| 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 |
| 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 |
| 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 |
| 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} |
| 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 } |
| 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}. |
| 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 |
| 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, $x = (q_1 , \ldots |
| 223 |
< |
,q_f ,p_1 , \ldots ,p_f )$, with a unique set of values of $6f$ |
| 224 |
< |
coordinates and momenta is a phase space vector. |
| 225 |
< |
|
| 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. |
| 226 |
|
%%%fix me |
| 227 |
< |
A microscopic state or microstate of a classical system is |
| 228 |
< |
specification of the complete phase space vector of a system at any |
| 237 |
< |
instant in time. An ensemble is defined as a collection of systems |
| 238 |
< |
sharing one or more macroscopic characteristics but each being in a |
| 239 |
< |
unique microstate. The complete ensemble is specified by giving all |
| 240 |
< |
systems or microstates consistent with the common macroscopic |
| 241 |
< |
characteristics of the ensemble. Although the state of each |
| 242 |
< |
individual system in the ensemble could be precisely described at |
| 243 |
< |
any instance in time by a suitable phase space vector, when using |
| 244 |
< |
ensembles for statistical purposes, there is no need to maintain |
| 245 |
< |
distinctions between individual systems, since the numbers of |
| 246 |
< |
systems at any time in the different states which correspond to |
| 247 |
< |
different regions of the phase space are more interesting. Moreover, |
| 248 |
< |
in the point of view of statistical mechanics, one would prefer to |
| 249 |
< |
use ensembles containing a large enough population of separate |
| 250 |
< |
members so that the numbers of systems in such different states can |
| 251 |
< |
be regarded as changing continuously as we traverse different |
| 252 |
< |
regions of the phase space. The condition of an ensemble at any time |
| 227 |
> |
|
| 228 |
> |
In statistical mechanics, the condition of an ensemble at any time |
| 229 |
|
can be regarded as appropriately specified by the density $\rho$ |
| 230 |
|
with which representative points are distributed over the phase |
| 231 |
|
space. The density distribution for an ensemble with $f$ degrees of |
| 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 |
| 310 |
< |
isolated and conserve energy, the Microcanonical ensemble (NVE) has |
| 311 |
< |
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 - |
| 348 |
|
\frac{{\partial \rho }}{{\partial p_i }}\dot p_i } \right)} = 0 . |
| 349 |
|
\label{introEquation:liouvilleTheorem} |
| 350 |
|
\end{equation} |
| 377 |
– |
|
| 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 |
| 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, |
| 526 |
< |
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 |
|
|
| 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 |
< |
|
| 555 |
< |
Another generalization of Hamiltonian dynamics is Poisson |
| 556 |
< |
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} |
| 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. |
| 606 |
– |
|
| 576 |
|
Let $\varphi$ be the flow of Hamiltonian vector field, $\varphi$ is |
| 577 |
|
a \emph{symplectic} flow if it satisfies, |
| 578 |
|
\begin{equation} |
| 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 |
| 625 |
< |
be volume-preserving. |
| 626 |
< |
|
| 627 |
< |
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 |
< |
|
| 636 |
< |
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 |
|
\[ |
| 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, |
| 649 |
– |
|
