31 |
|
$F_{ji}$ be the force that particle $j$ exerts on particle $i$. |
32 |
|
Newton's third law states that |
33 |
|
\begin{equation} |
34 |
< |
F_{ij} = -F_{ji} |
34 |
> |
F_{ij} = -F_{ji}. |
35 |
|
\label{introEquation:newtonThirdLaw} |
36 |
|
\end{equation} |
37 |
– |
|
37 |
|
Conservation laws of Newtonian Mechanics play very important roles |
38 |
|
in solving mechanics problems. The linear momentum of a particle is |
39 |
|
conserved if it is free or it experiences no force. The second |
72 |
|
\subsection{\label{introSection:lagrangian}Lagrangian Mechanics} |
73 |
|
|
74 |
|
Newtonian Mechanics suffers from two important limitations: motions |
75 |
< |
can only be described in cartesian coordinate systems. Moreover, It |
76 |
< |
become impossible to predict analytically the properties of the |
75 |
> |
can only be described in cartesian coordinate systems. Moreover, it |
76 |
> |
becomes impossible to predict analytically the properties of the |
77 |
|
system even if we know all of the details of the interaction. In |
78 |
|
order to overcome some of the practical difficulties which arise in |
79 |
|
attempts to apply Newton's equation to complex system, approximate |
84 |
|
|
85 |
|
Hamilton introduced the dynamical principle upon which it is |
86 |
|
possible to base all of mechanics and most of classical physics. |
87 |
< |
Hamilton's Principle may be stated as follows, |
88 |
< |
|
89 |
< |
The actual trajectory, along which a dynamical system may move from |
90 |
< |
one point to another within a specified time, is derived by finding |
91 |
< |
the path which minimizes the time integral of the difference between |
93 |
< |
the kinetic, $K$, and potential energies, $U$. |
87 |
> |
Hamilton's Principle may be stated as follows: the actual |
88 |
> |
trajectory, along which a dynamical system may move from one point |
89 |
> |
to another within a specified time, is derived by finding the path |
90 |
> |
which minimizes the time integral of the difference between the |
91 |
> |
kinetic, $K$, and potential energies, $U$. |
92 |
|
\begin{equation} |
93 |
|
\delta \int_{t_1 }^{t_2 } {(K - U)dt = 0} , |
94 |
|
\label{introEquation:halmitonianPrinciple1} |
95 |
|
\end{equation} |
98 |
– |
|
96 |
|
For simple mechanical systems, where the forces acting on the |
97 |
|
different parts are derivable from a potential, the Lagrangian |
98 |
|
function $L$ can be defined as the difference between the kinetic |
135 |
|
p_i = \frac{{\partial L}}{{\partial q_i }} |
136 |
|
\label{introEquation:generalizedMomentaDot} |
137 |
|
\end{equation} |
141 |
– |
|
138 |
|
With the help of the generalized momenta, we may now define a new |
139 |
|
quantity $H$ by the equation |
140 |
|
\begin{equation} |
142 |
|
\label{introEquation:hamiltonianDefByLagrangian} |
143 |
|
\end{equation} |
144 |
|
where $ \dot q_1 \ldots \dot q_f $ are generalized velocities and |
145 |
< |
$L$ is the Lagrangian function for the system. |
146 |
< |
|
151 |
< |
Differentiating Eq.~\ref{introEquation:hamiltonianDefByLagrangian}, |
152 |
< |
one can obtain |
145 |
> |
$L$ is the Lagrangian function for the system. Differentiating |
146 |
> |
Eq.~\ref{introEquation:hamiltonianDefByLagrangian}, one can obtain |
147 |
|
\begin{equation} |
148 |
|
dH = \sum\limits_k {\left( {p_k d\dot q_k + \dot q_k dp_k - |
149 |
|
\frac{{\partial L}}{{\partial q_k }}dq_k - \frac{{\partial |
174 |
|
t}} |
175 |
|
\label{introEquation:motionHamiltonianTime} |
176 |
|
\end{equation} |
183 |
– |
|
177 |
|
Eq.~\ref{introEquation:motionHamiltonianCoordinate} and |
178 |
|
Eq.~\ref{introEquation:motionHamiltonianMomentum} are Hamilton's |
179 |
|
equation of motion. Due to their symmetrical formula, they are also |
220 |
|
momentum variables. Consider a dynamic system of $f$ particles in a |
221 |
|
cartesian space, where each of the $6f$ coordinates and momenta is |
222 |
|
assigned to one of $6f$ mutually orthogonal axes, the phase space of |
223 |
< |
this system is a $6f$ dimensional space. A point, $x = (q_1 , \ldots |
224 |
< |
,q_f ,p_1 , \ldots ,p_f )$, with a unique set of values of $6f$ |
