trunk/tengDissertation/Introduction.tex

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

\section{\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.

\subsection{\label{introSection:newtonian}Newtonian Mechanics}
The discovery of Newton's three laws of mechanics which govern the
motion of particles is the foundation of the classical mechanics.
Newton’s first law defines a class of inertial frames. Inertial
frames are reference frames where a particle not interacting with
other bodies will move with constant speed in the same direction.
With respect to inertial frames Newton’s second law has the form
\begin{equation}
F = \frac {dp}{dt} = \frac {mv}{dt}
\label{introEquation:newtonSecondLaw}
\end{equation}
A point mass interacting with other bodies moves with the
acceleration along the direction of the force acting on it. Let
$F_ij$ be the force that particle $i$ exerts on particle $j$, and
$F_ji$ be the force that particle $j$ exerts on particle $i$.
Newton’s third law states that
\begin{equation}
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
conservation theorem concerns the angular momentum of a particle.
The angular momentum $L$ of a particle with respect to an origin
from which $r$ is measured is defined to be
\begin{equation}
L \equiv r \times p \label{introEquation:angularMomentumDefinition}
\end{equation}
The torque $\tau$ with respect to the same origin is defined to be
\begin{equation}
N \equiv r \times F \label{introEquation:torqueDefinition}
\end{equation}
Differentiating Eq.~\ref{introEquation:angularMomentumDefinition},
\[
\dot L = \frac{d}{{dt}}(r \times p) = (\dot r \times p) + (r \times
\dot p)
\]
since
\[
\dot r \times p = \dot r \times mv = m\dot r \times \dot r \equiv 0
\]
thus,
\begin{equation}
\dot L = r \times \dot p = N
\end{equation}
If there are no external torques acting on a body, the angular
momentum of it is conserved. The last conservation theorem state
that if all forces are conservative, Energy
\begin{equation}E = T + V \label{introEquation:energyConservation}
\end{equation}
 is conserved. All of these conserved quantities are
important factors to determine the quality of numerical integration
scheme for rigid body \cite{Dullweber1997}.

\subsection{\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.

\subsubsection{\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, $U$ \cite{tolman79}.
\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 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 } {L dt = 0} ,
\label{introEquation:halmitonianPrinciple2}
\end{equation}

\subsubsection{\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
\label{introEquation:eqMotionLagrangian}
\end{equation}
where $q_{i}$ is generalized coordinate and $\dot{q_{i}}$ is
generalized velocity.

\subsection{\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}
The Lagrange equations of motion are then expressed by
\begin{equation}
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}
H = \sum\limits_k {p_k \dot q_k }  - 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.

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
L}}{{\partial \dot q_k }}d\dot q_k } \right)}  - \frac{{\partial
L}}{{\partial t}}dt \label{introEquation:diffHamiltonian1}
\end{equation}
Making use of  Eq.~\ref{introEquation:generalizedMomenta}, the
second and fourth terms in the parentheses cancel. Therefore,
Eq.~\ref{introEquation:diffHamiltonian1} can be rewritten as
\begin{equation}
dH = \sum\limits_k {\left( {\dot q_k dp_k  - \dot p_k dq_k }
\right)}  - \frac{{\partial L}}{{\partial t}}dt
\label{introEquation:diffHamiltonian2}
\end{equation}
By identifying the coefficients of $dq_k$, $dp_k$ and dt, we can
find
\begin{equation}
\frac{{\partial H}}{{\partial p_k }} = q_k
\label{introEquation:motionHamiltonianCoordinate}
\end{equation}
\begin{equation}
\frac{{\partial H}}{{\partial q_k }} =  - p_k
\label{introEquation:motionHamiltonianMomentum}
\end{equation}
and
\begin{equation}
\frac{{\partial H}}{{\partial t}} =  - \frac{{\partial L}}{{\partial
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
known as the canonical equations of motions \cite{Goldstein01}.

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\cite{Marion90}.

In Newtonian Mechanics, a system described by conservative forces
conserves the total energy \ref{introEquation:energyConservation}.
It follows that Hamilton's equations of motion conserve the total
Hamiltonian.
\begin{equation}
\frac{{dH}}{{dt}} = \sum\limits_i {\left( {\frac{{\partial
H}}{{\partial q_i }}\dot q_i  + \frac{{\partial H}}{{\partial p_i
}}\dot p_i } \right)}  = \sum\limits_i {\left( {\frac{{\partial
H}}{{\partial q_i }}\frac{{\partial H}}{{\partial p_i }} -
\frac{{\partial H}}{{\partial p_i }}\frac{{\partial H}}{{\partial
q_i }}} \right) = 0}
\label{introEquation:conserveHalmitonian}
\end{equation}

