| 1262 |
|
In general, there are two ways to satisfy the holonomic constraints. |
| 1263 |
|
We can use constraint force provided by lagrange multiplier on the |
| 1264 |
|
normal manifold to keep the motion on constraint space. Or we can |
| 1265 |
< |
simply evolve the system in constraint manifold. The two method are |
| 1266 |
< |
proved to be equivalent. The holonomic constraint and equations of |
| 1267 |
< |
motions define a constraint manifold for rigid body |
| 1265 |
> |
simply evolve the system in constraint manifold. These two methods |
| 1266 |
> |
are proved to be equivalent. The holonomic constraint and equations |
| 1267 |
> |
of motions define a constraint manifold for rigid body |
| 1268 |
|
\[ |
| 1269 |
|
M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0} |
| 1270 |
|
\right\}. |
| 1477 |
|
\] |
| 1478 |
|
The equations of motion corresponding to potential energy and |
| 1479 |
|
kinetic energy are listed in the below table, |
| 1480 |
+ |
\begin{table} |
| 1481 |
+ |
\caption{Equations of motion due to Potential and Kinetic Energies} |
| 1482 |
|
\begin{center} |
| 1483 |
|
\begin{tabular}{|l|l|} |
| 1484 |
|
\hline |
| 1491 |
|
\hline |
| 1492 |
|
\end{tabular} |
| 1493 |
|
\end{center} |
| 1494 |
< |
A second-order symplectic method is now obtained by the composition |
| 1495 |
< |
of the flow maps, |
| 1494 |
> |
\end{table} |
| 1495 |
> |
A second-order symplectic method is now obtained by the |
| 1496 |
> |
composition of the flow maps, |
| 1497 |
|
\[ |
| 1498 |
|
\varphi _{\Delta t} = \varphi _{\Delta t/2,V} \circ \varphi |
| 1499 |
|
_{\Delta t,T} \circ \varphi _{\Delta t/2,V}. |
| 1789 |
|
when the system become more and more complicate. Instead, various |
| 1790 |
|
approaches based on hydrodynamics have been developed to calculate |
| 1791 |
|
the friction coefficients. The friction effect is isotropic in |
| 1792 |
< |
Equation, \zeta can be taken as a scalar. In general, friction |
| 1793 |
< |
tensor \Xi is a $6\times 6$ matrix given by |
| 1792 |
> |
Equation, $\zeta$ can be taken as a scalar. In general, friction |
| 1793 |
> |
tensor $\Xi$ is a $6\times 6$ matrix given by |
| 1794 |
|
\[ |
| 1795 |
|
\Xi = \left( {\begin{array}{*{20}c} |
| 1796 |
|
{\Xi _{}^{tt} } & {\Xi _{}^{rt} } \\ |
