--- trunk/tengDissertation/Introduction.tex	2006/04/04 21:32:58	2692
+++ trunk/tengDissertation/Introduction.tex	2006/04/13 04:47:47	2706
@@ -1,14 +1,8 @@
 \chapter{\label{chapt:introduction}INTRODUCTION AND THEORETICAL BACKGROUND}
 
-\section{\label{introSection:molecularDynamics}Molecular Dynamics}
+\section{\label{introSection:classicalMechanics}Classical
+Mechanics}
 
-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{introSection:classicalMechanics}Classical Mechanics}
-
 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
@@ -20,10 +14,64 @@ 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}
+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}
 
-\subsubsection{\label{introSection:lagrangian}Lagrangian Mechanics}
+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
+\begin{equation}E = T + V \label{introEquation:energyConservation}
+\end{equation}
+ 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
 describes their motion in special cartesian coordinate systems.
 Another limitation of Newtonian mechanics becomes obvious when we
@@ -35,7 +83,7 @@ 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
+\subsubsection{\label{introSection:halmiltonPrinciple}Hamilton's
 Principle}
 
 Hamilton introduced the dynamical principle upon which it is
@@ -45,10 +93,10 @@ 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} ,
-\lable{introEquation:halmitonianPrinciple1}
+\label{introEquation:halmitonianPrinciple1}
 \end{equation}
 
 For simple mechanical systems, where the forces acting on the
@@ -62,24 +110,24 @@ 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
+\subsubsection{\label{introSection:equationOfMotionLagrangian}The
 Equations of Motion in Lagrangian Mechanics}
 
-for a holonomic system of $f$ degrees of freedom, the equations of
+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}
+\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 +138,59 @@ 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 \cite{Goldstein01}.
+
 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
@@ -108,28 +200,1014 @@ 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}.
 
+In Newtonian Mechanics, a system described by conservative forces
+conserves the total energy \ref{introEquation:energyConservation}.
+It follows that Hamilton's equations of motion conserve the total
+Hamiltonian.
+\begin{equation}
+\frac{{dH}}{{dt}} = \sum\limits_i {\left( {\frac{{\partial
+H}}{{\partial q_i }}\dot q_i  + \frac{{\partial H}}{{\partial p_i
+}}\dot p_i } \right)}  = \sum\limits_i {\left( {\frac{{\partial
+H}}{{\partial q_i }}\frac{{\partial H}}{{\partial p_i }} -
+\frac{{\partial H}}{{\partial p_i }}\frac{{\partial H}}{{\partial
+q_i }}} \right) = 0} \label{introEquation:conserveHalmitonian}
+\end{equation}
 
-\subsubsection{\label{introSection:canonicalTransformation}Canonical Transformation}
+\section{\label{introSection:statisticalMechanics}Statistical
+Mechanics}
 
-\subsection{\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.
+Statistical Mechanics concepts and theorem presented in this
+dissertation.
 
-\subsubsection{\label{introSection::ensemble}Ensemble}
+\subsection{\label{introSection:ensemble}Phase Space and Ensemble}
 
-\subsubsection{\label{introSection:ergodic}The Ergodic Hypothesis}
+Mathematically, phase space is the space which represents all
+possible states. Each possible state of the system corresponds to
+one unique point in the phase space. For mechanical systems, the
+phase space usually consists of all possible values of position and
+momentum variables. Consider a dynamic system in a cartesian space,
+where each of the $6f$ coordinates and momenta is assigned to one of
+$6f$ mutually orthogonal axes, the phase space of this system is a
+$6f$ dimensional space. A point, $x = (q_1 , \ldots ,q_f ,p_1 ,
+\ldots ,p_f )$, with a unique set of values of $6f$ coordinates and
+momenta is a phase space vector.
 
-\subsection{\label{introSection:rigidBody}Dynamics of Rigid Bodies}
+A microscopic state or microstate of a classical system is
+specification of the complete phase space vector of a system at any
+instant in time. An ensemble is defined as a collection of systems
+sharing one or more macroscopic characteristics but each being in a
+unique microstate. The complete ensemble is specified by giving all
+systems or microstates consistent with the common macroscopic
+characteristics of the ensemble. Although the state of each
+individual system in the ensemble could be precisely described at
+any instance in time by a suitable phase space vector, when using
+ensembles for statistical purposes, there is no need to maintain
+distinctions between individual systems, since the numbers of
+systems at any time in the different states which correspond to
+different regions of the phase space are more interesting. Moreover,
+in the point of view of statistical mechanics, one would prefer to
+use ensembles containing a large enough population of separate
+members so that the numbers of systems in such different states can
+be regarded as changing continuously as we traverse different
+regions of the phase space. The condition of an ensemble at any time
+can be regarded as appropriately specified by the density $\rho$
+with which representative points are distributed over the phase
+space. The density of distribution for an ensemble with $f$ degrees
+of freedom is defined as,
+\begin{equation}
+\rho  = \rho (q_1 , \ldots ,q_f ,p_1 , \ldots ,p_f ,t).
+\label{introEquation:densityDistribution}
+\end{equation}
+Governed by the principles of mechanics, the phase points change
+their value which would change the density at any time at phase
+space. Hence, the density of distribution is also to be taken as a
+function of the time.
 
