--- trunk/tengDissertation/Introduction.tex	2006/04/06 22:06:50	2696
+++ trunk/tengDissertation/Introduction.tex	2006/04/07 05:03:54	2697
@@ -237,41 +237,6 @@ can be rewritten as,
 \label{introEquation:compactHamiltonian}
 \end{equation}
 
-\section{\label{introSection:geometricIntegratos}Geometric Integrators}
-
-\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 $(M, \omega)$ consisting of a \emph{differentiable
-manifold} $M$ and a close, non-degenerated, bilinear symplectic
-form, $\omega$. One of the motivation 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.
-
-A \emph{symplectomorphism} is also known as a \emph{canonical
-transformation}.
-
-Any real-valued differentiable function H on a symplectic manifold
-can serve as an energy function or Hamiltonian. Associated to any
-Hamiltonian is a Hamiltonian vector field; the integral curves of
-the Hamiltonian vector field are solutions to the Hamilton-Jacobi
-equations. The Hamiltonian vector field defines a flow on the
-symplectic manifold, called a Hamiltonian flow or symplectomorphism.
-By Liouville's theorem, Hamiltonian flows preserve the volume form
-on the phase space.
-
-\subsection{\label{Construction of Symplectic Methods}}
-
 \section{\label{introSection:statisticalMechanics}Statistical
 Mechanics}
 
@@ -312,7 +277,62 @@ will be the best choice\cite{Frenkel1996}.
 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. According to Liouville's theorem, the
+Hamiltonian \emph{flow} or \emph{symplectomorphism} generated by the
+Hamiltonian vector filed preserves the volume form on the phase
+space, which is the basis of classical statistical mechanics.
+
+\subsection{\label{introSection:exactFlow}The Exact Flow of ODE}
+
+\subsection{\label{introSection:hamiltonianSplitting}Hamiltonian Splitting}
+
 \section{\label{introSection:molecularDynamics}Molecular Dynamics}
 
 As a special discipline of molecular modeling, Molecular dynamics
