| 73 |
|
\subsection{\label{introSection:lagrangian}Lagrangian Mechanics} |
| 74 |
|
|
| 75 |
|
Newtonian Mechanics suffers from two important limitations: motions |
| 76 |
< |
can only be described in cartesian coordinate systems. Moreover, It |
| 77 |
< |
become impossible to predict analytically the properties of the |
| 76 |
> |
can only be described in cartesian coordinate systems. Moreover, it |
| 77 |
> |
becomes impossible to predict analytically the properties of the |
| 78 |
|
system even if we know all of the details of the interaction. In |
| 79 |
|
order to overcome some of the practical difficulties which arise in |
| 80 |
|
attempts to apply Newton's equation to complex system, approximate |
| 227 |
|
momentum variables. Consider a dynamic system of $f$ particles in a |
| 228 |
|
cartesian space, where each of the $6f$ coordinates and momenta is |
| 229 |
|
assigned to one of $6f$ mutually orthogonal axes, the phase space of |
| 230 |
< |
this system is a $6f$ dimensional space. A point, $x = (q_1 , \ldots |
| 231 |
< |
,q_f ,p_1 , \ldots ,p_f )$, with a unique set of values of $6f$ |
| 232 |
< |
coordinates and momenta is a phase space vector. |
| 233 |
< |
|
| 230 |
> |
this system is a $6f$ dimensional space. A point, $x = (\rightarrow |
| 231 |
> |
q_1 , \ldots ,\rightarrow q_f ,\rightarrow p_1 , \ldots ,\rightarrow |
| 232 |
> |
p_f )$, with a unique set of values of $6f$ coordinates and momenta |
| 233 |
> |
is a phase space vector. |
| 234 |
|
%%%fix me |
| 235 |
< |
A microscopic state or microstate of a classical system is |
| 236 |
< |
specification of the complete phase space vector of a system at any |
| 237 |
< |
instant in time. An ensemble is defined as a collection of systems |
| 238 |
< |
sharing one or more macroscopic characteristics but each being in a |
| 239 |
< |
unique microstate. The complete ensemble is specified by giving all |
| 240 |
< |
systems or microstates consistent with the common macroscopic |
| 241 |
< |
characteristics of the ensemble. Although the state of each |
| 242 |
< |
individual system in the ensemble could be precisely described at |
| 243 |
< |
any instance in time by a suitable phase space vector, when using |
| 244 |
< |
ensembles for statistical purposes, there is no need to maintain |
| 245 |
< |
distinctions between individual systems, since the numbers of |
| 246 |
< |
systems at any time in the different states which correspond to |
| 247 |
< |
different regions of the phase space are more interesting. Moreover, |
| 248 |
< |
in the point of view of statistical mechanics, one would prefer to |
| 249 |
< |
use ensembles containing a large enough population of separate |
| 250 |
< |
members so that the numbers of systems in such different states can |
| 251 |
< |
be regarded as changing continuously as we traverse different |
| 252 |
< |
regions of the phase space. The condition of an ensemble at any time |
| 235 |
> |
|
| 236 |
> |
In statistical mechanics, the condition of an ensemble at any time |
| 237 |
|
can be regarded as appropriately specified by the density $\rho$ |
| 238 |
|
with which representative points are distributed over the phase |
| 239 |
|
space. The density distribution for an ensemble with $f$ degrees of |
| 736 |
|
\begin{equation} |
| 737 |
|
\varphi _h^{ - 1} = \varphi _{ - h}. |
| 738 |
|
\label{introEquation:timeReversible} |
| 739 |
< |
\end{equation},appendixFig:architecture |
| 739 |
> |
\end{equation} |
| 740 |
|
|
| 741 |
|
\subsubsection{\label{introSection:exampleSplittingMethod}\textbf{Examples of the Splitting Method}} |
| 742 |
|
The classical equation for a system consisting of interacting |
| 1098 |
|
(r)}{\rho}. |
| 1099 |
|
\] |
| 1100 |
|
Note that the delta function can be replaced by a histogram in |
| 1101 |
< |
computer simulation. Figure |
| 1102 |
< |
