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# Line 31 | Line 31 | F_{ij} = -F_{ji}
31   $F_{ji}$ be the force that particle $j$ exerts on particle $i$.
32   Newton's third law states that
33   \begin{equation}
34 < F_{ij} = -F_{ji}
34 > F_{ij} = -F_{ji}.
35   \label{introEquation:newtonThirdLaw}
36   \end{equation}
37 –
37   Conservation laws of Newtonian Mechanics play very important roles
38   in solving mechanics problems. The linear momentum of a particle is
39   conserved if it is free or it experiences no force. The second
# Line 63 | Line 62 | that if all forces are conservative, Energy
62   \end{equation}
63   If there are no external torques acting on a body, the angular
64   momentum of it is conserved. The last conservation theorem state
65 < that if all forces are conservative, Energy
66 < \begin{equation}E = T + V \label{introEquation:energyConservation}
65 > that if all forces are conservative, energy is conserved,
66 > \begin{equation}E = T + V. \label{introEquation:energyConservation}
67   \end{equation}
68 < is conserved. All of these conserved quantities are
69 < important factors to determine the quality of numerical integration
70 < schemes for rigid bodies \cite{Dullweber1997}.
68 > All of these conserved quantities are important factors to determine
69 > the quality of numerical integration schemes for rigid bodies
70 > \cite{Dullweber1997}.
71  
72   \subsection{\label{introSection:lagrangian}Lagrangian Mechanics}
73  
74 < Newtonian Mechanics suffers from two important limitations: motions
75 < can only be described in cartesian coordinate systems. Moreover, It
76 < become impossible to predict analytically the properties of the
77 < system even if we know all of the details of the interaction. In
78 < order to overcome some of the practical difficulties which arise in
79 < attempts to apply Newton's equation to complex system, approximate
80 < numerical procedures may be developed.
74 > Newtonian Mechanics suffers from a important limitation: motions can
75 > only be described in cartesian coordinate systems which make it
76 > impossible to predict analytically the properties of the system even
77 > if we know all of the details of the interaction. In order to
78 > overcome some of the practical difficulties which arise in attempts
79 > to apply Newton's equation to complex system, approximate numerical
80 > procedures may be developed.
81  
82   \subsubsection{\label{introSection:halmiltonPrinciple}\textbf{Hamilton's
83   Principle}}
84  
85   Hamilton introduced the dynamical principle upon which it is
86   possible to base all of mechanics and most of classical physics.
87 < Hamilton's Principle may be stated as follows,
88 <
89 < The actual trajectory, along which a dynamical system may move from
90 < one point to another within a specified time, is derived by finding
91 < the path which minimizes the time integral of the difference between
93 < the kinetic, $K$, and potential energies, $U$.
87 > Hamilton's Principle may be stated as follows: the actual
88 > trajectory, along which a dynamical system may move from one point
89 > to another within a specified time, is derived by finding the path
90 > which minimizes the time integral of the difference between the
91 > kinetic, $K$, and potential energies, $U$,
92   \begin{equation}
93 < \delta \int_{t_1 }^{t_2 } {(K - U)dt = 0} ,
93 > \delta \int_{t_1 }^{t_2 } {(K - U)dt = 0}.
94   \label{introEquation:halmitonianPrinciple1}
95   \end{equation}
98 –
96   For simple mechanical systems, where the forces acting on the
97   different parts are derivable from a potential, the Lagrangian
98   function $L$ can be defined as the difference between the kinetic
99   energy of the system and its potential energy,
100   \begin{equation}
101 < L \equiv K - U = L(q_i ,\dot q_i ) ,
101 > L \equiv K - U = L(q_i ,\dot q_i ).
102   \label{introEquation:lagrangianDef}
103   \end{equation}
104 < then Eq.~\ref{introEquation:halmitonianPrinciple1} becomes
104 > Thus, Eq.~\ref{introEquation:halmitonianPrinciple1} becomes
105   \begin{equation}
106 < \delta \int_{t_1 }^{t_2 } {L dt = 0} ,
106 > \delta \int_{t_1 }^{t_2 } {L dt = 0} .
107   \label{introEquation:halmitonianPrinciple2}
108   \end{equation}
109  
# Line 138 | Line 135 | p_i  = \frac{{\partial L}}{{\partial q_i }}
135   p_i  = \frac{{\partial L}}{{\partial q_i }}
136   \label{introEquation:generalizedMomentaDot}
137   \end{equation}
141 –
138   With the help of the generalized momenta, we may now define a new
139   quantity $H$ by the equation
140   \begin{equation}
# Line 146 | Line 142 | $L$ is the Lagrangian function for the system.
142   \label{introEquation:hamiltonianDefByLagrangian}
143   \end{equation}
144   where $ \dot q_1  \ldots \dot q_f $ are generalized velocities and
145 < $L$ is the Lagrangian function for the system.