225 |
< |
coordinates and momenta is a phase space vector. |
226 |
< |
|
223 |
> |
this system is a $6f$ dimensional space. A point, $x = (\rightarrow |
224 |
> |
q_1 , \ldots ,\rightarrow q_f ,\rightarrow p_1 , \ldots ,\rightarrow |
225 |
> |
p_f )$, with a unique set of values of $6f$ coordinates and momenta |
226 |
> |
is a phase space vector. |
227 |
|
%%%fix me |
228 |
< |
A microscopic state or microstate of a classical system is |
229 |
< |
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 |
228 |
> |
|
229 |
> |
In statistical mechanics, the condition of an ensemble at any time |
230 |
|
can be regarded as appropriately specified by the density $\rho$ |
231 |
|
with which representative points are distributed over the phase |
232 |
|
space. The density distribution for an ensemble with $f$ degrees of |
281 |
|
statistical characteristics. As a function of macroscopic |
282 |
|
parameters, such as temperature \textit{etc}, the partition function |
283 |
|
can be used to describe the statistical properties of a system in |
284 |
< |
thermodynamic equilibrium. |
285 |
< |
|
286 |
< |
As an ensemble of systems, each of which is known to be thermally |
310 |
< |
isolated and conserve energy, the Microcanonical ensemble (NVE) has |
311 |
< |
a partition function like, |
284 |
> |
thermodynamic equilibrium. As an ensemble of systems, each of which |
285 |
> |
is known to be thermally isolated and conserve energy, the |
286 |
> |
Microcanonical ensemble (NVE) has a partition function like, |
287 |
|
\begin{equation} |
288 |
|
\Omega (N,V,E) = e^{\beta TS} \label{introEquation:NVEPartition}. |
289 |
|
\end{equation} |
578 |
|
The hidden geometric properties\cite{Budd1999, Marsden1998} of an |
579 |
|
ODE and its flow play important roles in numerical studies. Many of |
580 |
|
them can be found in systems which occur naturally in applications. |
606 |
– |
|
581 |
|
Let $\varphi$ be the flow of Hamiltonian vector field, $\varphi$ is |
582 |
|
a \emph{symplectic} flow if it satisfies, |
583 |
|
\begin{equation} |
591 |
|
\begin{equation} |
592 |
|
{\varphi '}^T J \varphi ' = J \circ \varphi |
593 |
|
\end{equation} |
594 |
< |
is the property that must be preserved by the integrator. |
595 |
< |
|
596 |
< |
It is possible to construct a \emph{volume-preserving} flow for a |
597 |
< |
source free ODE ($ \nabla \cdot f = 0 $), if the flow satisfies $ |
598 |
< |
\det d\varphi = 1$. One can show easily that a symplectic flow will |
625 |
< |
be volume-preserving. |
626 |
< |
|
627 |
< |
Changing the variables $y = h(x)$ in an ODE |
594 |
> |
is the property that must be preserved by the integrator. It is |
595 |
> |
possible to construct a \emph{volume-preserving} flow for a source |
596 |
> |
free ODE ($ \nabla \cdot f = 0 $), if the flow satisfies $ \det |
597 |
> |
d\varphi = 1$. One can show easily that a symplectic flow will be |
598 |
> |
volume-preserving. Changing the variables $y = h(x)$ in an ODE |
599 |
|
(Eq.~\ref{introEquation:ODE}) will result in a new system, |
600 |
|
\[ |
601 |
|
\dot y = \tilde f(y) = ((dh \cdot f)h^{ - 1} )(y). |
602 |
|
\] |
603 |
|
The vector filed $f$ has reversing symmetry $h$ if $f = - \tilde f$. |
604 |
|
In other words, the flow of this vector field is reversible if and |
605 |
< |
only if $ h \circ \varphi ^{ - 1} = \varphi \circ h $. |
606 |
< |
|
636 |
< |
A \emph{first integral}, or conserved quantity of a general |
605 |
> |
only if $ h \circ \varphi ^{ - 1} = \varphi \circ h $. A |
606 |
> |
\emph{first integral}, or conserved quantity of a general |
607 |
|
differential function is a function $ G:R^{2d} \to R^d $ which is |
608 |
|
constant for all solutions of the ODE $\frac{{dx}}{{dt}} = f(x)$ , |
609 |
|
\[ |
616 |
|
which is the condition for conserving \emph{first integral}. For a |
617 |
|
canonical Hamiltonian system, the time evolution of an arbitrary |
618 |
|
smooth function $G$ is given by, |
649 |
– |
|
619 |
|
\begin{eqnarray} |
620 |
|
\frac{{dG(x(t))}}{{dt}} & = & [\nabla _x G(x(t))]^T \dot x(t) \\ |
621 |
|