-\subsection{\label{introSection:correlationFunctions}Correlation Functions}
+The number of systems $\delta N$ at time $t$ can be determined by,
+\begin{equation}
+\delta N = \rho (q,p,t)dq_1  \ldots dq_f dp_1  \ldots dp_f.
+\label{introEquation:deltaN}
+\end{equation}
+Assuming a large enough population of systems are exploited, we can
+sufficiently approximate $\delta N$ without introducing
+discontinuity when we go from one region in the phase space to
+another. By integrating over the whole phase space,
+\begin{equation}
+N = \int { \ldots \int {\rho (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f
+\label{introEquation:totalNumberSystem}
+\end{equation}
+gives us an expression for the total number of the systems. Hence,
+the probability per unit in the phase space can be obtained by,
+\begin{equation}
+\frac{{\rho (q,p,t)}}{N} = \frac{{\rho (q,p,t)}}{{\int { \ldots \int
+{\rho (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}.
+\label{introEquation:unitProbability}
+\end{equation}
+With the help of Equation(\ref{introEquation:unitProbability}) and
+the knowledge of the system, it is possible to calculate the average
+value of any desired quantity which depends on the coordinates and
+momenta of the system. Even when the dynamics of the real system is
+complex, or stochastic, or even discontinuous, the average
+properties of the ensemble of possibilities as a whole may still
+remain well defined. For a classical system in thermal equilibrium
+with its environment, the ensemble average of a mechanical quantity,
+$\langle A(q , p) \rangle_t$, takes the form of an integral over the
+phase space of the system,
+\begin{equation}
+\langle  A(q , p) \rangle_t = \frac{{\int { \ldots \int {A(q,p)\rho
+(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}{{\int { \ldots \int {\rho
+(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}
+\label{introEquation:ensembelAverage}
+\end{equation}
+
+There are several different types of ensembles with different
+statistical characteristics. As a function of macroscopic
+parameters, such as temperature \textit{etc}, partition function can
+be used to describe the statistical properties of a system in
+thermodynamic equilibrium.
+
+As an ensemble of systems, each of which is known to be thermally
+isolated and conserve energy, Microcanonical ensemble(NVE) has a
+partition function like,
+\begin{equation}
+\Omega (N,V,E) = e^{\beta TS} \label{introEquation:NVEPartition}.
+\end{equation}
+A canonical ensemble(NVT)is an ensemble of systems, each of which
+can share its energy with a large heat reservoir. The distribution
+of the total energy amongst the possible dynamical states is given
+by the partition function,
+\begin{equation}
+\Omega (N,V,T) = e^{ - \beta A}
+\label{introEquation:NVTPartition}
+\end{equation}
+Here, $A$ is the Helmholtz free energy which is defined as $ A = U -
+TS$. Since most experiment are carried out under constant pressure
+condition, isothermal-isobaric ensemble(NPT) play a very important
+role in molecular simulation. The isothermal-isobaric ensemble allow
+the system to exchange energy with a heat bath of temperature $T$
+and to change the volume as well. Its partition function is given as
+\begin{equation}
+\Delta (N,P,T) =  - e^{\beta G}.
+ \label{introEquation:NPTPartition}
+\end{equation}
+Here, $G = U - TS + PV$, and $G$ is called Gibbs free energy.
+
+\subsection{\label{introSection:liouville}Liouville's theorem}
+
+The Liouville's theorem is the foundation on which statistical
+mechanics rests. It describes the time evolution of phase space
+distribution function. In order to calculate the rate of change of
+$\rho$, we begin from Equation(\ref{introEquation:deltaN}). If we
+consider the two faces perpendicular to the $q_1$ axis, which are
+located at $q_1$ and $q_1 + \delta q_1$, the number of phase points
+leaving the opposite face is given by the expression,
+\begin{equation}
+\left( {\rho  + \frac{{\partial \rho }}{{\partial q_1 }}\delta q_1 }
+\right)\left( {\dot q_1  + \frac{{\partial \dot q_1 }}{{\partial q_1
+}}\delta q_1 } \right)\delta q_2  \ldots \delta q_f \delta p_1
+\ldots \delta p_f .
+\end{equation}
+Summing all over the phase space, we obtain
+\begin{equation}
+\frac{{d(\delta N)}}{{dt}} =  - \sum\limits_{i = 1}^f {\left[ {\rho
+\left( {\frac{{\partial \dot q_i }}{{\partial q_i }} +
+\frac{{\partial \dot p_i }}{{\partial p_i }}} \right) + \left(
+{\frac{{\partial \rho }}{{\partial q_i }}\dot q_i  + \frac{{\partial
+\rho }}{{\partial p_i }}\dot p_i } \right)} \right]} \delta q_1
+\ldots \delta q_f \delta p_1  \ldots \delta p_f .
+\end{equation}
+Differentiating the equations of motion in Hamiltonian formalism
+(\ref{introEquation:motionHamiltonianCoordinate},
+\ref{introEquation:motionHamiltonianMomentum}), we can show,
+\begin{equation}
+\sum\limits_i {\left( {\frac{{\partial \dot q_i }}{{\partial q_i }}
++ \frac{{\partial \dot p_i }}{{\partial p_i }}} \right)}  = 0 ,
+\end{equation}
+which cancels the first terms of the right hand side. Furthermore,
+divining $ \delta q_1  \ldots \delta q_f \delta p_1  \ldots \delta
+p_f $ in both sides, we can write out Liouville's theorem in a
+simple form,
+\begin{equation}
+\frac{{\partial \rho }}{{\partial t}} + \sum\limits_{i = 1}^f
+{\left( {\frac{{\partial \rho }}{{\partial q_i }}\dot q_i  +
+\frac{{\partial \rho }}{{\partial p_i }}\dot p_i } \right)}  = 0 .
+\label{introEquation:liouvilleTheorem}
+\end{equation}
+
+Liouville's theorem states that the distribution function is
+constant along any trajectory in phase space. In classical
