| 450 |
|
popularity in the molecular dynamics community. This fact can be |
| 451 |
|
partly explained by its geometric nature. |
| 452 |
|
|
| 453 |
< |
\subsection{\label{introSection:symplecticManifold}Symplectic Manifolds} |
| 453 |
> |
\subsection{\label{introSection:symplecticManifold}Manifolds and Bundles} |
| 454 |
|
A \emph{manifold} is an abstract mathematical space. It looks |
| 455 |
|
locally like Euclidean space, but when viewed globally, it may have |
| 456 |
|
more complicated structure. A good example of manifold is the |
| 465 |
|
$\omega(\lambda_1x_1+\lambda_2x_2, y) = \lambda_1\omega(x_1, y)+ |
| 466 |
|
\lambda_2\omega(x_2, y)$, $\omega(x, y) = - \omega(y, x)$ and |
| 467 |
|
$\omega(x, x) = 0$.\cite{McDuff1998} The cross product operation in |
| 468 |
< |
vector field is an example of symplectic form. One of the |
| 469 |
< |
motivations to study \emph{symplectic manifolds} in Hamiltonian |
| 470 |
< |
Mechanics is that a symplectic manifold can represent all possible |
| 471 |
< |
configurations of the system and the phase space of the system can |
| 472 |
< |
be described by it's cotangent bundle.\cite{Jost2002} Every |
| 473 |
< |
symplectic manifold is even dimensional. For instance, in Hamilton |
| 474 |
< |
equations, coordinate and momentum always appear in pairs. |
| 468 |
> |
vector field is an example of symplectic form. |
| 469 |
> |
Given vector spaces $V$ and $W$ over same field $F$, $f: V \to W$ is a linear transformation if |
| 470 |
> |
\begin{eqnarray*} |
| 471 |
> |
f(x+y) & = & f(x) + f(y) \\ |
| 472 |
> |
f(ax) & = & af(x) |
| 473 |
> |
\end{eqnarray*} |
| 474 |
> |
are always satisfied for any two vectors $x$ and $y$ in $V$ and any scalar $a$ in $F$. One can define the dual vector space $V^*$ of $V$ if any two built-in linear transformations $\phi$ and $\psi$ in $V^*$ satisfy the following definition of addition and scalar multiplication: |
| 475 |
> |
\begin{eqnarray*} |
| 476 |
> |
(\phi+\psi)(x) & = & \phi(x)+\psi(x) \\ |
| 477 |
> |
(a\phi)(x) & = & a \phi(x) |
| 478 |
> |
\end{eqnarray*} |
| 479 |
> |
for all $a$ in $F$ and $x$ in $V$. For a manifold $M$, one can define a tangent vector of a tangent space $TM_q$ at every point $q$ |
| 480 |
> |
\begin{equation} |
| 481 |
> |
\dot q = \mathop {\lim }\limits_{t \to 0} \frac{{\phi (t) - \phi (0)}}{t} |
| 482 |
> |
\end{equation} |
| 483 |
> |
where $\phi(0)=q$ and $\phi(t) \in M$. One may also define a cotangent space $T^*M_q$ as the dual space of the tangent space $TM_q$. The tangent space and the cotangent space are isomorphic to each other, since they are both real vector spaces with same dimension. |
| 484 |
> |
The union of tangent spaces at every point of $M$ is called the tangent bundle of $M$ and is denoted by $TM$, while cotangent bundle $T^*M$ is defined as the union of the cotangent spaces to $M$.\cite{Jost2002} For a Hamiltonian system with configuration manifold $V$, the $(q,\dot q)$ phase space is the tangent bundle of the configuration manifold $V$, while the cotangent bundle is represented by $(q,p)$. |
| 485 |
|
|
| 486 |
|
\subsection{\label{introSection:ODE}Ordinary Differential Equations} |
| 487 |
|
|
| 538 |
|
\begin{equation} |
| 539 |
|
\varphi _\tau = \varphi _{ - \tau }^{ - 1}. |
| 540 |
|
\end{equation} |
| 531 |
– |
The exact propagator can also be written as an operator, |
| 532 |
– |
\begin{equation} |
| 533 |
– |
\varphi _\tau (x) = e^{\tau \sum\limits_i {f_i (x)\frac{\partial |
| 534 |
– |
}{{\partial x_i }}} } (x) \equiv \exp (\tau f)(x). |
| 535 |
– |
\label{introEquation:exponentialOperator} |
| 536 |
– |
\end{equation} |
| 541 |
|
In most cases, it is not easy to find the exact propagator |
| 542 |
|
$\varphi_\tau$. Instead, we use an approximate map, $\psi_\tau$, |
| 543 |
|
which is usually called an integrator. The order of an integrator |
| 627 |
|
variational methods can capture the decay of energy |
| 628 |
|
accurately.\cite{Kane2000} Since they are geometrically unstable |
| 629 |
|
against non-Hamiltonian perturbations, ordinary implicit Runge-Kutta |
| 630 |
< |
methods are not suitable for Hamiltonian system. Recently, various |
| 631 |
< |
high-order explicit Runge-Kutta methods \cite{Owren1992,Chen2003} |
| 632 |
< |
have been developed to overcome this instability. However, due to |
| 633 |
< |
computational penalty involved in implementing the Runge-Kutta |
| 634 |
< |
methods, they have not attracted much attention from the Molecular |
| 635 |
< |
Dynamics community. Instead, splitting methods have been widely |
| 636 |
< |
accepted since they exploit natural decompositions of the |
| 637 |
< |
system.\cite{McLachlan1998, Tuckerman1992} |
| 630 |
> |
methods are not suitable for Hamiltonian |
| 631 |
> |
system.\cite{Cartwright1992} Recently, various high-order explicit |
| 632 |
> |
Runge-Kutta methods \cite{Owren1992,Chen2003} have been developed to |
| 633 |
> |
overcome this instability. However, due to computational penalty |
| 634 |
> |
involved in implementing the Runge-Kutta methods, they have not |
| 635 |
> |
attracted much attention from the Molecular Dynamics community. |
| 636 |
> |
Instead, splitting methods have been widely accepted since they |
| 637 |
> |
exploit natural decompositions of the system.\cite{McLachlan1998, |
| 638 |
> |
Tuckerman1992} |
| 639 |
|
|
| 640 |
|
\subsubsection{\label{introSection:splittingMethod}\textbf{Splitting Methods}} |
| 641 |
|
|
| 658 |
|
problem. If $H_1$ and $H_2$ can be integrated using exact |
| 659 |
|
propagators $\varphi_1(t)$ and $\varphi_2(t)$, respectively, a |
| 660 |
|
simple first order expression is then given by the Lie-Trotter |
| 661 |
< |
formula |
| 661 |
> |
formula\cite{Trotter1959} |
| 662 |
|
\begin{equation} |
| 663 |
|
\varphi _h = \varphi _{1,h} \circ \varphi _{2,h}, |
| 664 |
|
\label{introEquation:firstOrderSplitting} |
| 1201 |
|
1,\tilde Q^T \tilde PJ^{ - 1} + J^{ - 1} P^T \tilde Q = 0} \right\} |
| 1202 |
|
\] |
| 1203 |
|
For a body fixed vector $X_i$ with respect to the center of mass of |
| 1204 |
< |
the rigid body, its corresponding lab fixed vector $X_0^{lab}$ is |
| 1204 |
> |
the rigid body, its corresponding lab fixed vector $X_i^{lab}$ is |
| 1205 |
|
given as |
| 1206 |
|
\begin{equation} |
| 1207 |
|
X_i^{lab} = Q X_i + q. |
