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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
75 > can only be described in cartesian coordinate systems. Moreover, it
76 > becomes 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
# Line 85 | Line 84 | Hamilton's Principle may be stated as follows,
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
# 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}
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 }
# 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
# Line 288 | Line 280 | thermodynamic equilibrium.
280   statistical characteristics. As a function of macroscopic
281   parameters, such as temperature \textit{etc}, the partition function
282   can be used to describe the statistical properties of a system in
283 < thermodynamic equilibrium.
284 <
285 < 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,
283 > thermodynamic equilibrium. As an ensemble of systems, each of which
284 > is known to be thermally isolated and conserve energy, the
285 > Microcanonical ensemble (NVE) has a partition function like,
286   \begin{equation}
287 < \Omega (N,V,E) = e^{\beta TS} \label{introEquation:NVEPartition}.
287 > \Omega (N,V,E) = e^{\beta TS}. \label{introEquation:NVEPartition}.
288   \end{equation}
289   A canonical ensemble (NVT)is an ensemble of systems, each of which
290   can share its energy with a large heat reservoir. The distribution
291   of the total energy amongst the possible dynamical states is given
292   by the partition function,
293   \begin{equation}
294 < \Omega (N,V,T) = e^{ - \beta A}
294 > \Omega (N,V,T) = e^{ - \beta A}.
295   \label{introEquation:NVTPartition}
296   \end{equation}
297   Here, $A$ is the Helmholtz free energy which is defined as $ A = U -
# Line 358 | Line 348 | simple form,
348   \frac{{\partial \rho }}{{\partial p_i }}\dot p_i } \right)}  = 0 .
349   \label{introEquation:liouvilleTheorem}
350   \end{equation}
361 –
351   Liouville's theorem states that the distribution function is
352   constant along any trajectory in phase space. In classical
353   statistical mechanics, since the number of members in an ensemble is
# Line 500 | Line 489 | an example of symplectic form.
489   $\omega(\lambda_1x_1+\lambda_2x_2, y) = \lambda_1\omega(x_1, y)+
490   \lambda_2\omega(x_2, y)$, $\omega(x, y) = - \omega(y, x)$ and
491   $\omega(x, x) = 0$. The cross product operation in vector field is
492 < an example of symplectic form.
493 <
494 < One of the motivations to study \emph{symplectic manifolds} in
495 < Hamiltonian Mechanics is that a symplectic manifold can represent
496 < all possible configurations of the system and the phase space of the
497 < system can be described by it's cotangent bundle. Every symplectic
498 < manifold is even dimensional. For instance, in Hamilton equations,
510 < coordinate and momentum always appear in pairs.
492 > an example of symplectic form. One of the motivations to study
493 > \emph{symplectic manifolds} in Hamiltonian Mechanics is that a
494 > symplectic manifold can represent all possible configurations of the
495 > system and the phase space of the system can be described by it's
496 > cotangent bundle. Every symplectic manifold is even dimensional. For
497 > instance, in Hamilton equations, coordinate and momentum always
498 > appear in pairs.
499  
500   \subsection{\label{introSection:ODE}Ordinary Differential Equations}
501  
# Line 534 | Line 522 | called a \emph{Hamiltonian vector field}.
522   \frac{d}{{dt}}x = J\nabla _x H(x)
523   \label{introEquation:compactHamiltonian}
524   \end{equation}In this case, $f$ is
525 < called a \emph{Hamiltonian vector field}.
526 <
539 < Another generalization of Hamiltonian dynamics is Poisson
540 < Dynamics\cite{Olver1986},
525 > called a \emph{Hamiltonian vector field}. Another generalization of
526 > Hamiltonian dynamics is Poisson Dynamics\cite{Olver1986},
527   \begin{equation}
528   \dot x = J(x)\nabla _x H \label{introEquation:poissonHamiltonian}
529   \end{equation}
# Line 587 | Line 573 | them can be found in systems which occur naturally in
573   The hidden geometric properties\cite{Budd1999, Marsden1998} of an
574   ODE and its flow play important roles in numerical studies. Many of
575   them can be found in systems which occur naturally in applications.
590 –
576   Let $\varphi$ be the flow of Hamiltonian vector field, $\varphi$ is
577   a \emph{symplectic} flow if it satisfies,
578   \begin{equation}
# Line 601 | Line 586 | is the property that must be preserved by the integrat
586   \begin{equation}
587   {\varphi '}^T J \varphi ' = J \circ \varphi
588   \end{equation}
589 < is the property that must be preserved by the integrator.
