| 312 |
|
\end{equation} |
| 313 |
|
where $x = x(q,p)^T$, this system is canonical Hamiltonian, if |
| 314 |
|
\begin{equation} |
| 315 |
< |
f(r) = J\nabla _x H(r) |
| 315 |
> |
f(r) = J\nabla _x H(r). |
| 316 |
|
\end{equation} |
| 317 |
|
$H = H (q, p)$ is Hamiltonian function and $J$ is the skew-symmetric |
| 318 |
|
matrix |
| 372 |
|
\end{equation} |
| 373 |
|
|
| 374 |
|
The hidden geometric properties of ODE and its flow play important |
| 375 |
< |
roles in numerical studies. The flow of a Hamiltonian vector field |
| 376 |
< |
on a symplectic manifold is a symplectomorphism. Let $\varphi$ be |
| 377 |
< |
the flow of Hamiltonian vector field, $\varphi$ is a |
| 378 |
< |
\emph{symplectic} flow if it satisfies, |
| 375 |
> |
roles in numerical studies. Let $\varphi$ be the flow of Hamiltonian |
| 376 |
> |
vector field, $\varphi$ is a \emph{symplectic} flow if it satisfies, |
| 377 |
|
\begin{equation} |
| 378 |
< |
d \varphi^T J d \varphi = J. |
| 378 |
> |
'\varphi^T J '\varphi = J. |
| 379 |
|
\end{equation} |
| 380 |
|
According to Liouville's theorem, the symplectic volume is invariant |
| 381 |
|
under a Hamiltonian flow, which is the basis for classical |
| 382 |
< |
statistical mechanics. As to the Poisson system, |
| 382 |
> |
statistical mechanics. Furthermore, the flow of a Hamiltonian vector |
| 383 |
> |
field on a symplectic manifold can be shown to be a |
| 384 |
> |
symplectomorphism. As to the Poisson system, |
| 385 |
|
\begin{equation} |
| 386 |
< |
d\varphi ^T Jd\varphi = J \circ \varphi |
| 386 |
> |
'\varphi ^T J '\varphi = J \circ \varphi |
| 387 |
|
\end{equation} |
| 388 |
|
is the property must be preserved by the integrator. It is possible |
| 389 |
|
to construct a \emph{volume-preserving} flow for a source free($ |
| 399 |
|
designing any numerical methods, one should always try to preserve |
| 400 |
|
the structural properties of the original ODE and its flow. |
| 401 |
|
|
| 402 |
< |
\subsection{\label{introSection:splittingAndComposition}Splitting and Composition Methods} |
| 402 |
> |
\subsection{\label{introSection:constructionSymplectic}Construction of Symplectic Methods} |
| 403 |
> |
A lot of well established and very effective numerical methods have |
| 404 |
> |
been successful precisely because of their symplecticities even |
| 405 |
> |
though this fact was not recognized when they were first |
| 406 |
> |
constructed. The most famous example is leapfrog methods in |
| 407 |
> |
molecular dynamics. In general, symplectic integrators can be |
| 408 |
> |
constructed using one of four different methods. |
| 409 |
> |
\begin{enumerate} |
| 410 |
> |
\item Generating functions |
| 411 |
> |
\item Variational methods |
| 412 |
> |
\item Runge-Kutta methods |
| 413 |
> |
\item Splitting methods |
| 414 |
> |
\end{enumerate} |
| 415 |
> |
|
| 416 |
> |
Generating function tends to lead to methods which are cumbersome |
| 417 |
> |
and difficult to use\cite{}. In dissipative systems, variational |
| 418 |
> |
methods can capture the decay of energy accurately\cite{}. Since |
| 419 |
> |
their geometrically unstable nature against non-Hamiltonian |
| 420 |
> |
perturbations, ordinary implicit Runge-Kutta methods are not |
| 421 |
> |
suitable for Hamiltonian system. Recently, various high-order |
| 422 |
> |
explicit Runge--Kutta methods have been developed to overcome this |
| 423 |
> |
instability \cite{}. However, due to computational penalty involved |
| 424 |
> |
in implementing the Runge-Kutta methods, they do not attract too |
| 425 |
> |
much attention from Molecular Dynamics community. Instead, splitting |
| 426 |
> |
have been widely accepted since they exploit natural decompositions |
| 427 |
> |
of the system\cite{Tuckerman92}. The main idea behind splitting |
| 428 |
> |
methods is to decompose the discrete $\varphi_h$ as a composition of |
| 429 |
> |
simpler flows, |
| 430 |
> |
\begin{equation} |
| 431 |
> |
\varphi _h = \varphi _{h_1 } \circ \varphi _{h_2 } \ldots \circ |
| 432 |
> |
\varphi _{h_n } |
| 433 |
> |
\label{introEquation:FlowDecomposition} |
| 434 |
> |
\end{equation} |
| 435 |
> |
where each of the sub-flow is chosen such that each represent a |
| 436 |
> |
simpler integration of the system. Let $\phi$ and $\psi$ both be |
| 437 |
> |
symplectic maps, it is easy to show that any composition of |
| 438 |
> |
symplectic flows yields a symplectic map, |
| 439 |
> |
\begin{equation} |
| 440 |
> |
(\varphi '\phi ')^T J\varphi '\phi ' = \phi '^T \varphi '^T J\varphi |
| 441 |
> |
'\phi ' = \phi '^T J\phi ' = J. |
| 442 |
> |
\label{introEquation:SymplecticFlowComposition} |
| 443 |
> |
\end{equation} |
| 444 |
> |
Suppose that a Hamiltonian system has a form with $H = T + V$ |
| 445 |
|
|
| 446 |
+ |
|
| 447 |
+ |
|
| 448 |
|
\section{\label{introSection:molecularDynamics}Molecular Dynamics} |
| 449 |
|
|
| 450 |
|
As a special discipline of molecular modeling, Molecular dynamics |