+statistical mechanics, since the number of particles in the system
+is huge, we may be able to believe the system is stationary,
+\begin{equation}
+\frac{{\partial \rho }}{{\partial t}} = 0.
+\label{introEquation:stationary}
+\end{equation}
+In such stationary system, the density of distribution $\rho$ can be
+connected to the Hamiltonian $H$ through Maxwell-Boltzmann
+distribution,
+\begin{equation}
+\rho  \propto e^{ - \beta H}
+\label{introEquation:densityAndHamiltonian}
+\end{equation}
+
+\subsubsection{\label{introSection:phaseSpaceConservation}Conservation of Phase Space}
+Lets consider a region in the phase space,
+\begin{equation}
+\delta v = \int { \ldots \int {dq_1 } ...dq_f dp_1 } ..dp_f .
+\end{equation}
+If this region is small enough, the density $\rho$ can be regarded
+as uniform over the whole phase space. Thus, the number of phase
+points inside this region is given by,
+\begin{equation}
+\delta N = \rho \delta v = \rho \int { \ldots \int {dq_1 } ...dq_f
+dp_1 } ..dp_f.
+\end{equation}
+
+\begin{equation}
+\frac{{d(\delta N)}}{{dt}} = \frac{{d\rho }}{{dt}}\delta v + \rho
+\frac{d}{{dt}}(\delta v) = 0.
+\end{equation}
+With the help of stationary assumption
+(\ref{introEquation:stationary}), we obtain the principle of the
+\emph{conservation of extension in phase space},
+\begin{equation}
+\frac{d}{{dt}}(\delta v) = \frac{d}{{dt}}\int { \ldots \int {dq_1 }
+...dq_f dp_1 } ..dp_f  = 0.
+\label{introEquation:volumePreserving}
+\end{equation}
+
+\subsubsection{\label{introSection:liouvilleInOtherForms}Liouville's Theorem in Other Forms}
+
+Liouville's theorem can be expresses in a variety of different forms
+which are convenient within different contexts. For any two function
+$F$ and $G$ of the coordinates and momenta of a system, the Poisson
+bracket ${F, G}$ is defined as
+\begin{equation}
+\left\{ {F,G} \right\} = \sum\limits_i {\left( {\frac{{\partial
+F}}{{\partial q_i }}\frac{{\partial G}}{{\partial p_i }} -
+\frac{{\partial F}}{{\partial p_i }}\frac{{\partial G}}{{\partial
+q_i }}} \right)}.
+\label{introEquation:poissonBracket}
+\end{equation}
+Substituting equations of motion in Hamiltonian formalism(
+\ref{introEquation:motionHamiltonianCoordinate} ,
+\ref{introEquation:motionHamiltonianMomentum} ) into
+(\ref{introEquation:liouvilleTheorem}), we can rewrite Liouville's
+theorem using Poisson bracket notion,
+\begin{equation}
+\left( {\frac{{\partial \rho }}{{\partial t}}} \right) =  - \left\{
+{\rho ,H} \right\}.
+\label{introEquation:liouvilleTheromInPoissin}
+\end{equation}
+Moreover, the Liouville operator is defined as
+\begin{equation}
+iL = \sum\limits_{i = 1}^f {\left( {\frac{{\partial H}}{{\partial
+p_i }}\frac{\partial }{{\partial q_i }} - \frac{{\partial
+H}}{{\partial q_i }}\frac{\partial }{{\partial p_i }}} \right)}
+\label{introEquation:liouvilleOperator}
+\end{equation}
+In terms of Liouville operator, Liouville's equation can also be
+expressed as
+\begin{equation}
+\left( {\frac{{\partial \rho }}{{\partial t}}} \right) =  - iL\rho
+\label{introEquation:liouvilleTheoremInOperator}
+\end{equation}
+
+\subsection{\label{introSection:ergodic}The Ergodic Hypothesis}
+
+Various thermodynamic properties can be calculated from Molecular
+Dynamics simulation. By comparing experimental values with the
+calculated properties, one can determine the accuracy of the
+simulation and the quality of the underlying model. However, both of
+experiment and computer simulation are usually performed during a
+certain time interval and the measurements are averaged over a
+period of them which is different from the average behavior of
+many-body system in Statistical Mechanics. Fortunately, Ergodic
+Hypothesis is proposed to make a connection between time average and
+ensemble average. It states that time average and average over the
+statistical ensemble are identical \cite{Frenkel1996, leach01:mm}.
+\begin{equation}
+\langle A(q , p) \rangle_t = \mathop {\lim }\limits_{t \to \infty }
+\frac{1}{t}\int\limits_0^t {A(q(t),p(t))dt = \int\limits_\Gamma
+{A(q(t),p(t))} } \rho (q(t), p(t)) dqdp
+\end{equation}
+where $\langle  A(q , p) \rangle_t$ is an equilibrium value of a
+physical quantity and $\rho (p(t), q(t))$ is the equilibrium
+distribution function. If an observation is averaged over a
+sufficiently long time (longer than relaxation time), all accessible
+microstates in phase space are assumed to be equally probed, giving
+a properly weighted statistical average. This allows the researcher
+freedom of choice when deciding how best to measure a given
+observable. In case an ensemble averaged approach sounds most
+reasonable, the Monte Carlo techniques\cite{metropolis:1949} can be
+utilized. Or if the system lends itself to a time averaging
+approach, the Molecular Dynamics techniques in
+Sec.~\ref{introSection:molecularDynamics} will be the best
+choice\cite{Frenkel1996}.
+
+\section{\label{introSection:geometricIntegratos}Geometric Integrators}
+A variety of numerical integrators were proposed to simulate the
+motions. They usually begin with an initial conditionals and move
+the objects in the direction governed by the differential equations.
+However, most of them ignore the hidden physical law contained
+within the equations. Since 1990, geometric integrators, which
+preserve various phase-flow invariants such as symplectic structure,
+volume and time reversal symmetry, are developed to address this
+issue. The velocity verlet method, which happens to be a simple
+example of symplectic integrator, continues to gain its popularity
+in molecular dynamics community. This fact can be partly explained
+by its geometric nature.