| 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} |
| 655 |
– |
|
| 656 |
– |
|
| 619 |
|
Using poisson bracket notion, Equation |
| 620 |
|
\ref{introEquation:firstIntegral1} can be rewritten as |
| 621 |
|
\[ |
| 628 |
|
\] |
| 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 $\{ H,H\} = |
| 631 |
< |
0$. |
| 670 |
< |
|
| 671 |
< |
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} |
| 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 |
< |
|
| 714 |
< |
Suppose that a Hamiltonian system takes the form, |
| 672 |
> |
simpler integration of the system. Suppose that a Hamiltonian system |
| 673 |
> |
takes the form, |
| 674 |
|
\[ |
| 675 |
|
H = H_1 + H_2. |
| 676 |
|
\] |
| 711 |
|
\begin{equation} |
| 712 |
|
\varphi _h^{ - 1} = \varphi _{ - h}. |
| 713 |
|
\label{introEquation:timeReversible} |
| 714 |
< |
\end{equation},appendixFig:architecture |
| 714 |
> |
\end{equation} |
| 715 |
|
|
| 716 |
|
\subsubsection{\label{introSection:exampleSplittingMethod}\textbf{Examples of the Splitting Method}} |
| 717 |
|
The classical equation for a system consisting of interacting |
| 760 |
|
|
| 761 |
|
\item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values. |
| 762 |
|
\end{enumerate} |
| 804 |
– |
|
| 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, |
| 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}. |
| 1107 |
– |
|
| 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 |
| 1189 |
|
Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0 . \\ |
| 1190 |
|
\label{introEquation:RBFirstOrderConstraint} |
| 1191 |
|
\end{equation} |
| 1235 |
– |
|
| 1192 |
|
Using Equation (\ref{introEquation:motionHamiltonianCoordinate}, |
| 1193 |
|
\ref{introEquation:motionHamiltonianMomentum}), one can write down |
| 1194 |
|
the equations of motion, |
| 1239 |
– |
|
| 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} |
| 1246 |
– |
|
| 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 |
| 1209 |
|
M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0} |
| 1210 |
|
\right\}. |
| 1211 |
|
\] |
| 1258 |
– |
|
| 1212 |
|
Unfortunately, this constraint manifold is not the cotangent bundle |
| 1213 |
< |
$T_{\star}SO(3)$. However, it turns out that under symplectic |
| 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 |
| 1215 |
|
transformation, the cotangent space and the phase space are |
| 1216 |
|
diffeomorphic. By introducing |
| 1217 |
|
\[ |
| 1223 |
|
T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \tilde Q = |
| 1224 |
|
1,\tilde Q^T \tilde PJ^{ - 1} + J^{ - 1} P^T \tilde Q = 0} \right\} |
| 1225 |
|
\] |
| 1272 |
– |
|
| 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 |
| 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 |
| 1294 |
< |
body, the angular momentum on the body fixed frame $\Pi = Q^t P$ is |
| 1295 |
< |
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 |
< |
\mathop \Pi \limits^ \bullet = J^{ - 1} \Pi ^T \Pi + Q^T \sum\limits_i {F_i (q,Q)X_i^T } - \Lambda \\ |
| 1250 |
< |
\mathop Q\limits^{{\rm{ }} \bullet } = 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 |
|
\[ |
| 1256 |
|
\begin{array}{l} |
| 1257 |
< |
\Pi J^{ - 1} + J^{ - 1} \Pi ^t = 0 \\ |
| 1258 |
< |
Q^T Q = 1 \\ |
| 1257 |
> |
\Pi J^{ - 1} + J^{ - 1} \Pi ^t = 0, \\ |
| 1258 |
> |
Q^T Q = 1 .\\ |
| 1259 |
|
\end{array} |
| 1260 |
|
\] |
| 1310 |
– |
|
| 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} |
| 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 |
+ |
\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 |
+ |
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} |
| 1330 |
– |
(\mathop \Pi \limits^ \bullet - \mathop \Pi \limits^ {\bullet ^T} |
| 1331 |
– |
){\rm{ }} = {\rm{ }}(\Pi - \Pi ^T ){\rm{ }}(J^{ - 1} \Pi + \Pi J^{ |
| 1332 |
– |
- 1} ) + \sum\limits_i {[Q^T F_i (r,Q)X_i^T - X_i F_i (r,Q)^T Q]} - |
| 1333 |
– |
