| 661 |
|
In other words, the flow of this vector field is reversible if and |
| 662 |
|
only if $ h \circ \varphi ^{ - 1} = \varphi \circ h $. |
| 663 |
|
|
| 664 |
< |
When designing any numerical methods, one should always try to |
| 664 |
> |
A \emph{first integral}, or conserved quantity of a general |
| 665 |
> |
differential function is a function $ G:R^{2d} \to R^d $ which is |
| 666 |
> |
constant for all solutions of the ODE $\frac{{dx}}{{dt}} = f(x)$ , |
| 667 |
> |
\[ |
| 668 |
> |
\frac{{dG(x(t))}}{{dt}} = 0. |
| 669 |
> |
\] |
| 670 |
> |
Using chain rule, one may obtain, |
| 671 |
> |
\[ |
| 672 |
> |
\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \bullet \nabla G, |
| 673 |
> |
\] |
| 674 |
> |
which is the condition for conserving \emph{first integral}. For a |
| 675 |
> |
canonical Hamiltonian system, the time evolution of an arbitrary |
| 676 |
> |
smooth function $G$ is given by, |
| 677 |
> |
\begin{equation} |
| 678 |
> |
\begin{array}{c} |
| 679 |
> |
\frac{{dG(x(t))}}{{dt}} = [\nabla _x G(x(t))]^T \dot x(t) \\ |
| 680 |
> |
= [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). \\ |
| 681 |
> |
\end{array} |
| 682 |
> |
\label{introEquation:firstIntegral1} |
| 683 |
> |
\end{equation} |
| 684 |
> |
Using poisson bracket notion, Equation |
| 685 |
> |
\ref{introEquation:firstIntegral1} can be rewritten as |
| 686 |
> |
\[ |
| 687 |
> |
\frac{d}{{dt}}G(x(t)) = \left\{ {G,H} \right\}(x(t)). |
| 688 |
> |
\] |
| 689 |
> |
Therefore, the sufficient condition for $G$ to be the \emph{first |
| 690 |
> |
integral} of a Hamiltonian system is |
| 691 |
> |
\[ |
| 692 |
> |
\left\{ {G,H} \right\} = 0. |
| 693 |
> |
\] |
| 694 |
> |
As well known, the Hamiltonian (or energy) H of a Hamiltonian system |
| 695 |
> |
is a \emph{first integral}, which is due to the fact $\{ H,H\} = |
| 696 |
> |
0$. |
| 697 |
> |
|
| 698 |
> |
|
| 699 |
> |
When designing any numerical methods, one should always try to |
| 700 |
|
preserve the structural properties of the original ODE and its flow. |
| 701 |
|
|
| 702 |
|
\subsection{\label{introSection:constructionSymplectic}Construction of Symplectic Methods} |
| 907 |
|
|
| 908 |
|
\subsection{\label{introSec:mdInit}Initialization} |
| 909 |
|
|
| 910 |
+ |
\subsection{\label{introSec:forceEvaluation}Force Evaluation} |
| 911 |
+ |
|
| 912 |
|
\subsection{\label{introSection:mdIntegration} Integration of the Equations of Motion} |
| 913 |
|
|
| 914 |
|
\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} |
| 915 |
|
|
| 916 |
< |
A rigid body is a body in which the distance between any two given |
| 917 |
< |
points of a rigid body remains constant regardless of external |
| 918 |
< |
forces exerted on it. A rigid body therefore conserves its shape |
| 919 |
< |
during its motion. |
| 916 |
> |
Rigid bodies are frequently involved in the modeling of different |
| 917 |
> |
areas, from engineering, physics, to chemistry. For example, |
| 918 |
> |
missiles and vehicle are usually modeled by rigid bodies. The |
| 919 |
> |
movement of the objects in 3D gaming engine or other physics |
| 920 |
> |
simulator is governed by the rigid body dynamics. In molecular |
| 921 |
> |
simulation, rigid body is used to simplify the model in |
| 922 |
> |
protein-protein docking study{\cite{Gray03}}. |
| 923 |
|
|
| 924 |
< |
Applications of dynamics of rigid bodies. |
| 924 |
> |
It is very important to develop stable and efficient methods to |
| 925 |
> |
integrate the equations of motion of orientational degrees of |
| 926 |
> |
freedom. Euler angles are the nature choice to describe the |
| 927 |
> |
rotational degrees of freedom. However, due to its singularity, the |
| 928 |
> |
numerical integration of corresponding equations of motion is very |
| 929 |
> |
inefficient and inaccurate. Although an alternative integrator using |
| 930 |
> |
different sets of Euler angles can overcome this difficulty\cite{}, |
| 931 |
> |
the computational penalty and the lost of angular momentum |
| 932 |
> |
conservation still remain. A singularity free representation |
| 933 |
> |
utilizing quaternions was developed by Evans in 1977. Unfortunately, |
| 934 |
> |
this approach suffer from the nonseparable Hamiltonian resulted from |
| 935 |
> |
quaternion representation, which prevents the symplectic algorithm |
| 936 |
> |
to be utilized. Another different approach is to apply holonomic |
| 937 |
> |
constraints to the atoms belonging to the rigid body. Each atom |
| 938 |
> |
moves independently under the normal forces deriving from potential |
| 939 |
> |
energy and constraint forces which are used to guarantee the |
| 940 |
> |
rigidness. However, due to their iterative nature, SHAKE and Rattle |
| 941 |
> |
algorithm converge very slowly when the number of constraint |
| 942 |
> |
increases. |
| 943 |
|
|
| 944 |
+ |
The break through in geometric literature suggests that, in order to |
| 945 |
+ |
develop a long-term integration scheme, one should preserve the |
| 946 |
+ |
symplectic structure of the flow. Introducing conjugate momentum to |
| 947 |
+ |
rotation matrix $A$ and re-formulating Hamiltonian's equation, a |
| 948 |
+ |
symplectic integrator, RSHAKE, was proposed to evolve the |
| 949 |
+ |
Hamiltonian system in a constraint manifold by iteratively |
| 950 |
+ |
satisfying the orthogonality constraint $A_t A = 1$. An alternative |
| 951 |
+ |
method using quaternion representation was developed by Omelyan. |
| 952 |
+ |
However, both of these methods are iterative and inefficient. In |
| 953 |
+ |
this section, we will present a symplectic Lie-Poisson integrator |
| 954 |
+ |
for rigid body developed by Dullweber and his coworkers\cite{}. |
| 955 |
+ |
|
| 956 |
|
\subsection{\label{introSection:lieAlgebra}Lie Algebra} |
| 957 |
|
|
| 958 |
|
\subsection{\label{introSection:DLMMotionEquation}The Euler Equations of Rigid Body Motion} |