146 <
151 < Differentiating Eq.~\ref{introEquation:hamiltonianDefByLagrangian},
152 < one can obtain
145 > $L$ is the Lagrangian function for the system. Differentiating
146 > Eq.~\ref{introEquation:hamiltonianDefByLagrangian}, one can obtain
147   \begin{equation}
148   dH = \sum\limits_k {\left( {p_k d\dot q_k  + \dot q_k dp_k  -
149   \frac{{\partial L}}{{\partial q_k }}dq_k  - \frac{{\partial
150   L}}{{\partial \dot q_k }}d\dot q_k } \right)}  - \frac{{\partial
151 < L}}{{\partial t}}dt \label{introEquation:diffHamiltonian1}
151 > L}}{{\partial t}}dt . \label{introEquation:diffHamiltonian1}
152   \end{equation}
153 < Making use of  Eq.~\ref{introEquation:generalizedMomenta}, the
154 < second and fourth terms in the parentheses cancel. Therefore,
153 > Making use of Eq.~\ref{introEquation:generalizedMomenta}, the second
154 > and fourth terms in the parentheses cancel. Therefore,
155   Eq.~\ref{introEquation:diffHamiltonian1} can be rewritten as
156   \begin{equation}
157   dH = \sum\limits_k {\left( {\dot q_k dp_k  - \dot p_k dq_k }
158 < \right)}  - \frac{{\partial L}}{{\partial t}}dt
158 > \right)}  - \frac{{\partial L}}{{\partial t}}dt .
159   \label{introEquation:diffHamiltonian2}
160   \end{equation}
161   By identifying the coefficients of $dq_k$, $dp_k$ and dt, we can
# Line 180 | Line 174 | t}}
174   t}}
175   \label{introEquation:motionHamiltonianTime}
176   \end{equation}
177 <
184 < Eq.~\ref{introEquation:motionHamiltonianCoordinate} and
177 > where Eq.~\ref{introEquation:motionHamiltonianCoordinate} and
178   Eq.~\ref{introEquation:motionHamiltonianMomentum} are Hamilton's
179   equation of motion. Due to their symmetrical formula, they are also
180   known as the canonical equations of motions \cite{Goldstein2001}.
# Line 195 | Line 188 | only works with 1st-order differential equations\cite{
188   statistical mechanics and quantum mechanics, since it treats the
189   coordinate and its time derivative as independent variables and it
190   only works with 1st-order differential equations\cite{Marion1990}.
198 –
191   In Newtonian Mechanics, a system described by conservative forces
192 < conserves the total energy \ref{introEquation:energyConservation}.
193 < It follows that Hamilton's equations of motion conserve the total
194 < Hamiltonian.
192 > conserves the total energy
193 > (Eq.~\ref{introEquation:energyConservation}). It follows that
194 > Hamilton's equations of motion conserve the total Hamiltonian
195   \begin{equation}
196   \frac{{dH}}{{dt}} = \sum\limits_i {\left( {\frac{{\partial
197   H}}{{\partial q_i }}\dot q_i  + \frac{{\partial H}}{{\partial p_i
198   }}\dot p_i } \right)}  = \sum\limits_i {\left( {\frac{{\partial
199   H}}{{\partial q_i }}\frac{{\partial H}}{{\partial p_i }} -
200   \frac{{\partial H}}{{\partial p_i }}\frac{{\partial H}}{{\partial
201 < q_i }}} \right) = 0} \label{introEquation:conserveHalmitonian}
201 > q_i }}} \right) = 0}. \label{introEquation:conserveHalmitonian}
202   \end{equation}
203  
204   \section{\label{introSection:statisticalMechanics}Statistical
# Line 227 | Line 219 | this system is a $6f$ dimensional space. A point, $x =
219   momentum variables. Consider a dynamic system of $f$ particles in a
220   cartesian space, where each of the $6f$ coordinates and momenta is
221   assigned to one of $6f$ mutually orthogonal axes, the phase space of
222 < this system is a $6f$ dimensional space. A point, $x = (\rightarrow
223 < q_1 , \ldots ,\rightarrow q_f ,\rightarrow p_1 , \ldots ,\rightarrow
224 < p_f )$, with a unique set of values of $6f$ coordinates and momenta
225 < is a phase space vector.