590 <
591 < It is possible to construct a \emph{volume-preserving} flow for a
592 < source free ODE ($ \nabla \cdot f = 0 $), if the flow satisfies $
593 < \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
589 > is the property that must be preserved by the integrator. It is
590 > possible to construct a \emph{volume-preserving} flow for a source
591 > free ODE ($ \nabla \cdot f = 0 $), if the flow satisfies $ \det
592 > d\varphi  = 1$. One can show easily that a symplectic flow will be
593 > volume-preserving. Changing the variables $y = h(x)$ in an ODE
594   (Eq.~\ref{introEquation:ODE}) will result in a new system,
595   \[
596   \dot y = \tilde f(y) = ((dh \cdot f)h^{ - 1} )(y).
597   \]
598   The vector filed $f$ has reversing symmetry $h$ if $f = - \tilde f$.
599   In other words, the flow of this vector field is reversible if and
600 < only if $ h \circ \varphi ^{ - 1}  = \varphi  \circ h $.
601 <
620 < A \emph{first integral}, or conserved quantity of a general
600 > only if $ h \circ \varphi ^{ - 1}  = \varphi  \circ h $. A
601 > \emph{first integral}, or conserved quantity of a general
602   differential function is a function $ G:R^{2d}  \to R^d $ which is
603   constant for all solutions of the ODE $\frac{{dx}}{{dt}} = f(x)$ ,
604   \[
# Line 630 | Line 611 | smooth function $G$ is given by,
611   which is the condition for conserving \emph{first integral}. For a
612   canonical Hamiltonian system, the time evolution of an arbitrary
613   smooth function $G$ is given by,
633 –
614   \begin{eqnarray}
615   \frac{{dG(x(t))}}{{dt}} & = & [\nabla _x G(x(t))]^T \dot x(t) \\
616                          & = & [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). \\
617   \label{introEquation:firstIntegral1}
618   \end{eqnarray}
639 –
640 –
619   Using poisson bracket notion, Equation
620   \ref{introEquation:firstIntegral1} can be rewritten as
621   \[
# Line 650 | Line 628 | is a \emph{first integral}, which is due to the fact $
628   \]
629   As well known, the Hamiltonian (or energy) H of a Hamiltonian system
630   is a \emph{first integral}, which is due to the fact $\{ H,H\}  =
631 < 0$.
654 <
655 < When designing any numerical methods, one should always try to
631 > 0$. When designing any numerical methods, one should always try to
632   preserve the structural properties of the original ODE and its flow.
633  
634   \subsection{\label{introSection:constructionSymplectic}Construction of Symplectic Methods}
# Line 693 | Line 669 | simpler integration of the system.
669   \label{introEquation:FlowDecomposition}
670   \end{equation}
671   where each of the sub-flow is chosen such that each represent a
672 < simpler integration of the system.
673 <
698 < Suppose that a Hamiltonian system takes the form,
672 > simpler integration of the system. Suppose that a Hamiltonian system
673 > takes the form,
674   \[
675   H = H_1 + H_2.
676   \]
# Line 785 | Line 760 | the equations of motion would follow:
760  
761   \item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values.
762   \end{enumerate}
788 –
763   By simply switching the order of the propagators in the splitting
764   and composing a new integrator, the \emph{position verlet}
765   integrator, can be generated,
# Line 1088 | Line 1062 | expressed in terms of radial distribution function \ci
1062   to justify the correctness of a liquid model. Moreover, various
1063   equilibrium thermodynamic and structural properties can also be
1064   expressed in terms of radial distribution function \cite{Allen1987}.
