--- 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}