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# Line 7 | Line 7 | dynamical information.
7   biological systems, providing structural, thermodynamic and
8   dynamical information.
9  
10 < \subsection{\label{introSection:classicalMechanics}Classical Mechanics}
10 > \section{\label{introSection:classicalMechanics}Classical
11 > Mechanics}
12  
13   Closely related to Classical Mechanics, Molecular Dynamics
14   simulations are carried out by integrating the equations of motion
# Line 20 | Line 21 | sufficient to predict the future behavior of the syste
21   when further combine with the laws of mechanics will also be
22   sufficient to predict the future behavior of the system.
23  
24 < \subsubsection{\label{introSection:newtonian}Newtonian Mechanics}
24 > \subsection{\label{introSection:newtonian}Newtonian Mechanics}
25  
26 < \subsubsection{\label{introSection:lagrangian}Lagrangian Mechanics}
26 > \subsection{\label{introSection:lagrangian}Lagrangian Mechanics}
27  
28   Newtonian Mechanics suffers from two important limitations: it
29   describes their motion in special cartesian coordinate systems.
# Line 35 | Line 36 | system, alternative procedures may be developed.
36   which arise in attempts to apply Newton's equation to complex
37   system, alternative procedures may be developed.
38  
39 < \subsubsubsection{\label{introSection:halmiltonPrinciple}Hamilton's
39 > \subsection{\label{introSection:halmiltonPrinciple}Hamilton's
40   Principle}
41  
42   Hamilton introduced the dynamical principle upon which it is
# Line 48 | Line 49 | the kinetic, $K$, and potential energies, $U$.
49   the kinetic, $K$, and potential energies, $U$.
50   \begin{equation}
51   \delta \int_{t_1 }^{t_2 } {(K - U)dt = 0} ,
52 < \lable{introEquation:halmitonianPrinciple1}
52 > \label{introEquation:halmitonianPrinciple1}
53   \end{equation}
54  
55   For simple mechanical systems, where the forces acting on the
# Line 62 | Line 63 | then Eq.~\ref{introEquation:halmitonianPrinciple1} bec
63   \end{equation}
64   then Eq.~\ref{introEquation:halmitonianPrinciple1} becomes
65   \begin{equation}
66 < \delta \int_{t_1 }^{t_2 } {K dt = 0} ,
67 < \lable{introEquation:halmitonianPrinciple2}
66 > \delta \int_{t_1 }^{t_2 } {L dt = 0} ,
67 > \label{introEquation:halmitonianPrinciple2}
68   \end{equation}
69  
70 < \subsubsubsection{\label{introSection:equationOfMotionLagrangian}The
70 > \subsection{\label{introSection:equationOfMotionLagrangian}The
71   Equations of Motion in Lagrangian Mechanics}
72  
73   for a holonomic system of $f$ degrees of freedom, the equations of
# Line 74 | Line 75 | motion in the Lagrangian form is
75   \begin{equation}
76   \frac{d}{{dt}}\frac{{\partial L}}{{\partial \dot q_i }} -
77   \frac{{\partial L}}{{\partial q_i }} = 0,{\rm{ }}i = 1, \ldots,f
78 < \lable{introEquation:eqMotionLagrangian}
78 > \label{introEquation:eqMotionLagrangian}
79   \end{equation}
80   where $q_{i}$ is generalized coordinate and $\dot{q_{i}}$ is
81   generalized velocity.
82  
83 < \subsubsection{\label{introSection:hamiltonian}Hamiltonian Mechanics}
83 > \subsection{\label{introSection:hamiltonian}Hamiltonian Mechanics}
84  
85   Arising from Lagrangian Mechanics, Hamiltonian Mechanics was
86   introduced by William Rowan Hamilton in 1833 as a re-formulation of
# Line 90 | Line 91 | p_i = \frac{\partial L}{\partial \dot q_i}
91   p_i = \frac{\partial L}{\partial \dot q_i}
92   \label{introEquation:generalizedMomenta}
93   \end{equation}
94 < With the help of these momenta, we may now define a new quantity $H$
94 < by the equation
94 > The Lagrange equations of motion are then expressed by
95   \begin{equation}
96 < H = p_1 \dot q_1  +  \ldots  + p_f \dot q_f  - L,
96 > p_i  = \frac{{\partial L}}{{\partial q_i }}
97 > \label{introEquation:generalizedMomentaDot}
98 > \end{equation}
99 >
100 > With the help of the generalized momenta, we may now define a new
101 > quantity $H$ by the equation
102 > \begin{equation}
103 > H = \sum\limits_k {p_k \dot q_k }  - L ,
104   \label{introEquation:hamiltonianDefByLagrangian}
105   \end{equation}
106   where $ \dot q_1  \ldots \dot q_f $ are generalized velocities and
107   $L$ is the Lagrangian function for the system.
