitraryOrigin}, |
| 1667 |
> |
we can easily find out that the translational resistance tensor is |
| 1668 |
> |
origin independent, while the rotational resistance tensor and |
| 1669 |
> |
translation-rotation coupling resistance tensor do depend on the |
| 1670 |
> |
origin. Given resistance tensor at an arbitrary origin $O$, and a |
| 1671 |
> |
vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can |
| 1672 |
> |
obtain the resistance tensor at $P$ by |
| 1673 |
> |
\begin{equation} |
| 1674 |
> |
\begin{array}{l} |
| 1675 |
> |
\Xi _P^{tt} = \Xi _O^{tt} \\ |
| 1676 |
> |
\Xi _P^{tr} = \Xi _P^{rt} = \Xi _O^{tr} - U_{OP} \Xi _O^{tt} \\ |
| 1677 |
> |
\Xi _P^{rr} = \Xi _O^{rr} - U_{OP} \Xi _O^{tt} U_{OP} + \Xi _O^{tr} U_{OP} - U_{OP} \Xi _O^{tr} ^{^T } \\ |
| 1678 |
> |
\end{array} |
| 1679 |
> |
\label{introEquation:resistanceTensorTransformation} |
| 1680 |
> |
\end{equation} |
| 1681 |
> |
where |
| 1682 |
> |
\[ |
| 1683 |
> |
U_{OP} = \left( {\begin{array}{*{20}c} |
| 1684 |
> |
0 & { - z_{OP} } & {y_{OP} } \\ |
| 1685 |
> |
{z_i } & 0 & { - x_{OP} } \\ |
| 1686 |
> |
{ - y_{OP} } & {x_{OP} } & 0 \\ |
| 1687 |
> |
\end{array}} \right) |
| 1688 |
> |
\] |
| 1689 |
> |
Using Equations \ref{introEquation:definitionCR} and |
| 1690 |
> |
\ref{introEquation:resistanceTensorTransformation}, one can locate |
| 1691 |
> |
the position of center of resistance, |
| 1692 |
> |
\[ |
| 1693 |
> |
\left( \begin{array}{l} |
| 1694 |
> |
x_{OR} \\ |
| 1695 |
> |
y_{OR} \\ |
| 1696 |
> |
z_{OR} \\ |
| 1697 |
> |
\end{array} \right) = \left( {\begin{array}{*{20}c} |
| 1698 |
> |
{(\Xi _O^{rr} )_{yy} + (\Xi _O^{rr} )_{zz} } & { - (\Xi _O^{rr} )_{xy} } & { - (\Xi _O^{rr} )_{xz} } \\ |
| 1699 |
> |
{ - (\Xi _O^{rr} )_{xy} } & {(\Xi _O^{rr} )_{zz} + (\Xi _O^{rr} )_{xx} } & { - (\Xi _O^{rr} )_{yz} } \\ |
| 1700 |
> |
{ - (\Xi _O^{rr} )_{xz} } & { - (\Xi _O^{rr} )_{yz} } & {(\Xi _O^{rr} )_{xx} + (\Xi _O^{rr} )_{yy} } \\ |
| 1701 |
> |
\end{array}} \right)^{ - 1} \left( \begin{array}{l} |
| 1702 |
> |
(\Xi _O^{tr} )_{yz} - (\Xi _O^{tr} )_{zy} \\ |
| 1703 |
> |
(\Xi _O^{tr} )_{zx} - (\Xi _O^{tr} )_{xz} \\ |
| 1704 |
> |
(\Xi _O^{tr} )_{xy} - (\Xi _O^{tr} )_{yx} \\ |
| 1705 |
> |
\end{array} \right). |
| 1706 |
> |
\] |
| 1707 |
> |
where $x_OR$, $y_OR$, $z_OR$ are the components of the vector |
| 1708 |
> |
joining center of resistance $R$ and origin $O$. |
| 1709 |
> |
|
| 1710 |
> |
%\section{\label{introSection:correlationFunctions}Correlation Functions} |