222 > this system is a $6f$ dimensional space. A point, $x =
223 > (\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}
224 > \over q} _1 , \ldots
225 > ,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}
226 > \over q} _f
227 > ,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}
228 > \over p} _1  \ldots
229 > ,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}
230 > \over p} _f )$ , with a unique set of values of $6f$ coordinates and
231 > momenta is a phase space vector.
232   %%%fix me
233  
234   In statistical mechanics, the condition of an ensemble at any time
# Line 288 | Line 286 | thermodynamic equilibrium.
286   statistical characteristics. As a function of macroscopic
287   parameters, such as temperature \textit{etc}, the partition function
288   can be used to describe the statistical properties of a system in
289 < thermodynamic equilibrium.
290 <
291 < As an ensemble of systems, each of which is known to be thermally
294 < isolated and conserve energy, the Microcanonical ensemble (NVE) has
295 < a partition function like,
289 > thermodynamic equilibrium. As an ensemble of systems, each of which
290 > is known to be thermally isolated and conserve energy, the
291 > Microcanonical ensemble (NVE) has a partition function like,
292   \begin{equation}
293 < \Omega (N,V,E) = e^{\beta TS} \label{introEquation:NVEPartition}.
293 > \Omega (N,V,E) = e^{\beta TS}. \label{introEquation:NVEPartition}
294   \end{equation}
295   A canonical ensemble (NVT)is an ensemble of systems, each of which
296   can share its energy with a large heat reservoir. The distribution
297   of the total energy amongst the possible dynamical states is given
298   by the partition function,
299   \begin{equation}
300 < \Omega (N,V,T) = e^{ - \beta A}
300 > \Omega (N,V,T) = e^{ - \beta A}.
301   \label{introEquation:NVTPartition}
302   \end{equation}
303   Here, $A$ is the Helmholtz free energy which is defined as $ A = U -
# Line 358 | Line 354 | simple form,
354   \frac{{\partial \rho }}{{\partial p_i }}\dot p_i } \right)}  = 0 .
355   \label{introEquation:liouvilleTheorem}
356   \end{equation}
361 –
357   Liouville's theorem states that the distribution function is
358   constant along any trajectory in phase space. In classical
359   statistical mechanics, since the number of members in an ensemble is
# Line 500 | Line 495 | an example of symplectic form.
495   $\omega(\lambda_1x_1+\lambda_2x_2, y) = \lambda_1\omega(x_1, y)+
496   \lambda_2\omega(x_2, y)$, $\omega(x, y) = - \omega(y, x)$ and
497   $\omega(x, x) = 0$. The cross product operation in vector field is
498 < an example of symplectic form.
499 <
500 < One of the motivations to study \emph{symplectic manifolds} in
501 < Hamiltonian Mechanics is that a symplectic manifold can represent
502 < all possible configurations of the system and the phase space of the
503 < system can be described by it's cotangent bundle. Every symplectic
504 < manifold is even dimensional. For instance, in Hamilton equations,
510 < coordinate and momentum always appear in pairs.
498 > an example of symplectic form. One of the motivations to study
499 > \emph{symplectic manifolds} in Hamiltonian Mechanics is that a
500 > symplectic manifold can represent all possible configurations of the
501 > system and the phase space of the system can be described by it's
502 > cotangent bundle. Every symplectic manifold is even dimensional. For
503 > instance, in Hamilton equations, coordinate and momentum always
504 > appear in pairs.
505  
506   \subsection{\label{introSection:ODE}Ordinary Differential Equations}
507  
# Line 534 | Line 528 | called a \emph{Hamiltonian vector field}.
528   \frac{d}{{dt}}x = J\nabla _x H(x)
529   \label{introEquation:compactHamiltonian}
530   \end{equation}In this case, $f$ is
531 < called a \emph{Hamiltonian vector field}.
532 <
539 < Another generalization of Hamiltonian dynamics is Poisson
540 < Dynamics\cite{Olver1986},
531 > called a \emph{Hamiltonian vector field}. Another generalization of
532 > Hamiltonian dynamics is Poisson Dynamics\cite{Olver1986},
533   \begin{equation}
534   \dot x = J(x)\nabla _x H \label{introEquation:poissonHamiltonian}
535   \end{equation}
# Line 587 | Line 579 | them can be found in systems which occur naturally in
579   The hidden geometric properties\cite{Budd1999, Marsden1998} of an
580   ODE and its flow play important roles in numerical studies. Many of
581   them can be found in systems which occur naturally in applications.
