trunk/tengDissertation/Introduction.tex

\chapter{\label{chapt:introduction}INTRODUCTION AND THEORETICAL BACKGROUND}

\section{\label{introSection:molecularDynamics}Molecular Dynamics}

As a special discipline of molecular modeling, Molecular dynamics
has proven to be a powerful tool for studying the functions of
biological systems, providing structural, thermodynamic and
dynamical information.

\subsection{\label{introSection:classicalMechanics}Classical Mechanics}

Closely related to Classical Mechanics, Molecular Dynamics
simulations are carried out by integrating the equations of motion
for a given system of particles. There are three fundamental ideas
behind classical mechanics. Firstly, One can determine the state of
a mechanical system at any time of interest; Secondly, all the
mechanical properties of the system at that time can be determined
by combining the knowledge of the properties of the system with the
specification of this state; Finally, the specification of the state
when further combine with the laws of mechanics will also be
sufficient to predict the future behavior of the system.

\subsubsection{\label{introSection:newtonian}Newtonian Mechanics}

\subsubsection{\label{introSection:lagrangian}Lagrangian Mechanics}

Newtonian Mechanics suffers from two important limitations: it
describes their motion in special cartesian coordinate systems.
Another limitation of Newtonian mechanics becomes obvious when we
try to describe systems with large numbers of particles. It becomes
very difficult to predict the properties of the system by carrying
out calculations involving the each individual interaction between
all the particles, even if we know all of the details of the
interaction. In order to overcome some of the practical difficulties
which arise in attempts to apply Newton's equation to complex
system, alternative procedures may be developed.

\subsubsubsection{\label{introSection:halmiltonPrinciple}Hamilton's
Principle}

Hamilton introduced the dynamical principle upon which it is
possible to base all of mechanics and, indeed, most of classical
physics. Hamilton's Principle may be stated as follow,

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} ,
\lable{introEquation:halmitonianPrinciple1}
\end{equation}

For simple mechanical systems, where the forces acting on the
different part are derivable from a potential and the velocities are
small compared with that of light, the Lagrangian function $L$ can
be define as the difference between the kinetic energy of the system
and its potential energy,
\begin{equation}
L \equiv K - U = L(q_i ,\dot q_i ) ,
\label{introEquation:lagrangianDef}
\end{equation}
then Eq.~\ref{introEquation:halmitonianPrinciple1} becomes
\begin{equation}
\delta \int_{t_1 }^{t_2 } {K dt = 0} ,
\lable{introEquation:halmitonianPrinciple2}
\end{equation}

\subsubsubsection{\label{introSection:equationOfMotionLagrangian}The
Equations of Motion in Lagrangian Mechanics}

for a holonomic system of $f$ degrees of freedom, the equations of
motion in the Lagrangian form is
\begin{equation}
\frac{d}{{dt}}\frac{{\partial L}}{{\partial \dot q_i }} -
\frac{{\partial L}}{{\partial q_i }} = 0,{\rm{ }}i = 1, \ldots,f
\lable{introEquation:eqMotionLagrangian}
\end{equation}
where $q_{i}$ is generalized coordinate and $\dot{q_{i}}$ is
generalized velocity.

\subsubsection{\label{introSection:hamiltonian}Hamiltonian Mechanics}

Arising from Lagrangian Mechanics, Hamiltonian Mechanics was
introduced by William Rowan Hamilton in 1833 as a re-formulation of
classical mechanics. If the potential energy of a system is
independent of generalized velocities, the generalized momenta can
be defined as
\begin{equation}
p_i = \frac{\partial L}{\partial \dot q_i}
\label{introEquation:generalizedMomenta}
\end{equation}
With the help of these momenta, we may now define a new quantity $H$
by the equation
\begin{equation}
H = p_1 \dot q_1  +  \ldots  + p_f \dot q_f  - L,
\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.