108  
109 + Differentiating Eq.~\ref{introEquation:hamiltonianDefByLagrangian},
110 + one can obtain
111 + \begin{equation}
112 + dH = \sum\limits_k {\left( {p_k d\dot q_k  + \dot q_k dp_k  -
113 + \frac{{\partial L}}{{\partial q_k }}dq_k  - \frac{{\partial
114 + L}}{{\partial \dot q_k }}d\dot q_k } \right)}  - \frac{{\partial
115 + L}}{{\partial t}}dt \label{introEquation:diffHamiltonian1}
116 + \end{equation}
117 + Making use of  Eq.~\ref{introEquation:generalizedMomenta}, the
118 + second and fourth terms in the parentheses cancel. Therefore,
119 + Eq.~\ref{introEquation:diffHamiltonian1} can be rewritten as
120 + \begin{equation}
121 + dH = \sum\limits_k {\left( {\dot q_k dp_k  - \dot p_k dq_k }
122 + \right)}  - \frac{{\partial L}}{{\partial t}}dt
123 + \label{introEquation:diffHamiltonian2}
124 + \end{equation}
125 + By identifying the coefficients of $dq_k$, $dp_k$ and dt, we can
126 + find
127 + \begin{equation}
128 + \frac{{\partial H}}{{\partial p_k }} = q_k
129 + \label{introEquation:motionHamiltonianCoordinate}
130 + \end{equation}
131 + \begin{equation}
132 + \frac{{\partial H}}{{\partial q_k }} =  - p_k
133 + \label{introEquation:motionHamiltonianMomentum}
134 + \end{equation}
135 + and
136 + \begin{equation}
137 + \frac{{\partial H}}{{\partial t}} =  - \frac{{\partial L}}{{\partial
138 + t}}
139 + \label{introEquation:motionHamiltonianTime}
140 + \end{equation}
141 +
142 + Eq.~\ref{introEquation:motionHamiltonianCoordinate} and
143 + Eq.~\ref{introEquation:motionHamiltonianMomentum} are Hamilton's
144 + equation of motion. Due to their symmetrical formula, they are also
145 + known as the canonical equations of motions.
146 +
147   An important difference between Lagrangian approach and the
148   Hamiltonian approach is that the Lagrangian is considered to be a
149   function of the generalized velocities $\dot q_i$ and the
# Line 110 | Line 155 | equations.
155   independent variables and it only works with 1st-order differential
156   equations.
157  
158 + \subsection{\label{introSection:poissonBrackets}Poisson Brackets}
159  
160 < \subsubsection{\label{introSection:canonicalTransformation}Canonical Transformation}
160 > \subsection{\label{introSection:canonicalTransformation}Canonical
161 > Transformation}
162  
163 < \subsection{\label{introSection:statisticalMechanics}Statistical Mechanics}
163 > \section{\label{introSection:statisticalMechanics}Statistical
164 > Mechanics}
165  
166   The thermodynamic behaviors and properties  of Molecular Dynamics
167   simulation are governed by the principle of Statistical Mechanics.
168   The following section will give a brief introduction to some of the
169   Statistical Mechanics concepts presented in this dissertation.
170  
171 < \subsubsection{\label{introSection::ensemble}Ensemble}
171 > \subsection{\label{introSection::ensemble}Ensemble}
172  
173 < \subsubsection{\label{introSection:ergodic}The Ergodic Hypothesis}
173 > \subsection{\label{introSection:ergodic}The Ergodic Hypothesis}
174  
175 < \subsection{\label{introSection:rigidBody}Dynamics of Rigid Bodies}
175 > \section{\label{introSection:rigidBody}Dynamics of Rigid Bodies}
176  
177 < \subsection{\label{introSection:correlationFunctions}Correlation Functions}
177 > \section{\label{introSection:correlationFunctions}Correlation Functions}
178  
179   \section{\label{introSection:langevinDynamics}Langevin Dynamics}
180  

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