& = & [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). \\ |
622 |
|
\label{introEquation:firstIntegral1} |
623 |
|
\end{eqnarray} |
655 |
– |
|
656 |
– |
|
624 |
|
Using poisson bracket notion, Equation |
625 |
|
\ref{introEquation:firstIntegral1} can be rewritten as |
626 |
|
\[ |
633 |
|
\] |
634 |
|
As well known, the Hamiltonian (or energy) H of a Hamiltonian system |
635 |
|
is a \emph{first integral}, which is due to the fact $\{ H,H\} = |
636 |
< |
0$. |
670 |
< |
|
671 |
< |
When designing any numerical methods, one should always try to |
636 |
> |
0$. When designing any numerical methods, one should always try to |
637 |
|
preserve the structural properties of the original ODE and its flow. |
638 |
|
|
639 |
|
\subsection{\label{introSection:constructionSymplectic}Construction of Symplectic Methods} |
674 |
|
\label{introEquation:FlowDecomposition} |
675 |
|
\end{equation} |
676 |
|
where each of the sub-flow is chosen such that each represent a |
677 |
< |
simpler integration of the system. |
678 |
< |
|
714 |
< |
Suppose that a Hamiltonian system takes the form, |
677 |
> |
simpler integration of the system. Suppose that a Hamiltonian system |
678 |
> |
takes the form, |
679 |
|
\[ |
680 |
|
H = H_1 + H_2. |
681 |
|
\] |
716 |
|
\begin{equation} |
717 |
|
\varphi _h^{ - 1} = \varphi _{ - h}. |
718 |
|
\label{introEquation:timeReversible} |
719 |
< |
\end{equation},appendixFig:architecture |
719 |
> |
\end{equation} |
720 |
|
|
721 |
|
\subsubsection{\label{introSection:exampleSplittingMethod}\textbf{Examples of the Splitting Method}} |
722 |
|
The classical equation for a system consisting of interacting |
1221 |
|
\] |
1222 |
|
|
1223 |
|
Unfortunately, this constraint manifold is not the cotangent bundle |
1224 |
< |
$T_{\star}SO(3)$. However, it turns out that under symplectic |
1224 |
> |
$T^* SO(3)$ which can be consider as a symplectic manifold on Lie |
1225 |
> |
rotation group $SO(3)$. However, it turns out that under symplectic |
1226 |
|
transformation, the cotangent space and the phase space are |
1227 |
|
diffeomorphic. By introducing |
1228 |
|
\[ |
1260 |
|
introduced to rewrite the equations of motion, |
1261 |
|
\begin{equation} |
1262 |
|
\begin{array}{l} |
1263 |
< |
\mathop \Pi \limits^ \bullet = J^{ - 1} \Pi ^T \Pi + Q^T \sum\limits_i {F_i (q,Q)X_i^T } - \Lambda \\ |
1264 |
< |
\mathop Q\limits^{{\rm{ }} \bullet } = Q\Pi {\rm{ }}J^{ - 1} \\ |
1263 |
> |
\dot \Pi = J^{ - 1} \Pi ^T \Pi + Q^T \sum\limits_i {F_i (q,Q)X_i^T } - \Lambda \\ |
1264 |
> |
\dot Q = Q\Pi {\rm{ }}J^{ - 1} \\ |
1265 |
|
\end{array} |
1266 |
|
\label{introEqaution:RBMotionPI} |
1267 |
|
\end{equation} |
1291 |
|
\] |
1292 |
|
Using \ref{introEqaution:RBMotionPI}, one can construct a skew |
1293 |
|
matrix, |
1294 |
< |
\begin{equation} |
1295 |
< |
(\mathop \Pi \limits^ \bullet - \mathop \Pi \limits^ {\bullet ^T} |
1296 |
< |
){\rm{ }} = {\rm{ }}(\Pi - \Pi ^T ){\rm{ }}(J^{ - 1} \Pi + \Pi J^{ |
1297 |
< |
- 1} ) + \sum\limits_i {[Q^T F_i (r,Q)X_i^T - X_i F_i (r,Q)^T Q]} - |
1298 |
< |
(\Lambda - \Lambda ^T ) . \label{introEquation:skewMatrixPI} |
1299 |
< |
\end{equation} |
1294 |
> |
|
1295 |
> |
\begin{eqnarray*} |
1296 |
> |
(\dot \Pi - \dot \Pi ^T ){\rm{ }} = {\rm{ }}(\Pi - \Pi ^T ){\rm{ |
1297 |
> |
}}(J^{ - 1} \Pi + \Pi J^{ - 1} ) + \sum\limits_i {[Q^T F_i |
1298 |
> |
(r,Q)X_i^T - X_i F_i (r,Q)^T Q]} - (\Lambda - \Lambda ^T ). |
1299 |
> |
\label{introEquation:skewMatrixPI} |
1300 |
> |
\end{eqnarray*} |
1301 |
> |
|
1302 |
|
Since $\Lambda$ is symmetric, the last term of Equation |
1303 |
|
\ref{introEquation:skewMatrixPI} is zero, which implies the Lagrange |
1304 |
|
multiplier $\Lambda$ is absent from the equations of motion. This |
1430 |
|
The equations of motion corresponding to potential energy and |
1431 |
|
kinetic energy are listed in the below table, |
1432 |
|
\begin{table} |
1433 |
< |
\caption{Equations of motion due to Potential and Kinetic Energies} |
1433 |
> |
\caption{EQUATIONS OF MOTION DUE TO POTENTIAL AND KINETIC ENERGIES} |
1434 |
|
\begin{center} |
1435 |
|
\begin{tabular}{|l|l|} |
1436 |
|
\hline |