When studying Hamiltonian system, it is more convenient to use
notation
\begin{equation}
r = r(q,p)^T
\end{equation}
and to introduce a $2n \times 2n$ canonical structure matrix $J$,
\begin{equation}
J = \left( {\begin{array}{*{20}c}
   0 & I  \\
   { - I} & 0  \\
\end{array}} \right)
\label{introEquation:canonicalMatrix}
\end{equation}
where $I$ is a $n \times n$ identity matrix and $J$ is a
skew-symmetric matrix ($ J^T  =  - J $). Thus, Hamiltonian system
can be rewritten as,
\begin{equation}
\frac{d}{{dt}}r = J\nabla _r H(r)
\label{introEquation:compactHamiltonian}
\end{equation}

\section{\label{introSection:geometricIntegratos}Geometric Integrators}

\subsection{\label{introSection:symplecticManifold}Symplectic Manifold}
A \emph{manifold} is an abstract mathematical space. It locally
looks like Euclidean space, but when viewed globally, it may have
more complicate structure. A good example of manifold is the surface
of Earth. It seems to be flat locally, but it is round if viewed as
a whole. A \emph{differentiable manifold} (also known as
\emph{smooth manifold}) is a manifold with an open cover in which
the covering neighborhoods are all smoothly isomorphic to one
another. In other words,it is possible to apply calculus on
\emph{differentiable manifold}. A \emph{symplectic manifold} is
defined as a pair $(M, \omega)$ consisting of a \emph{differentiable
manifold} $M$ and a close, non-degenerated, bilinear symplectic
form, $\omega$. One of the motivation to study \emph{symplectic
manifold} in Hamiltonian Mechanics is that a symplectic manifold can
represent all possible configurations of the system and the phase
space of the system can be described by it's cotangent bundle. Every
symplectic manifold is even dimensional. For instance, in Hamilton
equations, coordinate and momentum always appear in pairs.

A \emph{symplectomorphism} is also known as a \emph{canonical
transformation}.

Any real-valued differentiable function H on a symplectic manifold
can serve as an energy function or Hamiltonian. Associated to any
Hamiltonian is a Hamiltonian vector field; the integral curves of
the Hamiltonian vector field are solutions to the Hamilton-Jacobi
equations. The Hamiltonian vector field defines a flow on the
symplectic manifold, called a Hamiltonian flow or symplectomorphism.
By Liouville's theorem, Hamiltonian flows preserve the volume form
on the phase space.

\subsection{\label{Construction of Symplectic Methods}}

\section{\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.

\subsection{\label{introSection:ensemble}Ensemble and Phase Space}

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

Various thermodynamic properties can be calculated from Molecular
Dynamics simulation. By comparing experimental values with the
calculated properties, one can determine the accuracy of the
simulation and the quality of the underlying model. However, both of
experiment and computer simulation are usually performed during a
certain time interval and the measurements are averaged over a
period of them which is different from the average behavior of
many-body system in Statistical Mechanics. Fortunately, Ergodic
Hypothesis is proposed to make a connection between time average and
ensemble average. It states that time average and average over the
statistical ensemble are identical \cite{Frenkel1996, leach01:mm}.
\begin{equation}
\langle A \rangle_t = \mathop {\lim }\limits_{t \to \infty }
\frac{1}{t}\int\limits_0^t {A(p(t),q(t))dt = \int\limits_\Gamma
{A(p(t),q(t))} } \rho (p(t), q(t)) dpdq
\end{equation}
where $\langle A \rangle_t$ is an equilibrium value of a physical
quantity and $\rho (p(t), q(t))$ is the equilibrium distribution
function. If an observation is averaged over a sufficiently long
time (longer than relaxation time), all accessible microstates in
phase space are assumed to be equally probed, giving a properly
weighted statistical average. This allows the researcher freedom of
choice when deciding how best to measure a given observable. In case
an ensemble averaged approach sounds most reasonable, the Monte
Carlo techniques\cite{metropolis:1949} can be utilized. Or if the
system lends itself to a time averaging approach, the Molecular
Dynamics techniques in Sec.~\ref{introSection:molecularDynamics}
will be the best choice\cite{Frenkel1996}.

\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{introSec:mdInit}Initialization}

\subsection{\label{introSection:mdIntegration} Integration of the Equations of Motion}

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

A rigid body is a body in which the distance between any two given
points of a rigid body remains constant regardless of external
forces exerted on it. A rigid body therefore conserves its shape
during its motion.

Applications of dynamics of rigid bodies.

