| 861 |
|
\varphi _{b_m h}^2 \circ \varphi _{a_m h}^1 \circ \varphi _{b_{m - |
| 862 |
|
1} h}^2 \circ \ldots \circ \varphi _{a_1 h}^1 . |
| 863 |
|
\end{equation} |
| 864 |
< |
Careful choice of coefficient $a_1 \ldot b_m$ will lead to higher |
| 864 |
> |
Careful choice of coefficient $a_1 \ldots b_m$ will lead to higher |
| 865 |
|
order method. Yoshida proposed an elegant way to compose higher |
| 866 |
|
order methods based on symmetric splitting\cite{Yoshida1990}. Given |
| 867 |
|
a symmetric second order base method $ \varphi _h^{(2)} $, a |
| 1256 |
|
Using Equation (\ref{introEquation:motionHamiltonianCoordinate}, |
| 1257 |
|
\ref{introEquation:motionHamiltonianMomentum}), one can write down |
| 1258 |
|
the equations of motion, |
| 1259 |
< |
\[ |
| 1260 |
< |
\begin{array}{c} |
| 1261 |
< |
\frac{{dq}}{{dt}} = \frac{p}{m} \label{introEquation:RBMotionPosition}\\ |
| 1262 |
< |
\frac{{dp}}{{dt}} = - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\ |
| 1263 |
< |
\frac{{dQ}}{{dt}} = PJ^{ - 1} \label{introEquation:RBMotionRotation}\\ |
| 1264 |
< |
\frac{{dP}}{{dt}} = - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}\\ |
| 1265 |
< |
\end{array} |
| 1266 |
< |
\] |
| 1259 |
> |
|
| 1260 |
> |
\begin{eqnarray} |
| 1261 |
> |
\frac{{dq}}{{dt}} & = & \frac{p}{m} \label{introEquation:RBMotionPosition}\\ |
| 1262 |
> |
\frac{{dp}}{{dt}} & = & - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\ |
| 1263 |
> |
\frac{{dQ}}{{dt}} & = & PJ^{ - 1} \label{introEquation:RBMotionRotation}\\ |
| 1264 |
> |
\frac{{dP}}{{dt}} & = & - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP} |
| 1265 |
> |
\end{eqnarray} |
| 1266 |
|
|
| 1267 |
|
In general, there are two ways to satisfy the holonomic constraints. |
| 1268 |
|
We can use constraint force provided by lagrange multiplier on the |