(\Lambda - \Lambda ^T ) . \label{introEquation:skewMatrixPI} |
| 1334 |
– |
\end{equation} |
| 1335 |
– |
Since $\Lambda$ is symmetric, the last term of Equation |
| 1336 |
– |
\ref{introEquation:skewMatrixPI} is zero, which implies the Lagrange |
| 1337 |
– |
multiplier $\Lambda$ is absent from the equations of motion. This |
| 1338 |
– |
unique property eliminates the requirement of iterations which can |
| 1339 |
– |
not be avoided in other methods\cite{Kol1997, Omelyan1998}. |
| 1340 |
– |
|
| 1341 |
– |
Applying the hat-map isomorphism, we obtain the equation of motion |
| 1342 |
– |
for angular momentum on body frame |
| 1343 |
– |
\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 }. |
| 1296 |
|
\label{introEquation:bodyAngularMotion} |
| 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 |
| 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} |
| 1379 |
– |
|
| 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 |
| 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 |
| 1376 |
|
\circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi |
| 1377 |
|
_1 }. |
| 1378 |
|
\] |
| 1430 |
– |
|
| 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 |
| 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 |
| 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, |
| 1414 |
|
\begin{table} |
| 1415 |
< |
\caption{Equations of motion due to Potential and Kinetic Energies} |
| 1415 |
> |
\caption{EQUATIONS OF MOTION DUE TO POTENTIAL AND KINETIC ENERGIES} |
| 1416 |
|
\begin{center} |
| 1417 |
|
\begin{tabular}{|l|l|} |
| 1418 |
|
\hline |
| 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 } \\ |
| 1514 |
< |
\circ \varphi _{\Delta t/2,\tau } \circ \varphi _{\Delta t/2,F} .\\ |
| 1515 |
< |
\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 |
| 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 |
| 1565 |
< |
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 |
| 1530 |
|
\frac{{g_\alpha }}{{m_\alpha w_\alpha ^2 }}x} \right). |
| 1531 |
|
\label{introEquation:bathMotionGLE} |
| 1532 |
|
\end{equation} |
| 1587 |
– |
|
| 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 |
| 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 |
< |
|
| 1598 |
< |
Let $f(t)$ be a function defined on $ [0,\infty ) $. The Laplace |
| 1599 |
< |
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 |
| 1605 |
– |
|
| 1548 |
|
\begin{eqnarray*} |
| 1549 |
|
L(x + y) & = & L(x) + L(y) \\ |
| 1550 |
|
L(ax) & = & aL(x) \\ |
| 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*} |
| 1613 |
– |
|
| 1614 |
– |
|
| 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*} |
| 1620 |
– |
|
| 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*} |
| 1626 |
– |
|
| 1566 |
|
With the help of some relatively important inverse Laplace |
| 1567 |
|
transformations: |
| 1568 |
|
\[ |
| 1572 |
|
L(1) = \frac{1}{p} \\ |
| 1573 |
|
\end{array} |
| 1574 |
|
\] |
| 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 |
| 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 |
|
\[ |
| 1650 |
|
m\ddot x = - \frac{\partial }{{\partial x}}\left( {W(x) + |
| 1651 |
|
\frac{1}{2}\xi _0 (x - x_0 )^2 } \right) + R(t), |
| 1662 |
|
\int_0^t {\xi (t)\dot x(t - \tau )d\tau } = 2\xi _0 \int_0^t |
| 1663 |
|
{\delta (t)\dot x(t - \tau )d\tau } = \xi _0 \dot x(t), |
| 1664 |
|
\] |
| 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} |
| 1685 |
|
R(t) = \sum\limits_{\alpha = 1}^N {g_\alpha q_\alpha (t)}. |
| 1686 |
|
\] |
| 1687 |
|
And since the $q$ coordinates are harmonic oscillators, |
| 1749 |
– |
|
| 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) \\ |
| 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*} |
| 1758 |
– |
|
| 1696 |
|
Thus, we recover the \emph{second fluctuation dissipation theorem} |
| 1697 |
|
\begin{equation} |
| 1698 |
|
\xi (t) = \left\langle {R(t)R(0)} \right\rangle |
