--- trunk/tengDissertation/Introduction.tex	2006/04/05 03:44:32	2693
+++ trunk/tengDissertation/Introduction.tex	2006/04/05 21:00:19	2694
@@ -1,12 +1,5 @@
 \chapter{\label{chapt:introduction}INTRODUCTION AND THEORETICAL BACKGROUND}
 
-\section{\label{introSection:molecularDynamics}Molecular Dynamics}
-
-As a special discipline of molecular modeling, Molecular dynamics
-has proven to be a powerful tool for studying the functions of
-biological systems, providing structural, thermodynamic and
-dynamical information.
-
 \section{\label{introSection:classicalMechanics}Classical
 Mechanics}
 
@@ -22,7 +15,59 @@ sufficient to predict the future behavior of the syste
 sufficient to predict the future behavior of the system.
 
 \subsection{\label{introSection:newtonian}Newtonian Mechanics}
+The discovery of Newton's three laws of mechanics which govern the
+motion of particles is the foundation of the classical mechanics.
+Newton��s first law defines a class of inertial frames. Inertial
+frames are reference frames where a particle not interacting with
+other bodies will move with constant speed in the same direction.
+With respect to inertial frames Newton��s second law has the form
+\begin{equation}
+F = \frac {dp}{dt} = \frac {mv}{dt}
+\label{introEquation:newtonSecondLaw}
+\end{equation}
+A point mass interacting with other bodies moves with the
+acceleration along the direction of the force acting on it. Let
+$F_ij$ be the force that particle $i$ exerts on particle $j$, and
+$F_ji$ be the force that particle $j$ exerts on particle $i$.
+Newton��s third law states that
+\begin{equation}
+F_ij = -F_ji
+ \label{introEquation:newtonThirdLaw}
+\end{equation}
 
+Conservation laws of Newtonian Mechanics play very important roles
+in solving mechanics problems. The linear momentum of a particle is
+conserved if it is free or it experiences no force. The second
+conservation theorem concerns the angular momentum of a particle.
+The angular momentum $L$ of a particle with respect to an origin
+from which $r$ is measured is defined to be
+\begin{equation}
+L \equiv r \times p \label{introEquation:angularMomentumDefinition}
+\end{equation}
+The torque $\tau$ with respect to the same origin is defined to be
+\begin{equation}
+N \equiv r \times F \label{introEquation:torqueDefinition}
+\end{equation}
+Differentiating Eq.~\ref{introEquation:angularMomentumDefinition},
+\[
+\dot L = \frac{d}{{dt}}(r \times p) = (\dot r \times p) + (r \times
+\dot p)
+\]
+since
+\[
+\dot r \times p = \dot r \times mv = m\dot r \times \dot r \equiv 0
+\]
+thus,
+\begin{equation}
+\dot L = r \times \dot p = N
+\end{equation}
+If there are no external torques acting on a body, the angular
+momentum of it is conserved. The last conservation theorem state
+that if all forces are conservative, Energy $E = T + V$ is
+conserved. All of these conserved quantities are important factors
+to determine the quality of numerical integration scheme for rigid
+body \cite{Dullweber1997}.
+
 \subsection{\label{introSection:lagrangian}Lagrangian Mechanics}
 
 Newtonian Mechanics suffers from two important limitations: it
@@ -36,7 +81,7 @@ system, alternative procedures may be developed.
 which arise in attempts to apply Newton's equation to complex
 system, alternative procedures may be developed.
 
-\subsection{\label{introSection:halmiltonPrinciple}Hamilton's
+\subsubsection{\label{introSection:halmiltonPrinciple}Hamilton's
 Principle}
 
 Hamilton introduced the dynamical principle upon which it is
@@ -46,7 +91,7 @@ the kinetic, $K$, and potential energies, $U$.
 The actual trajectory, along which a dynamical system may move from
 one point to another within a specified time, is derived by finding
 the path which minimizes the time integral of the difference between
-the kinetic, $K$, and potential energies, $U$.
+the kinetic, $K$, and potential energies, $U$ \cite{tolman79}.
 \begin{equation}
 \delta \int_{t_1 }^{t_2 } {(K - U)dt = 0} ,
 \label{introEquation:halmitonianPrinciple1}
@@ -67,7 +112,7 @@ then Eq.~\ref{introEquation:halmitonianPrinciple1} bec
 \label{introEquation:halmitonianPrinciple2}
 \end{equation}
 