\subsection{\label{introSection:lieAlgebra}Lie Algebra}

\subsection{\label{introSection:DLMMotionEquation}The Euler Equations of Rigid Body Motion}

\subsection{\label{introSection:otherRBMotionEquation}Other Formulations for Rigid Body Motion}

%\subsection{\label{introSection:poissonBrackets}Poisson Brackets}

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

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

\subsection{\label{introSection:LDIntroduction}Introduction and application of Langevin Dynamics}

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

\begin{equation}
H = \frac{{p^2 }}{{2m}} + U(x) + H_B  + \Delta U(x,x_1 , \ldots x_N)
\label{introEquation:bathGLE}
\end{equation}
where $H_B$ is harmonic bath Hamiltonian,
\[
H_B  =\sum\limits_{\alpha  = 1}^N {\left\{ {\frac{{p_\alpha ^2
}}{{2m_\alpha  }} + \frac{1}{2}m_\alpha  w_\alpha ^2 } \right\}}
\]
and $\Delta U$ is bilinear system-bath coupling,
\[
\Delta U =  - \sum\limits_{\alpha  = 1}^N {g_\alpha  x_\alpha  x}
\]
Completing the square,
\[
H_B  + \Delta U = \sum\limits_{\alpha  = 1}^N {\left\{
{\frac{{p_\alpha ^2 }}{{2m_\alpha  }} + \frac{1}{2}m_\alpha
w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha
w_\alpha ^2 }}x} \right)^2 } \right\}}  - \sum\limits_{\alpha  =
1}^N {\frac{{g_\alpha ^2 }}{{2m_\alpha  w_\alpha ^2 }}} x^2
\]
and putting it back into Eq.~\ref{introEquation:bathGLE},
\[
H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha  = 1}^N
{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha  }} + \frac{1}{2}m_\alpha
w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha
w_\alpha ^2 }}x} \right)^2 } \right\}}
\]
where
\[
W(x) = U(x) - \sum\limits_{\alpha  = 1}^N {\frac{{g_\alpha ^2
}}{{2m_\alpha  w_\alpha ^2 }}} x^2
\]
Since the first two terms of the new Hamiltonian depend only on the
system coordinates, we can get the equations of motion for
Generalized Langevin Dynamics by Hamilton's equations
\ref{introEquation:motionHamiltonianCoordinate,
introEquation:motionHamiltonianMomentum},
\begin{align}
\dot p &=  - \frac{{\partial H}}{{\partial x}}
       &= m\ddot x
       &= - \frac{{\partial W(x)}}{{\partial x}} - \sum\limits_{\alpha  = 1}^N {g_\alpha  \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha  w_\alpha ^2 }}x} \right)}
\label{introEq:Lp5}
\end{align}
, and
\begin{align}
\dot p_\alpha   &=  - \frac{{\partial H}}{{\partial x_\alpha  }}
                &= m\ddot x_\alpha
                &= \- m_\alpha  w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha}}{{m_\alpha  w_\alpha ^2 }}x} \right)
\end{align}

\subsection{\label{introSection:laplaceTransform}The Laplace Transform}

\[
L(x) = \int_0^\infty  {x(t)e^{ - pt} dt}
\]

\[
L(x + y) = L(x) + L(y)
\]

\[
L(ax) = aL(x)
\]

\[
L(\dot x) = pL(x) - px(0)
\]

\[
L(\ddot x) = p^2 L(x) - px(0) - \dot x(0)
\]

\[
L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right) = G(p)H(p)
\]

Some relatively important transformation,
\[
L(\cos at) = \frac{p}{{p^2  + a^2 }}
\]

\[
L(\sin at) = \frac{a}{{p^2  + a^2 }}
\]

\[
L(1) = \frac{1}{p}
\]

First, the bath coordinates,
\[
p^2 L(x_\alpha  ) - px_\alpha  (0) - \dot x_\alpha  (0) =  - \omega
_\alpha ^2 L(x_\alpha  ) + \frac{{g_\alpha  }}{{\omega _\alpha
}}L(x)
\]
\[
L(x_\alpha  ) = \frac{{\frac{{g_\alpha  }}{{\omega _\alpha  }}L(x) +
px_\alpha  (0) + \dot x_\alpha  (0)}}{{p^2  + \omega _\alpha ^2 }}
\]
Then, the system coordinates,
\begin{align}
mL(\ddot x) &=  - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} -
\sum\limits_{\alpha  = 1}^N {\left\{ {\frac{{\frac{{g_\alpha
}}{{\omega _\alpha  }}L(x) + px_\alpha  (0) + \dot x_\alpha
(0)}}{{p^2  + \omega _\alpha ^2 }} - \frac{{g_\alpha ^2 }}{{m_\alpha
}}\omega _\alpha ^2 L(x)} \right\}}
%
 &= - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} -
 \sum\limits_{\alpha  = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha  \omega _\alpha ^2 }}\frac{p}{{p^2  + \omega _\alpha ^2 }}pL(x)
 - \frac{p}{{p^2  + \omega _\alpha ^2 }}g_\alpha  x_\alpha  (0)
 - \frac{1}{{p^2  + \omega _\alpha ^2 }}g_\alpha  \dot x_\alpha  (0)} \right\}}
\end{align}
Then, the inverse transform,