590 –
582   Let $\varphi$ be the flow of Hamiltonian vector field, $\varphi$ is
583   a \emph{symplectic} flow if it satisfies,
584   \begin{equation}
# Line 601 | Line 592 | is the property that must be preserved by the integrat
592   \begin{equation}
593   {\varphi '}^T J \varphi ' = J \circ \varphi
594   \end{equation}
595 < is the property that must be preserved by the integrator.
596 <
597 < It is possible to construct a \emph{volume-preserving} flow for a
598 < source free ODE ($ \nabla \cdot f = 0 $), if the flow satisfies $
599 < \det d\varphi  = 1$. One can show easily that a symplectic flow will
609 < be volume-preserving.
610 <
611 < Changing the variables $y = h(x)$ in an ODE
595 > is the property that must be preserved by the integrator. It is
596 > possible to construct a \emph{volume-preserving} flow for a source
597 > free ODE ($ \nabla \cdot f = 0 $), if the flow satisfies $ \det
598 > d\varphi  = 1$. One can show easily that a symplectic flow will be
599 > volume-preserving. Changing the variables $y = h(x)$ in an ODE
600   (Eq.~\ref{introEquation:ODE}) will result in a new system,
601   \[
602   \dot y = \tilde f(y) = ((dh \cdot f)h^{ - 1} )(y).
603   \]
604   The vector filed $f$ has reversing symmetry $h$ if $f = - \tilde f$.
605   In other words, the flow of this vector field is reversible if and
606 < only if $ h \circ \varphi ^{ - 1}  = \varphi  \circ h $.
607 <
620 < A \emph{first integral}, or conserved quantity of a general
606 > only if $ h \circ \varphi ^{ - 1}  = \varphi  \circ h $. A
607 > \emph{first integral}, or conserved quantity of a general
608   differential function is a function $ G:R^{2d}  \to R^d $ which is
609   constant for all solutions of the ODE $\frac{{dx}}{{dt}} = f(x)$ ,
610   \[
# Line 630 | Line 617 | smooth function $G$ is given by,
617   which is the condition for conserving \emph{first integral}. For a
618   canonical Hamiltonian system, the time evolution of an arbitrary
619   smooth function $G$ is given by,
633 –
620   \begin{eqnarray}
621   \frac{{dG(x(t))}}{{dt}} & = & [\nabla _x G(x(t))]^T \dot x(t) \\
622                          & = & [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). \\
623   \label{introEquation:firstIntegral1}
624   \end{eqnarray}
639 –
640 –
625   Using poisson bracket notion, Equation
626   \ref{introEquation:firstIntegral1} can be rewritten as
627   \[
# Line 650 | Line 634 | is a \emph{first integral}, which is due to the fact $
634   \]
635   As well known, the Hamiltonian (or energy) H of a Hamiltonian system
636   is a \emph{first integral}, which is due to the fact $\{ H,H\}  =
637 < 0$.
654 <
655 < When designing any numerical methods, one should always try to
637 > 0$. When designing any numerical methods, one should always try to
638   preserve the structural properties of the original ODE and its flow.
639  
640   \subsection{\label{introSection:constructionSymplectic}Construction of Symplectic Methods}
# Line 693 | Line 675 | simpler integration of the system.
675   \label{introEquation:FlowDecomposition}
676   \end{equation}
677   where each of the sub-flow is chosen such that each represent a
678 < simpler integration of the system.
679 <
698 < Suppose that a Hamiltonian system takes the form,
678 > simpler integration of the system. Suppose that a Hamiltonian system
679 > takes the form,
680   \[
681   H = H_1 + H_2.
682   \]
# Line 785 | Line 766 | the equations of motion would follow:
766  
767   \item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values.
768   \end{enumerate}
788 –
769   By simply switching the order of the propagators in the splitting
770   and composing a new integrator, the \emph{position verlet}
771   integrator, can be generated,
# Line 1088 | Line 1068 | expressed in terms of radial distribution function \ci
1068   to justify the correctness of a liquid model. Moreover, various
1069   equilibrium thermodynamic and structural properties can also be
1070   expressed in terms of radial distribution function \cite{Allen1987}.
