| 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 |