\begin{align}
m\ddot x &=  - \frac{{\partial W(x)}}{{\partial x}} -
\sum\limits_{\alpha  = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2
}}{{m_\alpha  \omega _\alpha ^2 }}} \right)\int_0^t {\cos (\omega
_\alpha  t)\dot x(t - \tau )d\tau  - \left[ {g_\alpha  x_\alpha  (0)
- \frac{{g_\alpha  }}{{m_\alpha  \omega _\alpha  }}} \right]\cos
(\omega _\alpha  t) - \frac{{g_\alpha  \dot x_\alpha  (0)}}{{\omega
_\alpha  }}\sin (\omega _\alpha  t)} } \right\}}
%
&= - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t
{\sum\limits_{\alpha  = 1}^N {\left( { - \frac{{g_\alpha ^2
}}{{m_\alpha  \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha
t)\dot x(t - \tau )d} \tau }  + \sum\limits_{\alpha  = 1}^N {\left\{
{\left[ {g_\alpha  x_\alpha  (0) - \frac{{g_\alpha  }}{{m_\alpha
\omega _\alpha  }}} \right]\cos (\omega _\alpha  t) +
\frac{{g_\alpha  \dot x_\alpha  (0)}}{{\omega _\alpha  }}\sin
(\omega _\alpha  t)} \right\}}
\end{align}

\begin{equation}
m\ddot x =  - \frac{{\partial W}}{{\partial x}} - \int_0^t {\xi
(t)\dot x(t - \tau )d\tau }  + R(t)
\label{introEuqation:GeneralizedLangevinDynamics}
\end{equation}
%where $ {\xi (t)}$ is friction kernel, $R(t)$ is random force and
%$W$ is the potential of mean force. $W(x) =  - kT\ln p(x)$
\[
\xi (t) = \sum\limits_{\alpha  = 1}^N {\left( { - \frac{{g_\alpha ^2
}}{{m_\alpha  \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha  t)}
\]
For an infinite harmonic bath, we can use the spectral density and
an integral over frequencies.

\[
R(t) = \sum\limits_{\alpha  = 1}^N {\left( {g_\alpha  x_\alpha  (0)
- \frac{{g_\alpha ^2 }}{{m_\alpha  \omega _\alpha ^2 }}x(0)}
\right)\cos (\omega _\alpha  t)}  + \frac{{\dot x_\alpha
(0)}}{{\omega _\alpha  }}\sin (\omega _\alpha  t)
\]
The random forces depend only on initial conditions.

\subsubsection{\label{introSection:secondFluctuationDissipation}The Second Fluctuation Dissipation Theorem}
So we can define a new set of coordinates,
\[
q_\alpha  (t) = x_\alpha  (t) - \frac{1}{{m_\alpha  \omega _\alpha
^2 }}x(0)
\]
This makes
\[
R(t) = \sum\limits_{\alpha  = 1}^N {g_\alpha  q_\alpha  (t)}
\]
And since the $q$ coordinates are harmonic oscillators,
\[
\begin{array}{l}
 \left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle  = \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha  t) \\
 \left\langle {q_\alpha  (t)q_\beta  (0)} \right\rangle  = \delta _{\alpha \beta } \left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle  \\
 \end{array}
\]

\begin{align}
\left\langle {R(t)R(0)} \right\rangle  &= \sum\limits_\alpha
{\sum\limits_\beta  {g_\alpha  g_\beta  \left\langle {q_\alpha
(t)q_\beta  (0)} \right\rangle } }
%
&= \sum\limits_\alpha  {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)}
\right\rangle \cos (\omega _\alpha  t)}
%
&= kT\xi (t)
\end{align}

\begin{equation}
\xi (t) = \left\langle {R(t)R(0)} \right\rangle
\label{introEquation:secondFluctuationDissipation}
\end{equation}

\section{\label{introSection:hydroynamics}Hydrodynamics}

\subsection{\label{introSection:frictionTensor} Friction Tensor}
\subsection{\label{introSection:analyticalApproach}Analytical
Approach}

\subsection{\label{introSection:approximationApproach}Approximation
Approach}

\subsection{\label{introSection:centersRigidBody}Centers of Rigid
Body}
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