+
+\subsection{\label{introSection:symplecticManifold}Symplectic Manifold}
+A \emph{manifold} is an abstract mathematical space. It locally
+looks like Euclidean space, but when viewed globally, it may have
+more complicate structure. A good example of manifold is the surface
+of Earth. It seems to be flat locally, but it is round if viewed as
+a whole. A \emph{differentiable manifold} (also known as
+\emph{smooth manifold}) is a manifold with an open cover in which
+the covering neighborhoods are all smoothly isomorphic to one
+another. In other words,it is possible to apply calculus on
+\emph{differentiable manifold}. A \emph{symplectic manifold} is
+defined as a pair $(M, \omega)$ which consisting of a
+\emph{differentiable manifold} $M$ and a close, non-degenerated,
+bilinear symplectic form, $\omega$. A symplectic form on a vector
+space $V$ is a function $\omega(x, y)$ which satisfies
+$\omega(\lambda_1x_1+\lambda_2x_2, y) = \lambda_1\omega(x_1, y)+
+\lambda_2\omega(x_2, y)$, $\omega(x, y) = - \omega(y, x)$ and
+$\omega(x, x) = 0$. Cross product operation in vector field is an
+example of symplectic form.
+
+One of the motivations to study \emph{symplectic manifold} in
+Hamiltonian Mechanics is that a symplectic manifold can represent
+all possible configurations of the system and the phase space of the
+system can be described by it's cotangent bundle. Every symplectic
+manifold is even dimensional. For instance, in Hamilton equations,
+coordinate and momentum always appear in pairs.
+
+Let  $(M,\omega)$ and $(N, \eta)$ be symplectic manifolds. A map
+\[
+f : M \rightarrow N
+\]
+is a \emph{symplectomorphism} if it is a \emph{diffeomorphims} and
+the \emph{pullback} of $\eta$ under f is equal to $\omega$.
+Canonical transformation is an example of symplectomorphism in
+classical mechanics.
+
+\subsection{\label{introSection:ODE}Ordinary Differential Equations}
+
+For a ordinary differential system defined as
+\begin{equation}
+\dot x = f(x)
+\end{equation}
+where $x = x(q,p)^T$, this system is canonical Hamiltonian, if
+\begin{equation}
+f(r) = J\nabla _x H(r).
+\end{equation}
+$H = H (q, p)$ is Hamiltonian function and $J$ is the skew-symmetric
+matrix
+\begin{equation}
+J = \left( {\begin{array}{*{20}c}
+   0 & I  \\
+   { - I} & 0  \\
+\end{array}} \right)
+\label{introEquation:canonicalMatrix}
+\end{equation}
+where $I$ is an identity matrix. Using this notation, Hamiltonian
+system can be rewritten as,
+\begin{equation}
+\frac{d}{{dt}}x = J\nabla _x H(x)
+\label{introEquation:compactHamiltonian}
+\end{equation}In this case, $f$ is
+called a \emph{Hamiltonian vector field}.
+
+Another generalization of Hamiltonian dynamics is Poisson Dynamics,
+\begin{equation}
+\dot x = J(x)\nabla _x H \label{introEquation:poissonHamiltonian}
+\end{equation}
+The most obvious change being that matrix $J$ now depends on $x$.
+The free rigid body is an example of Poisson system (actually a
+Lie-Poisson system) with Hamiltonian function of angular kinetic
+energy.
+\begin{equation}
+J(\pi ) = \left( {\begin{array}{*{20}c}
+   0 & {\pi _3 } & { - \pi _2 }  \\
+   { - \pi _3 } & 0 & {\pi _1 }  \\
+   {\pi _2 } & { - \pi _1 } & 0  \\
+\end{array}} \right)
+\end{equation}
+
+\begin{equation}
+H = \frac{1}{2}\left( {\frac{{\pi _1^2 }}{{I_1 }} + \frac{{\pi _2^2
+}}{{I_2 }} + \frac{{\pi _3^2 }}{{I_3 }}} \right)
+\end{equation}
+
+\subsection{\label{introSection:exactFlow}Exact Flow}
+
+Let $x(t)$ be the exact solution of the ODE system,
+\begin{equation}
+\frac{{dx}}{{dt}} = f(x) \label{introEquation:ODE}
+\end{equation}
+The exact flow(solution) $\varphi_\tau$ is defined by
+\[
+x(t+\tau) =\varphi_\tau(x(t))
+\]
+where $\tau$ is a fixed time step and $\varphi$ is a map from phase
+space to itself. The flow has the continuous group property,
+\begin{equation}
+\varphi _{\tau _1 }  \circ \varphi _{\tau _2 }  = \varphi _{\tau _1
++ \tau _2 } .
+\end{equation}
+In particular,
+\begin{equation}
+\varphi _\tau   \circ \varphi _{ - \tau }  = I
+\end{equation}
+Therefore, the exact flow is self-adjoint,
+\begin{equation}
+\varphi _\tau   = \varphi _{ - \tau }^{ - 1}.
+\end{equation}
+The exact flow can also be written in terms of the of an operator,
+\begin{equation}
+\varphi _\tau  (x) = e^{\tau \sum\limits_i {f_i (x)\frac{\partial
+}{{\partial x_i }}} } (x) \equiv \exp (\tau f)(x).
+\label{introEquation:exponentialOperator}
+\end{equation}
+
+In most cases, it is not easy to find the exact flow $\varphi_\tau$.
+Instead, we use a approximate map, $\psi_\tau$, which is usually
+called integrator. The order of an integrator $\psi_\tau$ is $p$, if
+the Taylor series of $\psi_\tau$ agree to order $p$,
+\begin{equation}
+\psi_tau(x) = x + \tau f(x) + O(\tau^{p+1})
+\end{equation}
+
+\subsection{\label{introSection:geometricProperties}Geometric Properties}
+
+The hidden geometric properties of ODE and its flow play important
+roles in numerical studies. Many of them can be found in systems
+which occur naturally in applications.
+
+Let $\varphi$ be the flow of Hamiltonian vector field, $\varphi$ is
+a \emph{symplectic} flow if it satisfies,
+\begin{equation}
+{\varphi '}^T J \varphi ' = J.
+\end{equation}
+According to Liouville's theorem, the symplectic volume is invariant
+under a Hamiltonian flow, which is the basis for classical
+statistical mechanics. Furthermore, the flow of a Hamiltonian vector
+field on a symplectic manifold can be shown to be a
+symplectomorphism. As to the Poisson system,
+\begin{equation}
+{\varphi '}^T J \varphi ' = J \circ \varphi
+\end{equation}
+is the property must be preserved by the integrator.
 