An important difference between Lagrangian approach and the
Hamiltonian approach is that the Lagrangian is considered to be a
function of the generalized velocities $\dot q_i$ and the
generalized coordinates $q_i$, while the Hamiltonian is considered
to be a function of the generalized momenta $p_i$ and the conjugate
generalized coordinate $q_i$. Hamiltonian Mechanics is more
appropriate for application to statistical mechanics and quantum
mechanics, since it treats the coordinate and its time derivative as
independent variables and it only works with 1st-order differential
equations.


\subsubsection{\label{introSection:canonicalTransformation}Canonical Transformation}

\subsection{\label{introSection:statisticalMechanics}Statistical Mechanics}

The thermodynamic behaviors and properties  of Molecular Dynamics
simulation are governed by the principle of Statistical Mechanics.
The following section will give a brief introduction to some of the
Statistical Mechanics concepts presented in this dissertation.

\subsubsection{\label{introSection::ensemble}Ensemble}

\subsubsection{\label{introSection:ergodic}The Ergodic Hypothesis}

\subsection{\label{introSection:rigidBody}Dynamics of Rigid Bodies}

\subsection{\label{introSection:correlationFunctions}Correlation Functions}

\section{\label{introSection:langevinDynamics}Langevin Dynamics}

\subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics}

\subsection{\label{introSection:hydroynamics}Hydrodynamics}
Revision:	2692
Committed:	Tue Apr 4 21:32:58 2006 UTC (18 years, 3 months ago) by tim
Content type:	application/x-tex
File size:	5852 byte(s)
Log Message:	adding Lagrangian Mechanics and Hamiltonian Mechanics
#	Content
1	\chapter{\label{chapt:introduction}INTRODUCTION AND THEORETICAL BACKGROUND}
2
3	\section{\label{introSection:molecularDynamics}Molecular Dynamics}
4
5	As a special discipline of molecular modeling, Molecular dynamics
6	has proven to be a powerful tool for studying the functions of
7	biological systems, providing structural, thermodynamic and
8	dynamical information.
9
10	\subsection{\label{introSection:classicalMechanics}Classical Mechanics}
11
12	Closely related to Classical Mechanics, Molecular Dynamics
13	simulations are carried out by integrating the equations of motion
14	for a given system of particles. There are three fundamental ideas
15	behind classical mechanics. Firstly, One can determine the state of
16	a mechanical system at any time of interest; Secondly, all the
17	mechanical properties of the system at that time can be determined
18	by combining the knowledge of the properties of the system with the
19	specification of this state; Finally, the specification of the state
20	when further combine with the laws of mechanics will also be
21	sufficient to predict the future behavior of the system.
22
23	\subsubsection{\label{introSection:newtonian}Newtonian Mechanics}
24
25	\subsubsection{\label{introSection:lagrangian}Lagrangian Mechanics}
26
27	Newtonian Mechanics suffers from two important limitations: it
28	describes their motion in special cartesian coordinate systems.
29	Another limitation of Newtonian mechanics becomes obvious when we
30	try to describe systems with large numbers of particles. It becomes
31	very difficult to predict the properties of the system by carrying
32	out calculations involving the each individual interaction between
33	all the particles, even if we know all of the details of the
34	interaction. In order to overcome some of the practical difficulties
35	which arise in attempts to apply Newton's equation to complex
36	system, alternative procedures may be developed.
37
38	\subsubsubsection{\label{introSection:halmiltonPrinciple}Hamilton's
39	Principle}
40
41	Hamilton introduced the dynamical principle upon which it is
42	possible to base all of mechanics and, indeed, most of classical
43	physics. Hamilton's Principle may be stated as follow,
44