-\subsection{\label{introSection:equationOfMotionLagrangian}The
+\subsubsection{\label{introSection:equationOfMotionLagrangian}The
 Equations of Motion in Lagrangian Mechanics}
 
 for a holonomic system of $f$ degrees of freedom, the equations of
@@ -142,7 +187,7 @@ known as the canonical equations of motions.
 Eq.~\ref{introEquation:motionHamiltonianCoordinate} and
 Eq.~\ref{introEquation:motionHamiltonianMomentum} are Hamilton's
 equation of motion. Due to their symmetrical formula, they are also
-known as the canonical equations of motions.
+known as the canonical equations of motions \cite{Goldstein01}.
 
 An important difference between Lagrangian approach and the
 Hamiltonian approach is that the Lagrangian is considered to be a
@@ -153,17 +198,42 @@ equations.
 appropriate for application to statistical mechanics and quantum
 mechanics, since it treats the coordinate and its time derivative as
 independent variables and it only works with 1st-order differential
-equations.
+equations\cite{Marion90}.
 
-\subsection{\label{introSection:poissonBrackets}Poisson Brackets}
+When studying Hamiltonian system, it is more convenient to use
+notation
+\begin{equation}
+r = r(q,p)^T
+\end{equation}
+and to introduce a $2n \times 2n$ canonical structure matrix $J$,
+\begin{equation}
+J = \left( {\begin{array}{*{20}c}
+   0 & I  \\
+   { - I} & 0  \\
+\end{array}} \right)
+\label{introEquation:canonicalMatrix}
+\end{equation}
+Thus, Hamiltonian system can be rewritten as,
+\begin{equation}
+\frac{d}{{dt}}r = J\nabla _r H(r)
+\label{introEquation:compactHamiltonian}
+\end{equation}
+where $I$ is an identity matrix and $J$ is a skew-symmetrix matrix
+($ J^T  =  - J $).
 
-\subsection{\label{introSection:canonicalTransformation}Canonical
+%\subsection{\label{introSection:canonicalTransformation}Canonical
 Transformation}
 
+\section{\label{introSection:geometricIntegratos}Geometric Integrators}
+
+\subsection{\label{introSection:symplecticMaps}Symplectic Maps and Methods}
+
+\subsection{\label{Construction of Symplectic Methods}}
+
 \section{\label{introSection:statisticalMechanics}Statistical
 Mechanics}
 
-The thermodynamic behaviors and properties  of Molecular Dynamics
+The thermodynamic behaviors and properties of Molecular Dynamics
 simulation are governed by the principle of Statistical Mechanics.
 The following section will give a brief introduction to some of the
 Statistical Mechanics concepts presented in this dissertation.
@@ -172,8 +242,30 @@ Statistical Mechanics concepts presented in this disse
 
 \subsection{\label{introSection:ergodic}The Ergodic Hypothesis}
 
+
+\section{\label{introSection:molecularDynamics}Molecular Dynamics}
+
+As a special discipline of molecular modeling, Molecular dynamics
+has proven to be a powerful tool for studying the functions of
+biological systems, providing structural, thermodynamic and
+dynamical information.
+
+\subsection{\label{introSec:mdInit}Initialization}
+
+\subsection{\label{introSection:mdIntegration} Integration of the Equations of Motion}
+
 \section{\label{introSection:rigidBody}Dynamics of Rigid Bodies}
 
+A rigid body is a body in which the distance between any two given
+points of a rigid body remains constant regardless of external
+forces exerted on it. A rigid body therefore conserves its shape
+during its motion.
+
+Applications of dynamics of rigid bodies.
+
+
+%\subsection{\label{introSection:poissonBrackets}Poisson Brackets}
+
 \section{\label{introSection:correlationFunctions}Correlation Functions}
 
 \section{\label{introSection:langevinDynamics}Langevin Dynamics}
