--- trunk/tengDissertation/Langevin.tex	2006/06/11 02:06:01	2851
+++ trunk/tengDissertation/Langevin.tex	2006/06/11 02:23:00	2853
@@ -1,5 +1,5 @@
 
-\chapter{\label{chapt:methodology}Langevin Dynamics for Rigid Bodies of Arbitrary Shape}
+\chapter{\label{chapt:methodology}LANGEVIN DYNAMICS for RIGID BODIES of ARBITRARY SHAPE}
 
 \section{Introduction}
 
@@ -47,59 +47,6 @@ efficiently\cite{Sandu1999}.
 has been shown to be able to damp out the resonance artifact more
 efficiently\cite{Sandu1999}.
 
-%review rigid body dynamics
-Rigid bodies are frequently involved in the modeling of different
-areas, from engineering, physics, to chemistry. For example,
-missiles and vehicle are usually modeled by rigid bodies.  The
-movement of the objects in 3D gaming engine or other physics
-simulator is governed by the rigid body dynamics. In molecular
-simulation, rigid body is used to simplify the model in
-protein-protein docking study{\cite{Gray2003}}.
-
-It is very important to develop stable and efficient methods to
-integrate the equations of motion of orientational degrees of
-freedom. Euler angles are the nature choice to describe the
-rotational degrees of freedom. However, due to its singularity, the
-numerical integration of corresponding equations of motion is very
-inefficient and inaccurate. Although an alternative integrator using
-different sets of Euler angles can overcome this difficulty\cite{},
-the computational penalty and the lost of angular momentum
-conservation still remain. In 1977, a singularity free
-representation utilizing quaternions was developed by
-Evans\cite{Evans1977}. Unfortunately, this approach suffer from the
-nonseparable Hamiltonian resulted from quaternion representation,
-which prevents the symplectic algorithm to be utilized. Another
-different approach is to apply holonomic constraints to the atoms
-belonging to the rigid body\cite{}. Each atom moves independently
-under the normal forces deriving from potential energy and
-constraint forces which are used to guarantee the rigidness.
-However, due to their iterative nature, SHAKE and Rattle algorithm
-converge very slowly when the number of constraint increases.
-
-The break through in geometric literature suggests that, in order to
-develop a long-term integration scheme, one should preserve the
-geometric structure of the flow. Matubayasi and Nakahara developed a
-time-reversible integrator for rigid bodies in quaternion
-representation. Although it is not symplectic, this integrator still
-demonstrates a better long-time energy conservation than traditional
-methods because of the time-reversible nature. Extending
-Trotter-Suzuki to general system with a flat phase space, Miller and
-his colleagues devised an novel symplectic, time-reversible and
-volume-preserving integrator in quaternion representation. However,
-all of the integrators in quaternion representation suffer from the
-computational penalty of constructing a rotation matrix from
-quaternions to evolve coordinates and velocities at every time step.
-An alternative integration scheme utilizing rotation matrix directly
-is RSHAKE , in which a conjugate momentum to rotation matrix is
-introduced to re-formulate the Hamiltonian's equation and the
-Hamiltonian is evolved in a constraint manifold by iteratively
-satisfying the orthogonality constraint. However, RSHAKE is
-inefficient because of the iterative procedure. An extremely
-efficient integration scheme in rotation matrix representation,
-which also preserves the same structural properties of the
-Hamiltonian flow as Miller's integrator, is proposed by Dullweber,
-Leimkuhler and McLachlan (DLM).
-
 %review langevin/browninan dynamics for arbitrarily shaped rigid body
 Combining Langevin or Brownian dynamics with rigid body dynamics,
 one can study the slow processes in biomolecular systems. Modeling
@@ -153,7 +100,7 @@ sophisticated rigid body dynamics.
 
 \section{Method{\label{methodSec}}}
 
-\subsection{\label{introSection:frictionTensor} Friction Tensor}
+\subsection{\label{introSection:frictionTensor}Friction Tensor}
 Theoretically, the friction kernel can be determined using velocity
 autocorrelation function. However, this approach become impractical
 when the system become more and more complicate. Instead, various