+It is possible to construct a \emph{volume-preserving} flow for a
+source free($ \nabla \cdot f = 0 $) ODE, if the flow satisfies $
+\det d\varphi  = 1$. One can show easily that a symplectic flow will
+be volume-preserving.
+
+Changing the variables $y = h(x)$ in a ODE\ref{introEquation:ODE}
+will result in a new system,
+\[
+\dot y = \tilde f(y) = ((dh \cdot f)h^{ - 1} )(y).
+\]
+The vector filed $f$ has reversing symmetry $h$ if $f = - \tilde f$.
+In other words, the flow of this vector field is reversible if and
+only if $ h \circ \varphi ^{ - 1}  = \varphi  \circ h $.
+
+A \emph{first integral}, or conserved quantity of a general
+differential function is a function $ G:R^{2d}  \to R^d $ which is
+constant for all solutions of the ODE $\frac{{dx}}{{dt}} = f(x)$ ,
+\[
+\frac{{dG(x(t))}}{{dt}} = 0.
+\]
+Using chain rule, one may obtain,
+\[
+\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \bullet \nabla G,
+\]
+which is the condition for conserving \emph{first integral}. For a
+canonical Hamiltonian system, the time evolution of an arbitrary
+smooth function $G$ is given by,
+\begin{equation}
+\begin{array}{c}
+ \frac{{dG(x(t))}}{{dt}} = [\nabla _x G(x(t))]^T \dot x(t) \\
+  = [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). \\
+ \end{array}
+\label{introEquation:firstIntegral1}
+\end{equation}
+Using poisson bracket notion, Equation
+\ref{introEquation:firstIntegral1} can be rewritten as
+\[
+\frac{d}{{dt}}G(x(t)) = \left\{ {G,H} \right\}(x(t)).
+\]
+Therefore, the sufficient condition for $G$ to be the \emph{first
+integral} of a Hamiltonian system is
+\[
+\left\{ {G,H} \right\} = 0.
+\]
+As well known, the Hamiltonian (or energy) H of a Hamiltonian system
+is a \emph{first integral}, which is due to the fact $\{ H,H\}  =
+0$.
+
+
+ When designing any numerical methods, one should always try to
+preserve the structural properties of the original ODE and its flow.
+
+\subsection{\label{introSection:constructionSymplectic}Construction of Symplectic Methods}
+A lot of well established and very effective numerical methods have
+been successful precisely because of their symplecticities even
+though this fact was not recognized when they were first
+constructed. The most famous example is leapfrog methods in
+molecular dynamics. In general, symplectic integrators can be
+constructed using one of four different methods.
+\begin{enumerate}
+\item Generating functions
+\item Variational methods
+\item Runge-Kutta methods
+\item Splitting methods
+\end{enumerate}
+
+Generating function tends to lead to methods which are cumbersome
+and difficult to use. In dissipative systems, variational methods
+can capture the decay of energy accurately. Since their
+geometrically unstable nature against non-Hamiltonian perturbations,
+ordinary implicit Runge-Kutta methods are not suitable for
+Hamiltonian system. Recently, various high-order explicit
+Runge--Kutta methods have been developed to overcome this
+instability. However, due to computational penalty involved in
+implementing the Runge-Kutta methods, they do not attract too much
+attention from Molecular Dynamics community. Instead, splitting have
+been widely accepted since they exploit natural decompositions of
+the system\cite{Tuckerman92}.
+
+\subsubsection{\label{introSection:splittingMethod}Splitting Method}
+
+The main idea behind splitting methods is to decompose the discrete
+$\varphi_h$ as a composition of simpler flows,
+\begin{equation}
+\varphi _h  = \varphi _{h_1 }  \circ \varphi _{h_2 }  \ldots  \circ
+\varphi _{h_n }
+\label{introEquation:FlowDecomposition}
+\end{equation}
+where each of the sub-flow is chosen such that each represent a
+simpler integration of the system.
+
+Suppose that a Hamiltonian system takes the form,
+\[
+H = H_1 + H_2.
+\]
+Here, $H_1$ and $H_2$ may represent different physical processes of
+the system. For instance, they may relate to kinetic and potential
+energy respectively, which is a natural decomposition of the
+problem. If $H_1$ and $H_2$ can be integrated using exact flows
+$\varphi_1(t)$ and $\varphi_2(t)$, respectively, a simple first
+order is then given by the Lie-Trotter formula
+\begin{equation}
+\varphi _h  = \varphi _{1,h}  \circ \varphi _{2,h},
+\label{introEquation:firstOrderSplitting}
+\end{equation}
+where $\varphi _h$ is the result of applying the corresponding
+continuous $\varphi _i$ over a time $h$. By definition, as
+$\varphi_i(t)$ is the exact solution of a Hamiltonian system, it
+must follow that each operator $\varphi_i(t)$ is a symplectic map.
+It is easy to show that any composition of symplectic flows yields a
+symplectic map,
+\begin{equation}
+(\varphi '\phi ')^T J\varphi '\phi ' = \phi '^T \varphi '^T J\varphi
+'\phi ' = \phi '^T J\phi ' = J,
+ \label{introEquation:SymplecticFlowComposition}
+\end{equation}
+where $\phi$ and $\psi$ both are symplectic maps. Thus operator
+splitting in this context automatically generates a symplectic map.
+
+The Lie-Trotter splitting(\ref{introEquation:firstOrderSplitting})
