--- trunk/tengDissertation/Introduction.tex	2006/04/04 21:32:58	2692
+++ trunk/tengDissertation/Introduction.tex	2006/04/05 03:44:32	2693
@@ -7,7 +7,8 @@ dynamical information.
 biological systems, providing structural, thermodynamic and
 dynamical information.
 
-\subsection{\label{introSection:classicalMechanics}Classical Mechanics}
+\section{\label{introSection:classicalMechanics}Classical
+Mechanics}
 
 Closely related to Classical Mechanics, Molecular Dynamics
 simulations are carried out by integrating the equations of motion
@@ -20,9 +21,9 @@ sufficient to predict the future behavior of the syste
 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}
+\subsection{\label{introSection:newtonian}Newtonian Mechanics}
 
-\subsubsection{\label{introSection:lagrangian}Lagrangian Mechanics}
+\subsection{\label{introSection:lagrangian}Lagrangian Mechanics}
 
 Newtonian Mechanics suffers from two important limitations: it
 describes their motion in special cartesian coordinate systems.
@@ -35,7 +36,7 @@ system, alternative procedures may be developed.
 which arise in attempts to apply Newton's equation to complex
 system, alternative procedures may be developed.
 
-\subsubsubsection{\label{introSection:halmiltonPrinciple}Hamilton's
+\subsection{\label{introSection:halmiltonPrinciple}Hamilton's
 Principle}
 
 Hamilton introduced the dynamical principle upon which it is
@@ -48,7 +49,7 @@ the kinetic, $K$, and potential energies, $U$.
 the kinetic, $K$, and potential energies, $U$.
 \begin{equation}
 \delta \int_{t_1 }^{t_2 } {(K - U)dt = 0} ,
-\lable{introEquation:halmitonianPrinciple1}
+\label{introEquation:halmitonianPrinciple1}
 \end{equation}
 
 For simple mechanical systems, where the forces acting on the
@@ -62,11 +63,11 @@ then Eq.~\ref{introEquation:halmitonianPrinciple1} bec
 \end{equation}
 then Eq.~\ref{introEquation:halmitonianPrinciple1} becomes
 \begin{equation}
-\delta \int_{t_1 }^{t_2 } {K dt = 0} ,
-\lable{introEquation:halmitonianPrinciple2}
+\delta \int_{t_1 }^{t_2 } {L dt = 0} ,
+\label{introEquation:halmitonianPrinciple2}
 \end{equation}
 
-\subsubsubsection{\label{introSection:equationOfMotionLagrangian}The
+\subsection{\label{introSection:equationOfMotionLagrangian}The
 Equations of Motion in Lagrangian Mechanics}
 
 for a holonomic system of $f$ degrees of freedom, the equations of
@@ -74,12 +75,12 @@ 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}
+\label{introEquation:eqMotionLagrangian}
 \end{equation}
 where $q_{i}$ is generalized coordinate and $\dot{q_{i}}$ is
 generalized velocity.
 
-\subsubsection{\label{introSection:hamiltonian}Hamiltonian Mechanics}
+\subsection{\label{introSection:hamiltonian}Hamiltonian Mechanics}
 
 Arising from Lagrangian Mechanics, Hamiltonian Mechanics was
 introduced by William Rowan Hamilton in 1833 as a re-formulation of
@@ -90,15 +91,59 @@ With the help of these momenta, we may now define a ne
 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
+The Lagrange equations of motion are then expressed by
 \begin{equation}
-H = p_1 \dot q_1  +  \ldots  + p_f \dot q_f  - L,
+p_i  = \frac{{\partial L}}{{\partial q_i }}
+\label{introEquation:generalizedMomentaDot}
+\end{equation}
+
+With the help of the generalized momenta, we may now define a new
+quantity $H$ by the equation
+\begin{equation}
+H = \sum\limits_k {p_k \dot q_k }  - 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.
 
+Differentiating Eq.~\ref{introEquation:hamiltonianDefByLagrangian},
+one can obtain
+\begin{equation}
+dH = \sum\limits_k {\left( {p_k d\dot q_k  + \dot q_k dp_k  -
+\frac{{\partial L}}{{\partial q_k }}dq_k  - \frac{{\partial
+L}}{{\partial \dot q_k }}d\dot q_k } \right)}  - \frac{{\partial
+L}}{{\partial t}}dt \label{introEquation:diffHamiltonian1}
+\end{equation}
+Making use of  Eq.~\ref{introEquation:generalizedMomenta}, the
+second and fourth terms in the parentheses cancel. Therefore,
+Eq.~\ref{introEquation:diffHamiltonian1} can be rewritten as
+\begin{equation}
+dH = \sum\limits_k {\left( {\dot q_k dp_k  - \dot p_k dq_k }
+\right)}  - \frac{{\partial L}}{{\partial t}}dt
+\label{introEquation:diffHamiltonian2}
+\end{equation}
+By identifying the coefficients of $dq_k$, $dp_k$ and dt, we can
+find
+\begin{equation}
+\frac{{\partial H}}{{\partial p_k }} = q_k
+\label{introEquation:motionHamiltonianCoordinate}
+\end{equation}
+\begin{equation}
+\frac{{\partial H}}{{\partial q_k }} =  - p_k
+\label{introEquation:motionHamiltonianMomentum}
+\end{equation}
+and
+\begin{equation}
+\frac{{\partial H}}{{\partial t}} =  - \frac{{\partial L}}{{\partial
+t}}
+\label{introEquation:motionHamiltonianTime}
+\end{equation}
+
+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.
+
 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
@@ -110,23 +155,26 @@ equations.
 independent variables and it only works with 1st-order differential
 equations.
 
+\subsection{\label{introSection:poissonBrackets}Poisson Brackets}
 
-\subsubsection{\label{introSection:canonicalTransformation}Canonical Transformation}
+\subsection{\label{introSection:canonicalTransformation}Canonical
+Transformation}
 
-\subsection{\label{introSection:statisticalMechanics}Statistical Mechanics}
+\section{\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}
+\subsection{\label{introSection::ensemble}Ensemble}
 
-\subsubsection{\label{introSection:ergodic}The Ergodic Hypothesis}
+\subsection{\label{introSection:ergodic}The Ergodic Hypothesis}
 
-\subsection{\label{introSection:rigidBody}Dynamics of Rigid Bodies}
+\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies}
 
-\subsection{\label{introSection:correlationFunctions}Correlation Functions}
+\section{\label{introSection:correlationFunctions}Correlation Functions}
 
 \section{\label{introSection:langevinDynamics}Langevin Dynamics}
 