1091 –
1065   The pair distribution functions $g(r)$ gives the probability that a
1066   particle $i$ will be located at a distance $r$ from a another
1067   particle $j$ in the system
# Line 1216 | Line 1189 | Q^T PJ^{ - 1}  + J^{ - 1} P^T Q = 0 . \\
1189   Q^T PJ^{ - 1}  + J^{ - 1} P^T Q = 0 . \\
1190   \label{introEquation:RBFirstOrderConstraint}
1191   \end{equation}
1219 –
1192   Using Equation (\ref{introEquation:motionHamiltonianCoordinate},
1193   \ref{introEquation:motionHamiltonianMomentum}), one can write down
1194   the equations of motion,
1223 –
1195   \begin{eqnarray}
1196   \frac{{dq}}{{dt}} & = & \frac{p}{m} \label{introEquation:RBMotionPosition}\\
1197   \frac{{dp}}{{dt}} & = & - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\
1198   \frac{{dQ}}{{dt}} & = & PJ^{ - 1}  \label{introEquation:RBMotionRotation}\\
1199   \frac{{dP}}{{dt}} & = & - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}
1200   \end{eqnarray}
1230 –
1201   In general, there are two ways to satisfy the holonomic constraints.
1202   We can use a constraint force provided by a Lagrange multiplier on
1203   the normal manifold to keep the motion on constraint space. Or we
# Line 1239 | Line 1209 | M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1}  + J^{ - 1}
1209   M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1}  + J^{ - 1} P^T Q = 0}
1210   \right\}.
1211   \]
1242 –
1212   Unfortunately, this constraint manifold is not the cotangent bundle
1213   $T^* SO(3)$ which can be consider as a symplectic manifold on Lie
1214   rotation group $SO(3)$. However, it turns out that under symplectic
# Line 1254 | Line 1223 | T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \t
1223   T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \tilde Q =
1224   1,\tilde Q^T \tilde PJ^{ - 1}  + J^{ - 1} P^T \tilde Q = 0} \right\}
1225   \]
1257 –
1226   For a body fixed vector $X_i$ with respect to the center of mass of
1227   the rigid body, its corresponding lab fixed vector $X_0^{lab}$  is
1228   given as
# Line 1273 | Line 1241 | respectively.
1241   \[
1242   \nabla _Q V(q,Q) = F(q,Q)X_i^t
1243   \]
1244 < respectively.
1245 <
1246 < 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,
1244 > respectively. As a common choice to describe the rotation dynamics
1245 > of the rigid body, the angular momentum on the body fixed frame $\Pi
1246 > = Q^t P$ is introduced to rewrite the equations of motion,
1247   \begin{equation}
1248   \begin{array}{l}
1249 < \dot \Pi  = J^{ - 1} \Pi ^T \Pi  + Q^T \sum\limits_i {F_i (q,Q)X_i^T }  - \Lambda  \\
1250 < \dot Q  = Q\Pi {\rm{ }}J^{ - 1}  \\
1249 > \dot \Pi  = J^{ - 1} \Pi ^T \Pi  + Q^T \sum\limits_i {F_i (q,Q)X_i^T }  - \Lambda,  \\
1250 > \dot Q  = Q\Pi {\rm{ }}J^{ - 1},  \\
1251   \end{array}
1252   \label{introEqaution:RBMotionPI}
1253   \end{equation}
1254 < , as well as holonomic constraints,
1254 > as well as holonomic constraints,
1255   \[
1256   \begin{array}{l}
1257 < \Pi J^{ - 1}  + J^{ - 1} \Pi ^t  = 0 \\
1258 < Q^T Q = 1 \\
1257 > \Pi J^{ - 1}  + J^{ - 1} \Pi ^t  = 0, \\
1258 > Q^T Q = 1 .\\
1259   \end{array}
1260   \]
1295 –
1261   For a vector $v(v_1 ,v_2 ,v_3 ) \in R^3$ and a matrix $\hat v \in
1262   so(3)^ \star$, the hat-map isomorphism,
1263   \begin{equation}
# Line 1307 | Line 1272 | operations
1272   will let us associate the matrix products with traditional vector
1273   operations
1274   \[
1275 < \hat vu = v \times u
1275 > \hat vu = v \times u.
1276   \]
1277 < Using \ref{introEqaution:RBMotionPI}, one can construct a skew
1277 > Using Eq.~\ref{introEqaution:RBMotionPI}, one can construct a skew
1278   matrix,
1279 <
1280 < \begin{eqnarry*}
1281 < (\dot \Pi  - \dot \Pi ^T ){\rm{ }} = {\rm{ }}(\Pi  - \Pi ^T ){\rm{
1282 < }}(J^{ - 1} \Pi  + \Pi J^{ - 1} ) + \sum\limits_i {[Q^T F_i
1279 > \begin{eqnarray}
1280 > (\dot \Pi  - \dot \Pi ^T ){\rm{ }} &= &{\rm{ }}(\Pi  - \Pi ^T ){\rm{
1281 > }}(J^{ - 1} \Pi  + \Pi J^{ - 1} ) \notag \\
1282 > + \sum\limits_i {[Q^T F_i
1283   (r,Q)X_i^T  - X_i F_i (r,Q)^T Q]}  - (\Lambda  - \Lambda ^T ).
