--- trunk/tengDissertation/Introduction.tex	2006/06/27 02:42:30	2895
+++ trunk/tengDissertation/Introduction.tex	2006/06/27 03:22:42	2898
@@ -31,10 +31,9 @@ F_{ij} = -F_{ji}
 $F_{ji}$ be the force that particle $j$ exerts on particle $i$.
 Newton's third law states that
 \begin{equation}
-F_{ij} = -F_{ji}
+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
@@ -85,17 +84,15 @@ Hamilton's Principle may be stated as follows,
 
 Hamilton introduced the dynamical principle upon which it is
 possible to base all of mechanics and most of classical physics.
-Hamilton's Principle may be stated as follows,
-
-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, $U$.
+Hamilton's Principle may be stated as follows: 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, $U$.
 \begin{equation}
 \delta \int_{t_1 }^{t_2 } {(K - U)dt = 0} ,
 \label{introEquation:halmitonianPrinciple1}
 \end{equation}
-
 For simple mechanical systems, where the forces acting on the
 different parts are derivable from a potential, the Lagrangian
 function $L$ can be defined as the difference between the kinetic
@@ -138,7 +135,6 @@ p_i  = \frac{{\partial L}}{{\partial q_i }}
 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}
@@ -146,10 +142,8 @@ $L$ is the Lagrangian function for the system.
 \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
+$L$ 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
@@ -180,7 +174,6 @@ t}}
 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
@@ -288,11 +281,9 @@ thermodynamic equilibrium.
 statistical characteristics. As a function of macroscopic
 parameters, such as temperature \textit{etc}, the 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, the Microcanonical ensemble (NVE) has
-a partition function like,
+thermodynamic equilibrium. As an ensemble of systems, each of which
+is known to be thermally isolated and conserve energy, the
+Microcanonical ensemble (NVE) has a partition function like,
 \begin{equation}
 \Omega (N,V,E) = e^{\beta TS} \label{introEquation:NVEPartition}.
 \end{equation}
@@ -587,7 +578,6 @@ them can be found in systems which occur naturally in 
 The hidden geometric properties\cite{Budd1999, Marsden1998} of an
 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}
@@ -601,23 +591,19 @@ is the property that must be preserved by the integrat
 \begin{equation}
 {\varphi '}^T J \varphi ' = J \circ \varphi
 \end{equation}
-is the property that must be preserved by the integrator.
-
-It is possible to construct a \emph{volume-preserving} flow for a
-source free ODE ($ \nabla \cdot f = 0 $), 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 an ODE
+is the property that must be preserved by the integrator. It is
+possible to construct a \emph{volume-preserving} flow for a source
+free ODE ($ \nabla \cdot f = 0 $), 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 an ODE
 (Eq.~\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
+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 $\frac{{dx}}{{dt}} = f(x)$ ,
 \[
@@ -630,14 +616,11 @@ smooth function $G$ is given by,
 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{eqnarray}
 \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)). \\
 \label{introEquation:firstIntegral1}
 \end{eqnarray}
-
-
 Using poisson bracket notion, Equation
 \ref{introEquation:firstIntegral1} can be rewritten as
 \[
@@ -650,9 +633,7 @@ is a \emph{first integral}, which is due to the fact $
 \]
 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
+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}
@@ -693,9 +674,8 @@ simpler integration of the system.
 \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,
+simpler integration of the system. Suppose that a Hamiltonian system
+takes the form,
 \[
 H = H_1 + H_2.
 \]