+introduces local errors proportional to $h^2$, while Strang
+splitting gives a second-order decomposition,
+\begin{equation}
+\varphi _h  = \varphi _{1,h/2}  \circ \varphi _{2,h}  \circ \varphi
+_{1,h/2} , \label{introEquation:secondOrderSplitting}
+\end{equation}
+which has a local error proportional to $h^3$. Sprang splitting's
+popularity in molecular simulation community attribute to its
+symmetric property,
+\begin{equation}
+\varphi _h^{ - 1} = \varphi _{ - h}.
+\label{introEquation:timeReversible}
+\end{equation}
+
+\subsubsection{\label{introSection:exampleSplittingMethod}Example of Splitting Method}
+The classical equation for a system consisting of interacting
+particles can be written in Hamiltonian form,
+\[
+H = T + V
+\]
+where $T$ is the kinetic energy and $V$ is the potential energy.
+Setting $H_1 = T, H_2 = V$ and applying Strang splitting, one
+obtains the following:
+\begin{align}
+q(\Delta t) &= q(0) + \dot{q}(0)\Delta t +
+    \frac{F[q(0)]}{m}\frac{\Delta t^2}{2}, %
+\label{introEquation:Lp10a} \\%
+%
+\dot{q}(\Delta t) &= \dot{q}(0) + \frac{\Delta t}{2m}
+    \biggl [F[q(0)] + F[q(\Delta t)] \biggr]. %
+\label{introEquation:Lp10b}
+\end{align}
+where $F(t)$ is the force at time $t$. This integration scheme is
+known as \emph{velocity verlet} which is
+symplectic(\ref{introEquation:SymplecticFlowComposition}),
+time-reversible(\ref{introEquation:timeReversible}) and
+volume-preserving (\ref{introEquation:volumePreserving}). These
+geometric properties attribute to its long-time stability and its
+popularity in the community. However, the most commonly used
+velocity verlet integration scheme is written as below,
+\begin{align}
+\dot{q}\biggl (\frac{\Delta t}{2}\biggr ) &=
+    \dot{q}(0) + \frac{\Delta t}{2m}\, F[q(0)], \label{introEquation:Lp9a}\\%
+%
+q(\Delta t) &= q(0) + \Delta t\, \dot{q}\biggl (\frac{\Delta t}{2}\biggr ),%
+    \label{introEquation:Lp9b}\\%
+%
+\dot{q}(\Delta t) &= \dot{q}\biggl (\frac{\Delta t}{2}\biggr ) +
+    \frac{\Delta t}{2m}\, F[q(0)]. \label{introEquation:Lp9c}
+\end{align}
+From the preceding splitting, one can see that the integration of
+the equations of motion would follow:
+\begin{enumerate}
+\item calculate the velocities at the half step, $\frac{\Delta t}{2}$, from the forces calculated at the initial position.
+
+\item Use the half step velocities to move positions one whole step, $\Delta t$.
+
+\item Evaluate the forces at the new positions, $\mathbf{r}(\Delta t)$, and use the new forces to complete the velocity move.
+
+\item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values.
+\end{enumerate}
+
+Simply switching the order of splitting and composing, a new
+integrator, the \emph{position verlet} integrator, can be generated,
+\begin{align}
+\dot q(\Delta t) &= \dot q(0) + \Delta tF(q(0))\left[ {q(0) +
+\frac{{\Delta t}}{{2m}}\dot q(0)} \right], %
+\label{introEquation:positionVerlet1} \\%
+%
+q(\Delta t) &= q(0) + \frac{{\Delta t}}{2}\left[ {\dot q(0) + \dot
+q(\Delta t)} \right]. %
+ \label{introEquation:positionVerlet1}
+\end{align}
+
+\subsubsection{\label{introSection:errorAnalysis}Error Analysis and Higher Order Methods}
+
+Baker-Campbell-Hausdorff formula can be used to determine the local
+error of splitting method in terms of commutator of the
+operators(\ref{introEquation:exponentialOperator}) associated with
+the sub-flow. For operators $hX$ and $hY$ which are associate to
+$\varphi_1(t)$ and $\varphi_2(t$ respectively , we have
+\begin{equation}
+\exp (hX + hY) = \exp (hZ)
+\end{equation}
+where
+\begin{equation}
+hZ = hX + hY + \frac{{h^2 }}{2}[X,Y] + \frac{{h^3 }}{2}\left(
+{[X,[X,Y]] + [Y,[Y,X]]} \right) +  \ldots .
+\end{equation}
+Here, $[X,Y]$ is the commutators of operator $X$ and $Y$ given by
+\[
+[X,Y] = XY - YX .
+\]
+Applying Baker-Campbell-Hausdorff formula to Sprang splitting, we
+can obtain
+\begin{eqnarray*}
+\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2
+[X,Y]/4 + h^2 [Y,X]/4 \\ & & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\
+& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3 [X,[X,Y]]/24 & & \mbox{} +
+\ldots )
+\end{eqnarray*}
+Since \[ [X,Y] + [Y,X] = 0\] and \[ [X,X] = 0\], the dominant local
+error of Spring splitting is proportional to $h^3$. The same
+procedure can be applied to general splitting,  of the form
+\begin{equation}
+\varphi _{b_m h}^2  \circ \varphi _{a_m h}^1  \circ \varphi _{b_{m -
+1} h}^2  \circ  \ldots  \circ \varphi _{a_1 h}^1 .
+\end{equation}
+Careful choice of coefficient $a_1 ,\ldot , b_m$ will lead to higher
+order method. Yoshida proposed an elegant way to compose higher
+order methods based on symmetric splitting. Given a symmetric second
+order base method $ \varphi _h^{(2)} $, a fourth-order symmetric
+method can be constructed by composing,
+\[
+\varphi _h^{(4)}  = \varphi _{\alpha h}^{(2)}  \circ \varphi _{\beta
+h}^{(2)}  \circ \varphi _{\alpha h}^{(2)}
+\]
+where $ \alpha  =  - \frac{{2^{1/3} }}{{2 - 2^{1/3} }}$ and $ \beta