1091 –
1071   The pair distribution functions $g(r)$ gives the probability that a
1072   particle $i$ will be located at a distance $r$ from a another
1073   particle $j$ in the system
# Line 1216 | Line 1195 | Q^T PJ^{ - 1}  + J^{ - 1} P^T Q = 0 . \\
1195   Q^T PJ^{ - 1}  + J^{ - 1} P^T Q = 0 . \\
1196   \label{introEquation:RBFirstOrderConstraint}
1197   \end{equation}
1219 –
1198   Using Equation (\ref{introEquation:motionHamiltonianCoordinate},
1199   \ref{introEquation:motionHamiltonianMomentum}), one can write down
1200   the equations of motion,
1223 –
1201   \begin{eqnarray}
1202   \frac{{dq}}{{dt}} & = & \frac{p}{m} \label{introEquation:RBMotionPosition}\\
1203   \frac{{dp}}{{dt}} & = & - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\
1204   \frac{{dQ}}{{dt}} & = & PJ^{ - 1}  \label{introEquation:RBMotionRotation}\\
1205   \frac{{dP}}{{dt}} & = & - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}
1206   \end{eqnarray}
1230 –
1207   In general, there are two ways to satisfy the holonomic constraints.
1208   We can use a constraint force provided by a Lagrange multiplier on
1209   the normal manifold to keep the motion on constraint space. Or we
# Line 1239 | Line 1215 | M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1}  + J^{ - 1}
1215   M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1}  + J^{ - 1} P^T Q = 0}
1216   \right\}.
1217   \]
1242 –
1218   Unfortunately, this constraint manifold is not the cotangent bundle
1219   $T^* SO(3)$ which can be consider as a symplectic manifold on Lie
1220   rotation group $SO(3)$. However, it turns out that under symplectic
# Line 1254 | Line 1229 | T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \t
1229   T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \tilde Q =
1230   1,\tilde Q^T \tilde PJ^{ - 1}  + J^{ - 1} P^T \tilde Q = 0} \right\}
1231   \]
1257 –
1232   For a body fixed vector $X_i$ with respect to the center of mass of
1233   the rigid body, its corresponding lab fixed vector $X_0^{lab}$  is
1234   given as
# Line 1273 | Line 1247 | respectively.
1247   \[
1248   \nabla _Q V(q,Q) = F(q,Q)X_i^t
1249   \]
1250 < respectively.
1251 <
1252 < As a common choice to describe the rotation dynamics of the rigid
1279 < body, the angular momentum on the body fixed frame $\Pi  = Q^t P$ is
1280 < introduced to rewrite the equations of motion,
1250 > respectively. As a common choice to describe the rotation dynamics
1251 > of the rigid body, the angular momentum on the body fixed frame $\Pi
1252 > = Q^t P$ is introduced to rewrite the equations of motion,
1253   \begin{equation}
1254   \begin{array}{l}
1255 < \dot \Pi  = J^{ - 1} \Pi ^T \Pi  + Q^T \sum\limits_i {F_i (q,Q)X_i^T }  - \Lambda  \\
1256 < \dot Q  = Q\Pi {\rm{ }}J^{ - 1}  \\
1255 > \dot \Pi  = J^{ - 1} \Pi ^T \Pi  + Q^T \sum\limits_i {F_i (q,Q)X_i^T }  - \Lambda,  \\
1256 > \dot Q  = Q\Pi {\rm{ }}J^{ - 1},  \\
1257   \end{array}
1258   \label{introEqaution:RBMotionPI}
1259   \end{equation}
1260 < , as well as holonomic constraints,
1260 > as well as holonomic constraints,
1261   \[
1262   \begin{array}{l}
1263 < \Pi J^{ - 1}  + J^{ - 1} \Pi ^t  = 0 \\
1264 < Q^T Q = 1 \\
1263 > \Pi J^{ - 1}  + J^{ - 1} \Pi ^t  = 0, \\
1264 > Q^T Q = 1 .\\
1265   \end{array}
1266   \]
1295 –
1267   For a vector $v(v_1 ,v_2 ,v_3 ) \in R^3$ and a matrix $\hat v \in
1268   so(3)^ \star$, the hat-map isomorphism,
1269   \begin{equation}
# Line 1307 | Line 1278 | operations
1278   will let us associate the matrix products with traditional vector
1279   operations
1280   \[
1281 < \hat vu = v \times u
1281 > \hat vu = v \times u.
1282   \]
1283 < Using \ref{introEqaution:RBMotionPI}, one can construct a skew
1283 > Using Eq.~\ref{introEqaution:RBMotionPI}, one can construct a skew
1284   matrix,
1285 <
1286 < \begin{eqnarry*}
1287 < (\dot \Pi  - \dot \Pi ^T ){\rm{ }} = {\rm{ }}(\Pi  - \Pi ^T ){\rm{
1288 < }}(J^{ - 1} \Pi  + \Pi J^{ - 1} ) + \sum\limits_i {[Q^T F_i
1285 > \begin{eqnarray}
1286 > (\dot \Pi  - \dot \Pi ^T ){\rm{ }} &= &{\rm{ }}(\Pi  - \Pi ^T ){\rm{
1287 > }}(J^{ - 1} \Pi  + \Pi J^{ - 1} ) \notag \\
1288 > + \sum\limits_i {[Q^T F_i
1289   (r,Q)X_i^T  - X_i F_i (r,Q)^T Q]}  - (\Lambda  - \Lambda ^T ).
