| 846 |
|
\] |
| 847 |
|
Applying Baker-Campbell-Hausdorff formula to Sprang splitting, we |
| 848 |
|
can obtain |
| 849 |
< |
\begin{equation} |
| 850 |
< |
\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2 |
| 851 |
< |
[X,Y]/4 + h^2 [Y,X]/4 \\ & & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\ |
| 852 |
< |
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3 [X,[X,Y]]/24 & & \mbox{} + |
| 853 |
< |
\ldots ) |
| 854 |
< |
\end{equation} |
| 849 |
> |
\begin{eqnarray*} |
| 850 |
> |
\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2 [X,Y]/4 + h^2 [Y,X]/4 \\ |
| 851 |
> |
& & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\ |
| 852 |
> |
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3[X,[X,Y]]/24 + \ldots ) |
| 853 |
> |
\end{eqnarray*} |
| 854 |
|
Since \[ [X,Y] + [Y,X] = 0\] and \[ [X,X] = 0\], the dominant local |
| 855 |
|
error of Spring splitting is proportional to $h^3$. The same |
| 856 |
|
procedure can be applied to general splitting, of the form |
| 858 |
|
\varphi _{b_m h}^2 \circ \varphi _{a_m h}^1 \circ \varphi _{b_{m - |
| 859 |
|
1} h}^2 \circ \ldots \circ \varphi _{a_1 h}^1 . |
| 860 |
|
\end{equation} |
| 861 |
< |
Careful choice of coefficient $a_1 ,\ldot , b_m$ will lead to higher |
| 861 |
> |
Careful choice of coefficient $a_1 \ldot b_m$ will lead to higher |
| 862 |
|
order method. Yoshida proposed an elegant way to compose higher |
| 863 |
|
order methods based on symmetric splitting. Given a symmetric second |
| 864 |
|
order base method $ \varphi _h^{(2)} $, a fourth-order symmetric |
| 1261 |
|
In general, there are two ways to satisfy the holonomic constraints. |
| 1262 |
|
We can use constraint force provided by lagrange multiplier on the |
| 1263 |
|
normal manifold to keep the motion on constraint space. Or we can |
| 1264 |
< |
simply evolve the system in constraint manifold. The two method are |
| 1265 |
< |
proved to be equivalent. The holonomic constraint and equations of |
| 1266 |
< |
motions define a constraint manifold for rigid body |
| 1264 |
> |
simply evolve the system in constraint manifold. These two methods |
| 1265 |
> |
are proved to be equivalent. The holonomic constraint and equations |
| 1266 |
> |
of motions define a constraint manifold for rigid body |
| 1267 |
|
\[ |
| 1268 |
|
M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0} |
| 1269 |
|
\right\}. |
| 1476 |
|
\] |
| 1477 |
|
The equations of motion corresponding to potential energy and |
| 1478 |
|
kinetic energy are listed in the below table, |
| 1479 |
+ |
\begin{table} |
| 1480 |
+ |
\caption{Equations of motion due to Potential and Kinetic Energies} |
| 1481 |
|
\begin{center} |
| 1482 |
|
\begin{tabular}{|l|l|} |
| 1483 |
|
\hline |
| 1490 |
|
\hline |
| 1491 |
|
\end{tabular} |
| 1492 |
|
\end{center} |
| 1493 |
< |
A second-order symplectic method is now obtained by the composition |
| 1494 |
< |
of the flow maps, |
| 1493 |
> |
\end{table} |
| 1494 |
> |
A second-order symplectic method is now obtained by the |
| 1495 |
> |
composition of the flow maps, |
| 1496 |
|
\[ |
| 1497 |
|
\varphi _{\Delta t} = \varphi _{\Delta t/2,V} \circ \varphi |
| 1498 |
|
_{\Delta t,T} \circ \varphi _{\Delta t/2,V}. |
| 1788 |
|
when the system become more and more complicate. Instead, various |
| 1789 |
|
approaches based on hydrodynamics have been developed to calculate |
| 1790 |
|
the friction coefficients. The friction effect is isotropic in |
| 1791 |
< |
Equation, \zeta can be taken as a scalar. In general, friction |
| 1792 |
< |
tensor \Xi is a $6\times 6$ matrix given by |
| 1791 |
> |
Equation, $\zeta$ can be taken as a scalar. In general, friction |
| 1792 |
> |
tensor $\Xi$ is a $6\times 6$ matrix given by |
| 1793 |
|
\[ |
| 1794 |
|
\Xi = \left( {\begin{array}{*{20}c} |
| 1795 |
|
{\Xi _{}^{tt} } & {\Xi _{}^{rt} } \\ |