+= \frac{{2^{1/3} }}{{2 - 2^{1/3} }}$. Moreover, a symmetric
+integrator $ \varphi _h^{(2n + 2)}$ can be composed by
+\begin{equation}
+\varphi _h^{(2n + 2)}  = \varphi _{\alpha h}^{(2n)}  \circ \varphi
+_{\beta h}^{(2n)}  \circ \varphi _{\alpha h}^{(2n)}
+\end{equation}
+, if the weights are chosen as
+\[
+\alpha  =  - \frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }},\beta =
+\frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }} .
+\]
+
+\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{introSec:forceEvaluation}Force Evaluation}
+
+\subsection{\label{introSection:mdIntegration} Integration of the Equations of Motion}
+
+\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies}
+
+Rigid bodies are frequently involved in the modeling of different
+areas, from engineering, physics, to chemistry. For example,
+missiles and vehicle are usually modeled by rigid bodies.  The
+movement of the objects in 3D gaming engine or other physics
+simulator is governed by the rigid body dynamics. In molecular
+simulation, rigid body is used to simplify the model in
+protein-protein docking study{\cite{Gray03}}.
+
+It is very important to develop stable and efficient methods to
+integrate the equations of motion of orientational degrees of
+freedom. Euler angles are the nature choice to describe the
+rotational degrees of freedom. However, due to its singularity, the
+numerical integration of corresponding equations of motion is very
+inefficient and inaccurate. Although an alternative integrator using
+different sets of Euler angles can overcome this difficulty\cite{},
+the computational penalty and the lost of angular momentum
+conservation still remain. A singularity free representation
+utilizing quaternions was developed by Evans in 1977. Unfortunately,
+this approach suffer from the nonseparable Hamiltonian resulted from
+quaternion representation, which prevents the symplectic algorithm
+to be utilized. Another different approach is to apply holonomic
+constraints to the atoms belonging to the rigid body. Each atom
+moves independently under the normal forces deriving from potential
+energy and constraint forces which are used to guarantee the
+rigidness. However, due to their iterative nature, SHAKE and Rattle
+algorithm converge very slowly when the number of constraint
+increases.
+
+The break through in geometric literature suggests that, in order to
+develop a long-term integration scheme, one should preserve the
+symplectic structure of the flow. Introducing conjugate momentum to
+rotation matrix $A$ and re-formulating Hamiltonian's equation, a
+symplectic integrator, RSHAKE, was proposed to evolve the
+Hamiltonian system in a constraint manifold by iteratively
+satisfying the orthogonality constraint $A_t A = 1$. An alternative
+method using quaternion representation was developed by Omelyan.
+However, both of these methods are iterative and inefficient. In
+this section, we will present a symplectic Lie-Poisson integrator
+for rigid body developed by Dullweber and his coworkers\cite{}.
+
+\subsection{\label{introSection:lieAlgebra}Lie Algebra}
+
+\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Body}
+
+\begin{equation}
+H = \frac{1}{2}(p^T m^{ - 1} p) + \frac{1}{2}tr(PJ^{ - 1} P) +
+V(q,Q) + \frac{1}{2}tr[(QQ^T  - 1)\Lambda ].
+\label{introEquation:RBHamiltonian}
+\end{equation}
+Here, $q$ and $Q$  are the position and rotation matrix for the
+rigid-body, $p$ and $P$  are conjugate momenta to $q$  and $Q$ , and
+$J$, a diagonal matrix, is defined by
+\[
+I_{ii}^{ - 1}  = \frac{1}{2}\sum\limits_{i \ne j} {J_{jj}^{ - 1} }
+\]
+where $I_{ii}$ is the diagonal element of the inertia tensor. This
+constrained Hamiltonian equation subjects to a holonomic constraint,
+\begin{equation}
+Q^T Q = 1$, \label{introEquation:orthogonalConstraint}
+\end{equation}
+which is used to ensure rotation matrix's orthogonality.
+Differentiating \ref{introEquation:orthogonalConstraint} and using
+Equation \ref{introEquation:RBMotionMomentum}, one may obtain,
+\begin{equation}
+Q^t PJ^{ - 1}  + J^{ - 1} P^t Q = 0 . \\
+\label{introEquation:RBFirstOrderConstraint}
+\end{equation}
+
+Using Equation (\ref{introEquation:motionHamiltonianCoordinate},
+\ref{introEquation:motionHamiltonianMomentum}), one can write down
+the equations of motion,
+\[
+\begin{array}{c}
+ \frac{{dq}}{{dt}} = \frac{p}{m} \label{introEquation:RBMotionPosition}\\
+ \frac{{dp}}{{dt}} =  - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\
+ \frac{{dQ}}{{dt}} = PJ^{ - 1}  \label{introEquation:RBMotionRotation}\\
+ \frac{{dP}}{{dt}} =  - \nabla _q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}\\
+ \end{array}
+\]
+
+
+\[
+M = \left\{ {(Q,P):Q^T Q = 1,Q^t PJ^{ - 1}  + J^{ - 1} P^t Q = 0}
+\right\} .
+\]
+
+\subsection{\label{introSection:DLMMotionEquation}The Euler Equations of Rigid Body Motion}
+
+\subsection{\label{introSection:symplecticDiscretizationRB}Symplectic Discretization of Euler Equations}
+
+
 \section{\label{introSection:langevinDynamics}Langevin Dynamics}
 