1290   \label{introEquation:skewMatrixPI}
1291 < \end{eqnarray*}
1292 <
1293 < Since $\Lambda$ is symmetric, the last term of Equation
1294 < \ref{introEquation:skewMatrixPI} is zero, which implies the Lagrange
1295 < multiplier $\Lambda$ is absent from the equations of motion. This
1296 < unique property eliminates the requirement of iterations which can
1297 < not be avoided in other methods\cite{Kol1997, Omelyan1998}.
1298 <
1328 < Applying the hat-map isomorphism, we obtain the equation of motion
1329 < for angular momentum on body frame
1291 > \end{eqnarray}
1292 > Since $\Lambda$ is symmetric, the last term of
1293 > Eq.~\ref{introEquation:skewMatrixPI} is zero, which implies the
1294 > Lagrange multiplier $\Lambda$ is absent from the equations of
1295 > motion. This unique property eliminates the requirement of
1296 > iterations which can not be avoided in other methods\cite{Kol1997,
1297 > Omelyan1998}. Applying the hat-map isomorphism, we obtain the
1298 > equation of motion for angular momentum on body frame
1299   \begin{equation}
1300   \dot \pi  = \pi  \times I^{ - 1} \pi  + \sum\limits_i {\left( {Q^T
1301   F_i (r,Q)} \right) \times X_i }.
# Line 1335 | Line 1304 | given by
1304   In the same manner, the equation of motion for rotation matrix is
1305   given by
1306   \[
1307 < \dot Q = Qskew(I^{ - 1} \pi )
1307 > \dot Q = Qskew(I^{ - 1} \pi ).
1308   \]
1309  
1310   \subsection{\label{introSection:SymplecticFreeRB}Symplectic
# Line 1357 | Line 1326 | J(\pi ) = \left( {\begin{array}{*{20}c}
1326     0 & {\pi _3 } & { - \pi _2 }  \\
1327     { - \pi _3 } & 0 & {\pi _1 }  \\
1328     {\pi _2 } & { - \pi _1 } & 0  \\
1329 < \end{array}} \right)
1329 > \end{array}} \right).
1330   \end{equation}
1331   Thus, the dynamics of free rigid body is governed by
1332   \begin{equation}
1333 < \frac{d}{{dt}}\pi  = J(\pi )\nabla _\pi  T^r (\pi )
1333 > \frac{d}{{dt}}\pi  = J(\pi )\nabla _\pi  T^r (\pi ).
1334   \end{equation}
1366 –
1335   One may notice that each $T_i^r$ in Equation
1336   \ref{introEquation:rotationalKineticRB} can be solved exactly. For
1337   instance, the equations of motion due to $T_1^r$ are given by
# Line 1396 | Line 1364 | e^{\Delta tR_1 }  \approx (1 - \Delta tR_1 )^{ - 1} (1
1364   propagator,
1365   \[
1366   e^{\Delta tR_1 }  \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1
1367 < )
1367 > ).
1368   \]
1369   The flow maps for $T_2^r$ and $T_3^r$ can be found in the same
1370   manner. In order to construct a second-order symplectic method, we
# Line 1414 | Line 1382 | _1 }.
1382   \circ \varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t/2,\pi
1383   _1 }.
1384   \]
1417 –
1385   The non-canonical Lie-Poisson bracket ${F, G}$ of two function
1386   $F(\pi )$ and $G(\pi )$ is defined by
1387   \[
1388   \{ F,G\} (\pi ) = [\nabla _\pi  F(\pi )]^T J(\pi )\nabla _\pi  G(\pi
1389 < )
1389 > ).
1390   \]
1391   If the Poisson bracket of a function $F$ with an arbitrary smooth
1392   function $G$ is zero, $F$ is a \emph{Casimir}, which is the
# Line 1430 | Line 1397 | then by the chain rule
1397   then by the chain rule
1398   \[
1399   \nabla _\pi  F(\pi ) = S'(\frac{{\parallel \pi \parallel ^2
1400 < }}{2})\pi
1400 > }}{2})\pi.