1284   \label{introEquation:skewMatrixPI}
1285 < \end{eqnarray*}
1286 <
1287 < Since $\Lambda$ is symmetric, the last term of Equation
1288 < \ref{introEquation:skewMatrixPI} is zero, which implies the Lagrange
1289 < multiplier $\Lambda$ is absent from the equations of motion. This
1290 < unique property eliminates the requirement of iterations which can
1291 < not be avoided in other methods\cite{Kol1997, Omelyan1998}.
1292 <
1328 < Applying the hat-map isomorphism, we obtain the equation of motion
1329 < for angular momentum on body frame
1285 > \end{eqnarray}
1286 > Since $\Lambda$ is symmetric, the last term of
1287 > Eq.~\ref{introEquation:skewMatrixPI} is zero, which implies the
1288 > Lagrange multiplier $\Lambda$ is absent from the equations of
1289 > motion. This unique property eliminates the requirement of
1290 > iterations which can not be avoided in other methods\cite{Kol1997,
1291 > Omelyan1998}. Applying the hat-map isomorphism, we obtain the
1292 > equation of motion for angular momentum on body frame
1293   \begin{equation}
1294   \dot \pi  = \pi  \times I^{ - 1} \pi  + \sum\limits_i {\left( {Q^T
1295   F_i (r,Q)} \right) \times X_i }.
# Line 1335 | Line 1298 | given by
1298   In the same manner, the equation of motion for rotation matrix is
1299   given by
1300   \[
1301 < \dot Q = Qskew(I^{ - 1} \pi )
1301 > \dot Q = Qskew(I^{ - 1} \pi ).
1302   \]
1303  
1304   \subsection{\label{introSection:SymplecticFreeRB}Symplectic
# Line 1357 | Line 1320 | J(\pi ) = \left( {\begin{array}{*{20}c}
1320     0 & {\pi _3 } & { - \pi _2 }  \\
1321     { - \pi _3 } & 0 & {\pi _1 }  \\
1322     {\pi _2 } & { - \pi _1 } & 0  \\
1323 < \end{array}} \right)
1323 > \end{array}} \right).
1324   \end{equation}
1325   Thus, the dynamics of free rigid body is governed by
1326   \begin{equation}
1327 < \frac{d}{{dt}}\pi  = J(\pi )\nabla _\pi  T^r (\pi )
1327 > \frac{d}{{dt}}\pi  = J(\pi )\nabla _\pi  T^r (\pi ).
1328   \end{equation}
1366 –
1329   One may notice that each $T_i^r$ in Equation
1330   \ref{introEquation:rotationalKineticRB} can be solved exactly. For
1331   instance, the equations of motion due to $T_1^r$ are given by
# Line 1396 | Line 1358 | e^{\Delta tR_1 }  \approx (1 - \Delta tR_1 )^{ - 1} (1
1358   propagator,
1359   \[
1360   e^{\Delta tR_1 }  \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1
1361 < )
1361 > ).
1362   \]
1363   The flow maps for $T_2^r$ and $T_3^r$ can be found in the same
1364   manner. In order to construct a second-order symplectic method, we
# Line 1414 | Line 1376 | _1 }.
1376   \circ \varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t/2,\pi
1377   _1 }.
1378   \]
1417 –
1379   The non-canonical Lie-Poisson bracket ${F, G}$ of two function
1380   $F(\pi )$ and $G(\pi )$ is defined by
1381   \[
1382   \{ F,G\} (\pi ) = [\nabla _\pi  F(\pi )]^T J(\pi )\nabla _\pi  G(\pi
1383 < )
1383 > ).
1384   \]
1385   If the Poisson bracket of a function $F$ with an arbitrary smooth
1386   function $G$ is zero, $F$ is a \emph{Casimir}, which is the
# Line 1430 | Line 1391 | then by the chain rule
1391   then by the chain rule
1392   \[
1393   \nabla _\pi  F(\pi ) = S'(\frac{{\parallel \pi \parallel ^2
1394 < }}{2})\pi
1394 > }}{2})\pi.
