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 |