--- trunk/tengDissertation/Introduction.tex	2006/04/12 21:20:13	2705
+++ trunk/tengDissertation/Introduction.tex	2006/04/13 04:47:47	2706
@@ -315,8 +315,7 @@ partition function like,
 isolated and conserve energy, Microcanonical ensemble(NVE) has a
 partition function like,
 \begin{equation}
-\Omega (N,V,E) = e^{\beta TS}
-\label{introEqaution:NVEPartition}.
+\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
@@ -771,8 +770,7 @@ _{1,h/2} ,
 splitting gives a second-order decomposition,
 \begin{equation}
 \varphi _h  = \varphi _{1,h/2}  \circ \varphi _{2,h}  \circ \varphi
-_{1,h/2} ,
-\label{introEqaution:secondOrderSplitting}
+_{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
@@ -955,11 +953,54 @@ for rigid body developed by Dullweber and his coworker
 
 \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:otherRBMotionEquation}Other Formulations for Rigid Body Motion}
+\subsection{\label{introSection:symplecticDiscretizationRB}Symplectic Discretization of Euler Equations}
 
-\section{\label{introSection:correlationFunctions}Correlation Functions}
 
 \section{\label{introSection:langevinDynamics}Langevin Dynamics}
 
@@ -1168,3 +1209,5 @@ Body}
 
 \subsection{\label{introSection:centersRigidBody}Centers of Rigid
 Body}
+
+\section{\label{introSection:correlationFunctions}Correlation Functions}