--- trunk/tengDissertation/Introduction.tex	2006/04/07 22:05:48	2698
+++ trunk/tengDissertation/Introduction.tex	2006/04/10 05:35:55	2699
@@ -312,7 +312,7 @@ f(r) = J\nabla _x H(r)
 \end{equation}
 where $x = x(q,p)^T$, this system is canonical Hamiltonian, if
 \begin{equation}
-f(r) = J\nabla _x H(r)
+f(r) = J\nabla _x H(r).
 \end{equation}
 $H = H (q, p)$ is Hamiltonian function and $J$ is the skew-symmetric
 matrix
@@ -372,18 +372,18 @@ roles in numerical studies. The flow of a Hamiltonian 
 \end{equation}
 
 The hidden geometric properties of ODE and its flow play important
-roles in numerical studies. The flow of a Hamiltonian vector field
-on a symplectic manifold is a symplectomorphism. Let $\varphi$ be
-the flow of Hamiltonian vector field, $\varphi$ is a
-\emph{symplectic} flow if it satisfies,
+roles in numerical studies. Let $\varphi$ be the flow of Hamiltonian
+vector field, $\varphi$ is a \emph{symplectic} flow if it satisfies,
 \begin{equation}
-d \varphi^T J d \varphi = J.
+'\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. As to the Poisson system,
+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}
-d\varphi ^T Jd\varphi  = J \circ \varphi
+'\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($
@@ -399,8 +399,52 @@ the structural properties of the original ODE and its 
 designing any numerical methods, one should always try to preserve
 the structural properties of the original ODE and its flow.
 
-\subsection{\label{introSection:splittingAndComposition}Splitting and Composition Methods}
+\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\cite{}. In dissipative systems, variational
+methods can capture the decay of energy accurately\cite{}. 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 \cite{}. 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}. 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. Let $\phi$ and $\psi$ both be
+symplectic maps, 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}
+Suppose that a Hamiltonian system has a form with $H = T + V$
 
+
+
 \section{\label{introSection:molecularDynamics}Molecular Dynamics}
 
 As a special discipline of molecular modeling, Molecular dynamics