--- trunk/tengDissertation/Introduction.tex	2006/04/03 18:07:54	2685
+++ trunk/tengDissertation/Introduction.tex	2006/04/04 21:32:58	2692
@@ -1,15 +1,135 @@
 \chapter{\label{chapt:introduction}INTRODUCTION AND THEORETICAL BACKGROUND}
 
-\section{\label{introSection:classicalMechanics}Classical Mechanics}
+\section{\label{introSection:molecularDynamics}Molecular Dynamics}
 
-\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies}
+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:statisticalMechanics}Statistical Mechanics}
+\subsection{\label{introSection:classicalMechanics}Classical Mechanics}
 
-\section{\label{introSection:molecularDynamics}Molecular Dynamics}
+Closely related to Classical Mechanics, Molecular Dynamics
+simulations are carried out by integrating the equations of motion
+for a given system of particles. There are three fundamental ideas
+behind classical mechanics. Firstly, One can determine the state of
+a mechanical system at any time of interest; Secondly, all the
+mechanical properties of the system at that time can be determined
+by combining the knowledge of the properties of the system with the
+specification of this state; Finally, the specification of the state
+when further combine with the laws of mechanics will also be
+sufficient to predict the future behavior of the system.
 
+\subsubsection{\label{introSection:newtonian}Newtonian Mechanics}
+
+\subsubsection{\label{introSection:lagrangian}Lagrangian Mechanics}
+
+Newtonian Mechanics suffers from two important limitations: it
+describes their motion in special cartesian coordinate systems.
+Another limitation of Newtonian mechanics becomes obvious when we
+try to describe systems with large numbers of particles. It becomes
+very difficult to predict the properties of the system by carrying
+out calculations involving the each individual interaction between
+all the particles, even if we know all of the details of the
+interaction. In order to overcome some of the practical difficulties
+which arise in attempts to apply Newton's equation to complex
+system, alternative procedures may be developed.
+
+\subsubsubsection{\label{introSection:halmiltonPrinciple}Hamilton's
+Principle}
+
+Hamilton introduced the dynamical principle upon which it is
+possible to base all of mechanics and, indeed, most of classical
+physics. Hamilton's Principle may be stated as follow,
+
+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, $U$.
+\begin{equation}
+\delta \int_{t_1 }^{t_2 } {(K - U)dt = 0} ,
+\lable{introEquation:halmitonianPrinciple1}
+\end{equation}
+
+For simple mechanical systems, where the forces acting on the
+different part are derivable from a potential and the velocities are
+small compared with that of light, the Lagrangian function $L$ can
+be define as the difference between the kinetic energy of the system
+and its potential energy,
+\begin{equation}
+L \equiv K - U = L(q_i ,\dot q_i ) ,
+\label{introEquation:lagrangianDef}
+\end{equation}
+then Eq.~\ref{introEquation:halmitonianPrinciple1} becomes
+\begin{equation}
+\delta \int_{t_1 }^{t_2 } {K dt = 0} ,
+\lable{introEquation:halmitonianPrinciple2}
+\end{equation}
+
+\subsubsubsection{\label{introSection:equationOfMotionLagrangian}The
+Equations of Motion in Lagrangian Mechanics}
+
+for a holonomic system of $f$ degrees of freedom, the equations of
+motion in the Lagrangian form is
+\begin{equation}
+\frac{d}{{dt}}\frac{{\partial L}}{{\partial \dot q_i }} -
+\frac{{\partial L}}{{\partial q_i }} = 0,{\rm{ }}i = 1, \ldots,f
+\lable{introEquation:eqMotionLagrangian}
+\end{equation}
+where $q_{i}$ is generalized coordinate and $\dot{q_{i}}$ is
+generalized velocity.
+
+\subsubsection{\label{introSection:hamiltonian}Hamiltonian Mechanics}
+
+Arising from Lagrangian Mechanics, Hamiltonian Mechanics was
+introduced by William Rowan Hamilton in 1833 as a re-formulation of
+classical mechanics. If the potential energy of a system is
+independent of generalized velocities, the generalized momenta can
+be defined as
+\begin{equation}
+p_i = \frac{\partial L}{\partial \dot q_i}
+\label{introEquation:generalizedMomenta}
+\end{equation}
+With the help of these momenta, we may now define a new quantity $H$
+by the equation
+\begin{equation}
+H = p_1 \dot q_1  +  \ldots  + p_f \dot q_f  - L,
+\label{introEquation:hamiltonianDefByLagrangian}
+\end{equation}
+where $ \dot q_1  \ldots \dot q_f $ are generalized velocities and
+$L$ is the Lagrangian function for the system.
+
+An important difference between Lagrangian approach and the
+Hamiltonian approach is that the Lagrangian is considered to be a
+function of the generalized velocities $\dot q_i$ and the
+generalized coordinates $q_i$, while the Hamiltonian is considered
+to be a function of the generalized momenta $p_i$ and the conjugate
+generalized coordinate $q_i$. Hamiltonian Mechanics is more
+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.
+
+
+\subsubsection{\label{introSection:canonicalTransformation}Canonical Transformation}
+
+\subsection{\label{introSection:statisticalMechanics}Statistical Mechanics}
+
+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.
+
+\subsubsection{\label{introSection::ensemble}Ensemble}
+
+\subsubsection{\label{introSection:ergodic}The Ergodic Hypothesis}
+
+\subsection{\label{introSection:rigidBody}Dynamics of Rigid Bodies}
+
+\subsection{\label{introSection:correlationFunctions}Correlation Functions}
+
 \section{\label{introSection:langevinDynamics}Langevin Dynamics}
 
-\section{\label{introSection:hydroynamics}Hydrodynamics}
+\subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics}
 
-\section{\label{introSection:correlationFunctions}Correlation Functions}
+\subsection{\label{introSection:hydroynamics}Hydrodynamics}