1395   \]
1396 < Thus $ [\nabla _\pi  F(\pi )]^T J(\pi ) =  - S'(\frac{{\parallel \pi
1396 > Thus, $ [\nabla _\pi  F(\pi )]^T J(\pi ) =  - S'(\frac{{\parallel
1397 > \pi
1398   \parallel ^2 }}{2})\pi  \times \pi  = 0 $. This explicit
1399   Lie-Poisson integrator is found to be both extremely efficient and
1400   stable. These properties can be explained by the fact the small
# Line 1445 | Line 1407 | H = T(p,\pi ) + V(q,Q)
1407   The Hamiltonian of rigid body can be separated in terms of kinetic
1408   energy and potential energy,
1409   \[
1410 < H = T(p,\pi ) + V(q,Q)
1410 > H = T(p,\pi ) + V(q,Q).
1411   \]
1412   The equations of motion corresponding to potential energy and
1413   kinetic energy are listed in the below table,
# Line 1494 | Line 1456 | moving rigid bodies
1456   \]
1457   Finally, we obtain the overall symplectic propagators for freely
1458   moving rigid bodies
1459 < \begin{equation}
1460 < \begin{array}{c}
1461 < \varphi _{\Delta t}  = \varphi _{\Delta t/2,F}  \circ \varphi _{\Delta t/2,\tau }  \\
1462 <  \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}
1459 > \begin{eqnarray*}
1460 > \varphi _{\Delta t}  &=& \varphi _{\Delta t/2,F}  \circ \varphi _{\Delta t/2,\tau }  \\
1461 >  & & \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 }  \\
1462 >  & & \circ \varphi _{\Delta t/2,\tau }  \circ \varphi _{\Delta t/2,F}  .\\
1463   \label{introEquation:overallRBFlowMaps}
1464 < \end{equation}
1464 > \end{eqnarray*}
1465  
1466   \section{\label{introSection:langevinDynamics}Langevin Dynamics}
1467   As an alternative to newtonian dynamics, Langevin dynamics, which
# Line 1547 | Line 1507 | W(x) = U(x) - \sum\limits_{\alpha  = 1}^N {\frac{{g_\a
1507   \[
1508   W(x) = U(x) - \sum\limits_{\alpha  = 1}^N {\frac{{g_\alpha ^2
1509   }}{{2m_\alpha  w_\alpha ^2 }}} x^2
1510 < \] and combining the last two terms in Equation
1511 < \ref{introEquation:bathGLE}, we may rewrite the Harmonic bath
1552 < Hamiltonian as
1510 > \]
1511 > and combining the last two terms in Eq.~\ref{introEquation:bathGLE}, we may rewrite the Harmonic bath Hamiltonian as
1512   \[
1513   H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha  = 1}^N
1514   {\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha  }} + \frac{1}{2}m_\alpha
1515   w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha
1516 < w_\alpha ^2 }}x} \right)^2 } \right\}}
1516 > w_\alpha ^2 }}x} \right)^2 } \right\}}.
1517   \]
1518   Since the first two terms of the new Hamiltonian depend only on the
1519   system coordinates, we can get the equations of motion for
# Line 1571 | Line 1530 | m\ddot x_\alpha   =  - m_\alpha  w_\alpha ^2 \left( {x
1530   \frac{{g_\alpha  }}{{m_\alpha  w_\alpha ^2 }}x} \right).
1531   \label{introEquation:bathMotionGLE}
1532   \end{equation}
1574 –
1533   In order to derive an equation for $x$, the dynamics of the bath
1534   variables $x_\alpha$ must be solved exactly first. As an integral
1535   transform which is particularly useful in solving linear ordinary
# Line 1580 | Line 1538 | original problems.
1538   differential equations into simple algebra problems which can be
1539   solved easily. Then, by applying the inverse Laplace transform, also
1540   known as the Bromwich integral, we can retrieve the solutions of the
1541 < original problems.
