--- trunk/tengDissertation/Introduction.tex	2006/04/12 03:37:46	2702
+++ trunk/tengDissertation/Introduction.tex	2006/04/12 04:13:16	2703
@@ -635,7 +635,7 @@ a \emph{symplectic} flow if it satisfies,
 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.
+{\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
@@ -643,7 +643,7 @@ symplectomorphism. As to the Poisson system,
 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
+{\varphi '}^T J \varphi ' = J \circ \varphi
 \end{equation}
 is the property must be preserved by the integrator.
 
@@ -685,11 +685,11 @@ instability \cite{}. However, due to computational pen
 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}.
+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}
 
@@ -744,7 +744,7 @@ symmetric property,
 symmetric property,
 \begin{equation}
 \varphi _h^{ - 1} = \varphi _{ - h}.
-\lable{introEquation:timeReversible}
+\label{introEquation:timeReversible}
 \end{equation}
 
 \subsubsection{\label{introSection:exampleSplittingMethod}Example of Splitting Method}
@@ -802,7 +802,7 @@ q(\Delta t) = q(0) + \frac{{\Delta t}}{2}\left[ {\dot 
 \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) &= q(0) + \frac{{\Delta t}}{2}\left[ {\dot q(0) + \dot
 q(\Delta t)} \right]. %
  \label{introEquation:positionVerlet1}
 \end{align}
@@ -828,11 +828,12 @@ can obtain
 \]
 Applying Baker-Campbell-Hausdorff formula to Sprang splitting, we
 can obtain
-\begin{eqnarray}
+\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 +
-h^3 [Y,[Y,X]]/12 - h^3 [X,[X,Y]]/24 +  \ldots )
-\end{eqnarray}
+[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