\ref{introFigure:pairDistributionFunction} shows a typical pair |
| 1103 |
< |
distribution function for the liquid argon system. The occurrence of |
| 1120 |
< |
several peaks in the plot of $g(r)$ suggests that it is more likely |
| 1121 |
< |
to find particles at certain radial values than at others. This is a |
| 1122 |
< |
result of the attractive interaction at such distances. Because of |
| 1123 |
< |
the strong repulsive forces at short distance, the probability of |
| 1124 |
< |
locating particles at distances less than about 3.7{\AA} from each |
| 1125 |
< |
other is essentially zero. |
| 1101 |
> |
computer simulation. Peaks in $g(r)$ represent solvent shells, and |
| 1102 |
> |
the height of these peaks gradually decreases to 1 as the liquid of |
| 1103 |
> |
large distance approaches the bulk density. |
| 1104 |
|
|
| 1127 |
– |
%\begin{figure} |
| 1128 |
– |
%\centering |
| 1129 |
– |
%\includegraphics[width=\linewidth]{pdf.eps} |
| 1130 |
– |
%\caption[Pair distribution function for the liquid argon |
| 1131 |
– |
%]{Pair distribution function for the liquid argon} |
| 1132 |
– |
%\label{introFigure:pairDistributionFunction} |
| 1133 |
– |
%\end{figure} |
| 1105 |
|
|
| 1106 |
|
\subsubsection{\label{introSection:timeDependentProperties}\textbf{Time-dependent |
| 1107 |
|
Properties}} |
| 1241 |
|
\] |
| 1242 |
|
|
| 1243 |
|
Unfortunately, this constraint manifold is not the cotangent bundle |
| 1244 |
< |
$T_{\star}SO(3)$. However, it turns out that under symplectic |
| 1244 |
> |
$T^* SO(3)$ which can be consider as a symplectic manifold on Lie |
| 1245 |
> |
rotation group $SO(3)$. However, it turns out that under symplectic |
| 1246 |
|
transformation, the cotangent space and the phase space are |
| 1247 |
|
diffeomorphic. By introducing |
| 1248 |
|
\[ |
| 1280 |
|
introduced to rewrite the equations of motion, |
| 1281 |
|
\begin{equation} |
| 1282 |
|
\begin{array}{l} |
| 1283 |
< |
\mathop \Pi \limits^ \bullet = J^{ - 1} \Pi ^T \Pi + Q^T \sum\limits_i {F_i (q,Q)X_i^T } - \Lambda \\ |
| 1284 |
< |
\mathop Q\limits^{{\rm{ }} \bullet } = Q\Pi {\rm{ }}J^{ - 1} \\ |
| 1283 |
> |
\dot \Pi = J^{ - 1} \Pi ^T \Pi + Q^T \sum\limits_i {F_i (q,Q)X_i^T } - \Lambda \\ |
| 1284 |
> |
\dot Q = Q\Pi {\rm{ }}J^{ - 1} \\ |
| 1285 |
|
\end{array} |
| 1286 |
|
\label{introEqaution:RBMotionPI} |
| 1287 |
|
\end{equation} |
| 1311 |
|
\] |
| 1312 |
|
Using \ref{introEqaution:RBMotionPI}, one can construct a skew |
| 1313 |
|
matrix, |
| 1314 |
< |
\begin{equation} |
| 1315 |
< |
(\mathop \Pi \limits^ \bullet - \mathop \Pi \limits^ {\bullet ^T} |
| 1316 |
< |
){\rm{ }} = {\rm{ }}(\Pi - \Pi ^T ){\rm{ }}(J^{ - 1} \Pi + \Pi J^{ |
| 1317 |
< |
- 1} ) + \sum\limits_i {[Q^T F_i (r,Q)X_i^T - X_i F_i (r,Q)^T Q]} - |
| 1318 |
< |
(\Lambda - \Lambda ^T ) . \label{introEquation:skewMatrixPI} |
| 1319 |
< |
\end{equation} |
| 1314 |
> |
|
| 1315 |
> |
\begin{eqnarray*} |
| 1316 |
> |
(\dot \Pi - \dot \Pi ^T ){\rm{ }} = {\rm{ }}(\Pi - \Pi ^T ){\rm{ |
| 1317 |
> |
}}(J^{ - 1} \Pi + \Pi J^{ - 1} ) + \sum\limits_i {[Q^T F_i |
| 1318 |
> |
(r,Q)X_i^T - X_i F_i (r,Q)^T Q]} - (\Lambda - \Lambda ^T ). |
| 1319 |
> |
\label{introEquation:skewMatrixPI} |
| 1320 |
> |
\end{eqnarray*} |
| 1321 |
> |
|
| 1322 |
|
Since $\Lambda$ is symmetric, the last term of Equation |
| 1323 |
|
\ref{introEquation:skewMatrixPI} is zero, which implies the Lagrange |
| 1324 |
|
multiplier $\Lambda$ is absent from the equations of motion. This |
| 1450 |
|
The equations of motion corresponding to potential energy and |
| 1451 |
|
kinetic energy are listed in the below table, |
| 1452 |
|
\begin{table} |
| 1453 |
< |
\caption{Equations of motion due to Potential and Kinetic Energies} |
| 1453 |
> |
\caption{EQUATIONS OF MOTION DUE TO POTENTIAL AND KINETIC ENERGIES} |
| 1454 |
|
\begin{center} |
| 1455 |
|
\begin{tabular}{|l|l|} |
| 1456 |
|
\hline |