1401   \]
1402 < Thus $ [\nabla _\pi  F(\pi )]^T J(\pi ) =  - S'(\frac{{\parallel \pi
1402 > Thus, $ [\nabla _\pi  F(\pi )]^T J(\pi ) =  - S'(\frac{{\parallel
1403 > \pi
1404   \parallel ^2 }}{2})\pi  \times \pi  = 0 $. This explicit
1405   Lie-Poisson integrator is found to be both extremely efficient and
1406   stable. These properties can be explained by the fact the small
# Line 1445 | Line 1413 | H = T(p,\pi ) + V(q,Q)
1413   The Hamiltonian of rigid body can be separated in terms of kinetic
1414   energy and potential energy,
1415   \[
1416 < H = T(p,\pi ) + V(q,Q)
1416 > H = T(p,\pi ) + V(q,Q).
1417   \]
1418   The equations of motion corresponding to potential energy and
1419   kinetic energy are listed in the below table,
1420   \begin{table}
1421 < \caption{Equations of motion due to Potential and Kinetic Energies}
1421 > \caption{EQUATIONS OF MOTION DUE TO POTENTIAL AND KINETIC ENERGIES}
1422   \begin{center}
1423   \begin{tabular}{|l|l|}
1424    \hline
# Line 1494 | Line 1462 | moving rigid bodies
1462   \]
1463   Finally, we obtain the overall symplectic propagators for freely
1464   moving rigid bodies
1465 < \begin{equation}
1466 < \begin{array}{c}
1467 < \varphi _{\Delta t}  = \varphi _{\Delta t/2,F}  \circ \varphi _{\Delta t/2,\tau }  \\
1468 <  \circ \varphi _{\Delta t,T^t }  \circ \varphi _{\Delta t/2,\pi _1 }  \circ \varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t,\pi _3 }  \circ \varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t/2,\pi _1 }  \\
1501 <  \circ \varphi _{\Delta t/2,\tau }  \circ \varphi _{\Delta t/2,F}  .\\
1502 < \end{array}
1465 > \begin{eqnarray*}
1466 > \varphi _{\Delta t}  &=& \varphi _{\Delta t/2,F}  \circ \varphi _{\Delta t/2,\tau }  \\
1467 >  & & \circ \varphi _{\Delta t,T^t }  \circ \varphi _{\Delta t/2,\pi _1 }  \circ \varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t,\pi _3 }  \circ \varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t/2,\pi _1 }  \\
1468 >  & & \circ \varphi _{\Delta t/2,\tau }  \circ \varphi _{\Delta t/2,F}  .\\
1469   \label{introEquation:overallRBFlowMaps}
1470 < \end{equation}
1470 > \end{eqnarray*}
1471  
1472   \section{\label{introSection:langevinDynamics}Langevin Dynamics}
1473   As an alternative to newtonian dynamics, Langevin dynamics, which
# Line 1547 | Line 1513 | W(x) = U(x) - \sum\limits_{\alpha  = 1}^N {\frac{{g_\a
1513   \[
1514   W(x) = U(x) - \sum\limits_{\alpha  = 1}^N {\frac{{g_\alpha ^2
1515   }}{{2m_\alpha  w_\alpha ^2 }}} x^2
1516 < \] and combining the last two terms in Equation
1517 < \ref{introEquation:bathGLE}, we may rewrite the Harmonic bath
1552 < Hamiltonian as
1516 > \]
1517 > and combining the last two terms in Eq.~\ref{introEquation:bathGLE}, we may rewrite the Harmonic bath Hamiltonian as
1518   \[
1519   H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha  = 1}^N
1520   {\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha  }} + \frac{1}{2}m_\alpha
1521   w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha
1522 < w_\alpha ^2 }}x} \right)^2 } \right\}}
1522 > w_\alpha ^2 }}x} \right)^2 } \right\}}.
1523   \]
1524   Since the first two terms of the new Hamiltonian depend only on the
1525   system coordinates, we can get the equations of motion for
# Line 1571 | Line 1536 | m\ddot x_\alpha   =  - m_\alpha  w_\alpha ^2 \left( {x
1536   \frac{{g_\alpha  }}{{m_\alpha  w_\alpha ^2 }}x} \right).
1537   \label{introEquation:bathMotionGLE}
1538   \end{equation}
1574 –
1539   In order to derive an equation for $x$, the dynamics of the bath
1540   variables $x_\alpha$ must be solved exactly first. As an integral
1541   transform which is particularly useful in solving linear ordinary
# Line 1580 | Line 1544 | original problems.
1544   differential equations into simple algebra problems which can be
1545   solved easily. Then, by applying the inverse Laplace transform, also
1546   known as the Bromwich integral, we can retrieve the solutions of the
1547 < original problems.