+\subsection{\label{introSection:LDIntroduction}Introduction and application of Langevin Dynamics}
+
 \subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics}
 
-\subsection{\label{introSection:hydroynamics}Hydrodynamics}
+\begin{equation}
+H = \frac{{p^2 }}{{2m}} + U(x) + H_B  + \Delta U(x,x_1 , \ldots x_N)
+\label{introEquation:bathGLE}
+\end{equation}
+where $H_B$ is harmonic bath Hamiltonian,
+\[
+H_B  =\sum\limits_{\alpha  = 1}^N {\left\{ {\frac{{p_\alpha ^2
+}}{{2m_\alpha  }} + \frac{1}{2}m_\alpha  w_\alpha ^2 } \right\}}
+\]
+and $\Delta U$ is bilinear system-bath coupling,
+\[
+\Delta U =  - \sum\limits_{\alpha  = 1}^N {g_\alpha  x_\alpha  x}
+\]
+Completing the square,
+\[
+H_B  + \Delta U = \sum\limits_{\alpha  = 1}^N {\left\{
+{\frac{{p_\alpha ^2 }}{{2m_\alpha  }} + \frac{1}{2}m_\alpha
+w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha
+w_\alpha ^2 }}x} \right)^2 } \right\}}  - \sum\limits_{\alpha  =
+1}^N {\frac{{g_\alpha ^2 }}{{2m_\alpha  w_\alpha ^2 }}} x^2
+\]
+and putting it back into Eq.~\ref{introEquation:bathGLE},
+\[
+H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha  = 1}^N
+{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha  }} + \frac{1}{2}m_\alpha
+w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha
+w_\alpha ^2 }}x} \right)^2 } \right\}}
+\]
+where
+\[
+W(x) = U(x) - \sum\limits_{\alpha  = 1}^N {\frac{{g_\alpha ^2
+}}{{2m_\alpha  w_\alpha ^2 }}} x^2
+\]
+Since the first two terms of the new Hamiltonian depend only on the
+system coordinates, we can get the equations of motion for
+Generalized Langevin Dynamics by Hamilton's equations
+\ref{introEquation:motionHamiltonianCoordinate,
+introEquation:motionHamiltonianMomentum},
+\begin{align}
+\dot p &=  - \frac{{\partial H}}{{\partial x}}
+       &= m\ddot x
+       &= - \frac{{\partial W(x)}}{{\partial x}} - \sum\limits_{\alpha  = 1}^N {g_\alpha  \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha  w_\alpha ^2 }}x} \right)}
+\label{introEquation:Lp5}
+\end{align}
+, and
+\begin{align}
+\dot p_\alpha   &=  - \frac{{\partial H}}{{\partial x_\alpha  }}
+                &= m\ddot x_\alpha
+                &= \- m_\alpha  w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha}}{{m_\alpha  w_\alpha ^2 }}x} \right)
+\end{align}
+
+\subsection{\label{introSection:laplaceTransform}The Laplace Transform}
+
+\[
+L(x) = \int_0^\infty  {x(t)e^{ - pt} dt}
+\]
+
+\[
+L(x + y) = L(x) + L(y)
+\]
+
+\[
+L(ax) = aL(x)
+\]
+
+\[
+L(\dot x) = pL(x) - px(0)
+\]
+
+\[
+L(\ddot x) = p^2 L(x) - px(0) - \dot x(0)
+\]
+
+\[
+L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right) = G(p)H(p)
+\]
+
+Some relatively important transformation,
+\[
+L(\cos at) = \frac{p}{{p^2  + a^2 }}
+\]
+
+\[
+L(\sin at) = \frac{a}{{p^2  + a^2 }}
+\]
+
+\[
+L(1) = \frac{1}{p}
+\]
+
+First, the bath coordinates,
+\[
+p^2 L(x_\alpha  ) - px_\alpha  (0) - \dot x_\alpha  (0) =  - \omega
+_\alpha ^2 L(x_\alpha  ) + \frac{{g_\alpha  }}{{\omega _\alpha
+}}L(x)
+\]
+\[
+L(x_\alpha  ) = \frac{{\frac{{g_\alpha  }}{{\omega _\alpha  }}L(x) +
+px_\alpha  (0) + \dot x_\alpha  (0)}}{{p^2  + \omega _\alpha ^2 }}
+\]
+Then, the system coordinates,
+\begin{align}
+mL(\ddot x) &=  - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} -
+\sum\limits_{\alpha  = 1}^N {\left\{ {\frac{{\frac{{g_\alpha
+}}{{\omega _\alpha  }}L(x) + px_\alpha  (0) + \dot x_\alpha
+(0)}}{{p^2  + \omega _\alpha ^2 }} - \frac{{g_\alpha ^2 }}{{m_\alpha
+}}\omega _\alpha ^2 L(x)} \right\}}
+%
+ &= - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} -
+ \sum\limits_{\alpha  = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha  \omega _\alpha ^2 }}\frac{p}{{p^2  + \omega _\alpha ^2 }}pL(x)
+ - \frac{p}{{p^2  + \omega _\alpha ^2 }}g_\alpha  x_\alpha  (0)
+ - \frac{1}{{p^2  + \omega _\alpha ^2 }}g_\alpha  \dot x_\alpha  (0)} \right\}}
+\end{align}
+Then, the inverse transform,
+
+\begin{align}
+m\ddot x &=  - \frac{{\partial W(x)}}{{\partial x}} -
+\sum\limits_{\alpha  = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2
+}}{{m_\alpha  \omega _\alpha ^2 }}} \right)\int_0^t {\cos (\omega
+_\alpha  t)\dot x(t - \tau )d\tau  - \left[ {g_\alpha  x_\alpha  (0)
+- \frac{{g_\alpha  }}{{m_\alpha  \omega _\alpha  }}} \right]\cos
+(\omega _\alpha  t) - \frac{{g_\alpha  \dot x_\alpha  (0)}}{{\omega
+_\alpha  }}\sin (\omega _\alpha  t)} } \right\}}
+%
+&= - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t
+{\sum\limits_{\alpha  = 1}^N {\left( { - \frac{{g_\alpha ^2
+}}{{m_\alpha  \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha
+t)\dot x(t - \tau )d} \tau }  + \sum\limits_{\alpha  = 1}^N {\left\{
+{\left[ {g_\alpha  x_\alpha  (0) - \frac{{g_\alpha  }}{{m_\alpha
+\omega _\alpha  }}} \right]\cos (\omega _\alpha  t) +
+\frac{{g_\alpha  \dot x_\alpha  (0)}}{{\omega _\alpha  }}\sin
+(\omega _\alpha  t)} \right\}}
+\end{align}
+
+\begin{equation}
+m\ddot x =  - \frac{{\partial W}}{{\partial x}} - \int_0^t {\xi
+(t)\dot x(t - \tau )d\tau }  + R(t)
+\label{introEuqation:GeneralizedLangevinDynamics}
+\end{equation}
+%where $ {\xi (t)}$ is friction kernel, $R(t)$ is random force and
+%$W$ is the potential of mean force. $W(x) =  - kT\ln p(x)$
+\[
+\xi (t) = \sum\limits_{\alpha  = 1}^N {\left( { - \frac{{g_\alpha ^2
+}}{{m_\alpha  \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha  t)}
+\]
+For an infinite harmonic bath, we can use the spectral density and
+an integral over frequencies.
+
+\[
+R(t) = \sum\limits_{\alpha  = 1}^N {\left( {g_\alpha  x_\alpha  (0)
+- \frac{{g_\alpha ^2 }}{{m_\alpha  \omega _\alpha ^2 }}x(0)}
+\right)\cos (\omega _\alpha  t)}  + \frac{{\dot x_\alpha
+(0)}}{{\omega _\alpha  }}\sin (\omega _\alpha  t)
+\]
+The random forces depend only on initial conditions.
+
+\subsubsection{\label{introSection:secondFluctuationDissipation}The Second Fluctuation Dissipation Theorem}
+So we can define a new set of coordinates,
+\[
+q_\alpha  (t) = x_\alpha  (t) - \frac{1}{{m_\alpha  \omega _\alpha
+^2 }}x(0)
+\]
+This makes
+\[
+R(t) = \sum\limits_{\alpha  = 1}^N {g_\alpha  q_\alpha  (t)}
+\]
+And since the $q$ coordinates are harmonic oscillators,
+\[
+\begin{array}{l}
+ \left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle  = \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha  t) \\
+ \left\langle {q_\alpha  (t)q_\beta  (0)} \right\rangle  = \delta _{\alpha \beta } \left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle  \\
+ \end{array}
+\]
+
+\begin{align}
+\left\langle {R(t)R(0)} \right\rangle  &= \sum\limits_\alpha
+{\sum\limits_\beta  {g_\alpha  g_\beta  \left\langle {q_\alpha
+(t)q_\beta  (0)} \right\rangle } }
+%
+&= \sum\limits_\alpha  {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)}
+\right\rangle \cos (\omega _\alpha  t)}
+%
+&= kT\xi (t)
+\end{align}
+
+\begin{equation}
+\xi (t) = \left\langle {R(t)R(0)} \right\rangle
+\label{introEquation:secondFluctuationDissipation}
+\end{equation}
+
+\section{\label{introSection:hydroynamics}Hydrodynamics}
+
+\subsection{\label{introSection:frictionTensor} Friction Tensor}
+\subsection{\label{introSection:analyticalApproach}Analytical
+Approach}
+
+\subsection{\label{introSection:approximationApproach}Approximation
+Approach}
+
+\subsection{\label{introSection:centersRigidBody}Centers of Rigid
+Body}
+
+\section{\label{introSection:correlationFunctions}Correlation Functions}