1542 <
1585 < Let $f(t)$ be a function defined on $ [0,\infty ) $. The Laplace
1586 < transform of f(t) is a new function defined as
1541 > original problems. Let $f(t)$ be a function defined on $ [0,\infty )
1542 > $. The Laplace transform of f(t) is a new function defined as
1543   \[
1544   L(f(t)) \equiv F(p) = \int_0^\infty  {f(t)e^{ - pt} dt}
1545   \]
1546   where  $p$ is real and  $L$ is called the Laplace Transform
1547   Operator. Below are some important properties of Laplace transform
1592 –
1548   \begin{eqnarray*}
1549   L(x + y)  & = & L(x) + L(y) \\
1550   L(ax)     & = & aL(x) \\
# Line 1597 | Line 1552 | Operator. Below are some important properties of Lapla
1552   L(\ddot x)& = & p^2 L(x) - px(0) - \dot x(0) \\
1553   L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right)& = & G(p)H(p) \\
1554   \end{eqnarray*}
1600 –
1601 –
1555   Applying the Laplace transform to the bath coordinates, we obtain
1556   \begin{eqnarray*}
1557   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) \\
1558   L(x_\alpha  ) & = & \frac{{\frac{{g_\alpha  }}{{\omega _\alpha  }}L(x) + px_\alpha  (0) + \dot x_\alpha  (0)}}{{p^2  + \omega _\alpha ^2 }} \\
1559   \end{eqnarray*}
1607 –
1560   By the same way, the system coordinates become
1561   \begin{eqnarray*}
1562 < mL(\ddot x) & = & - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} \\
1563 <  & & \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\}}  \\
1562 > mL(\ddot x) & = &
1563 >  - \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\}}  \\
1564 >  & & - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}}
1565   \end{eqnarray*}
1613 –
1566   With the help of some relatively important inverse Laplace
1567   transformations:
1568   \[
# Line 1620 | Line 1572 | transformations:
1572   L(1) = \frac{1}{p} \\
1573   \end{array}
1574   \]
1575 < , we obtain
1575 > we obtain
1576   \begin{eqnarray*}
1577   m\ddot x & =  & - \frac{{\partial W(x)}}{{\partial x}} -
1578   \sum\limits_{\alpha  = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2
# Line 1693 | Line 1645 | and Equation \ref{introEuqation:GeneralizedLangevinDyn
1645   \[
1646   \int_0^t {\xi (t)\dot x(t - \tau )d\tau }  = \xi _0 (x(t) - x(0))
1647   \]
1648 < and Equation \ref{introEuqation:GeneralizedLangevinDynamics} becomes
1648 > and Eq.~\ref{introEuqation:GeneralizedLangevinDynamics} becomes
1649   \[
1650   m\ddot x =  - \frac{\partial }{{\partial x}}\left( {W(x) +
1651   \frac{1}{2}\xi _0 (x - x_0 )^2 } \right) + R(t),
# Line 1710 | Line 1662 | and Equation \ref{introEuqation:GeneralizedLangevinDyn
1662   \int_0^t {\xi (t)\dot x(t - \tau )d\tau }  = 2\xi _0 \int_0^t
1663   {\delta (t)\dot x(t - \tau )d\tau }  = \xi _0 \dot x(t),
1664   \]
1665 < and Equation \ref{introEuqation:GeneralizedLangevinDynamics} becomes
1665 > and Eq.~\ref{introEuqation:GeneralizedLangevinDynamics} becomes
1666   \begin{equation}
1667   m\ddot x =  - \frac{{\partial W(x)}}{{\partial x}} - \xi _0 \dot
1668   x(t) + R(t) \label{introEquation:LangevinEquation}
# Line 1733 | Line 1685 | And since the $q$ coordinates are harmonic oscillators
1685   R(t) = \sum\limits_{\alpha  = 1}^N {g_\alpha  q_\alpha  (t)}.
1686   \]
1687   And since the $q$ coordinates are harmonic oscillators,
1736 –
1688   \begin{eqnarray*}
1689   \left\langle {q_\alpha ^2 } \right\rangle  & = & \frac{{kT}}{{m_\alpha  \omega _\alpha ^2 }} \\
1690   \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 1693 | And since the $q$ coordinates are harmonic oscillators
1693    & = &\sum\limits_\alpha  {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha  t)}  \\
1694    & = &kT\xi (t) \\
1695   \end{eqnarray*}
1745 –
1696   Thus, we recover the \emph{second fluctuation dissipation theorem}
1697   \begin{equation}
1698   \xi (t) = \left\langle {R(t)R(0)} \right\rangle

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