1548 <
1585 < Let $f(t)$ be a function defined on $ [0,\infty ) $. The Laplace
1586 < transform of f(t) is a new function defined as
1547 > original problems. Let $f(t)$ be a function defined on $ [0,\infty )
1548 > $. The Laplace transform of f(t) is a new function defined as
1549   \[
1550   L(f(t)) \equiv F(p) = \int_0^\infty  {f(t)e^{ - pt} dt}
1551   \]
1552   where  $p$ is real and  $L$ is called the Laplace Transform
1553   Operator. Below are some important properties of Laplace transform
1592 –
1554   \begin{eqnarray*}
1555   L(x + y)  & = & L(x) + L(y) \\
1556   L(ax)     & = & aL(x) \\
# Line 1597 | Line 1558 | Operator. Below are some important properties of Lapla
1558   L(\ddot x)& = & p^2 L(x) - px(0) - \dot x(0) \\
1559   L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right)& = & G(p)H(p) \\
1560   \end{eqnarray*}
1600 –
1601 –
1561   Applying the Laplace transform to the bath coordinates, we obtain
1562   \begin{eqnarray*}
1563   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) \\
1564   L(x_\alpha  ) & = & \frac{{\frac{{g_\alpha  }}{{\omega _\alpha  }}L(x) + px_\alpha  (0) + \dot x_\alpha  (0)}}{{p^2  + \omega _\alpha ^2 }} \\
1565   \end{eqnarray*}
1607 –
1566   By the same way, the system coordinates become
1567   \begin{eqnarray*}
1568 < mL(\ddot x) & = & - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} \\
1569 <  & & \mbox{} - \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\}}  \\
1568 > mL(\ddot x) & = &
1569 >  - \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\}}  \\
1570 >  & & - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}}
1571   \end{eqnarray*}
1613 –
1572   With the help of some relatively important inverse Laplace
1573   transformations:
1574   \[
# Line 1620 | Line 1578 | transformations:
1578   L(1) = \frac{1}{p} \\
1579   \end{array}
1580   \]
1581 < , we obtain
1581 > we obtain
1582   \begin{eqnarray*}
1583   m\ddot x & =  & - \frac{{\partial W(x)}}{{\partial x}} -
1584   \sum\limits_{\alpha  = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2
# Line 1693 | Line 1651 | and Equation \ref{introEuqation:GeneralizedLangevinDyn
1651   \[
1652   \int_0^t {\xi (t)\dot x(t - \tau )d\tau }  = \xi _0 (x(t) - x(0))
1653   \]
1654 < and Equation \ref{introEuqation:GeneralizedLangevinDynamics} becomes
1654 > and Eq.~\ref{introEuqation:GeneralizedLangevinDynamics} becomes
1655   \[
1656   m\ddot x =  - \frac{\partial }{{\partial x}}\left( {W(x) +
1657   \frac{1}{2}\xi _0 (x - x_0 )^2 } \right) + R(t),
# Line 1710 | Line 1668 | and Equation \ref{introEuqation:GeneralizedLangevinDyn
1668   \int_0^t {\xi (t)\dot x(t - \tau )d\tau }  = 2\xi _0 \int_0^t
1669   {\delta (t)\dot x(t - \tau )d\tau }  = \xi _0 \dot x(t),
1670   \]
1671 < and Equation \ref{introEuqation:GeneralizedLangevinDynamics} becomes
1671 > and Eq.~\ref{introEuqation:GeneralizedLangevinDynamics} becomes
1672   \begin{equation}
1673   m\ddot x =  - \frac{{\partial W(x)}}{{\partial x}} - \xi _0 \dot
1674   x(t) + R(t) \label{introEquation:LangevinEquation}
# Line 1733 | Line 1691 | And since the $q$ coordinates are harmonic oscillators
1691   R(t) = \sum\limits_{\alpha  = 1}^N {g_\alpha  q_\alpha  (t)}.
1692   \]
1693   And since the $q$ coordinates are harmonic oscillators,
1736 –
1694   \begin{eqnarray*}
1695   \left\langle {q_\alpha ^2 } \right\rangle  & = & \frac{{kT}}{{m_\alpha  \omega _\alpha ^2 }} \\
1696   \left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle & = & \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha  t) \\
# Line 1742 | Line 1699 | And since the $q$ coordinates are harmonic oscillators
1699    & = &\sum\limits_\alpha  {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha  t)}  \\
1700    & = &kT\xi (t) \\
1701   \end{eqnarray*}
1745 –
1702   Thus, we recover the \emph{second fluctuation dissipation theorem}
1703   \begin{equation}
1704   \xi (t) = \left\langle {R(t)R(0)} \right\rangle

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