45	The actual trajectory, along which a dynamical system may move from
46	one point to another within a specified time, is derived by finding
47	the path which minimizes the time integral of the difference between
48	the kinetic, $K$, and potential energies, $U$.
49	\begin{equation}
50	\delta \int_{t_1 }^{t_2 } {(K - U)dt = 0} ,
51	\lable{introEquation:halmitonianPrinciple1}
52	\end{equation}
53
54	For simple mechanical systems, where the forces acting on the
55	different part are derivable from a potential and the velocities are
56	small compared with that of light, the Lagrangian function $L$ can
57	be define as the difference between the kinetic energy of the system
58	and its potential energy,
59	\begin{equation}
60	L \equiv K - U = L(q_i ,\dot q_i ) ,
61	\label{introEquation:lagrangianDef}
62	\end{equation}
63	then Eq.~\ref{introEquation:halmitonianPrinciple1} becomes
64	\begin{equation}
65	\delta \int_{t_1 }^{t_2 } {K dt = 0} ,
66	\lable{introEquation:halmitonianPrinciple2}
67	\end{equation}
68
69	\subsubsubsection{\label{introSection:equationOfMotionLagrangian}The
70	Equations of Motion in Lagrangian Mechanics}
71
72	for a holonomic system of $f$ degrees of freedom, the equations of
73	motion in the Lagrangian form is
74	\begin{equation}
75	\frac{d}{{dt}}\frac{{\partial L}}{{\partial \dot q_i }} -
76	\frac{{\partial L}}{{\partial q_i }} = 0,{\rm{ }}i = 1, \ldots,f
77	\lable{introEquation:eqMotionLagrangian}
78	\end{equation}
79	where $q_{i}$ is generalized coordinate and $\dot{q_{i}}$ is
80	generalized velocity.
81
82	\subsubsection{\label{introSection:hamiltonian}Hamiltonian Mechanics}
83
84	Arising from Lagrangian Mechanics, Hamiltonian Mechanics was
85	introduced by William Rowan Hamilton in 1833 as a re-formulation of
86	classical mechanics. If the potential energy of a system is
87	independent of generalized velocities, the generalized momenta can
88	be defined as
89	\begin{equation}
90	p_i = \frac{\partial L}{\partial \dot q_i}
91	\label{introEquation:generalizedMomenta}
92	\end{equation}
93	With the help of these momenta, we may now define a new quantity $H$
94	by the equation
95	\begin{equation}
96	H = p_1 \dot q_1 + \ldots + p_f \dot q_f - L,
97	\label{introEquation:hamiltonianDefByLagrangian}
98	\end{equation}
99	where $ \dot q_1 \ldots \dot q_f $ are generalized velocities and
100	$L$ is the Lagrangian function for the system.
101
102	An important difference between Lagrangian approach and the
103	Hamiltonian approach is that the Lagrangian is considered to be a
104	function of the generalized velocities $\dot q_i$ and the
105	generalized coordinates $q_i$, while the Hamiltonian is considered
106	to be a function of the generalized momenta $p_i$ and the conjugate
107	generalized coordinate $q_i$. Hamiltonian Mechanics is more
108	appropriate for application to statistical mechanics and quantum
109	mechanics, since it treats the coordinate and its time derivative as
110	independent variables and it only works with 1st-order differential
111	equations.
112
113
114	\subsubsection{\label{introSection:canonicalTransformation}Canonical Transformation}
115
116	\subsection{\label{introSection:statisticalMechanics}Statistical Mechanics}
117
118	The thermodynamic behaviors and properties of Molecular Dynamics
119	simulation are governed by the principle of Statistical Mechanics.
120	The following section will give a brief introduction to some of the
121	Statistical Mechanics concepts presented in this dissertation.
122
123	\subsubsection{\label{introSection::ensemble}Ensemble}
124
125	\subsubsection{\label{introSection:ergodic}The Ergodic Hypothesis}
126
127	\subsection{\label{introSection:rigidBody}Dynamics of Rigid Bodies}
128
129	\subsection{\label{introSection:correlationFunctions}Correlation Functions}
130
131	\section{\label{introSection:langevinDynamics}Langevin Dynamics}
132
133	\subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics}
134
135	\subsection{\label{introSection:hydroynamics}Hydrodynamics}