| 831 |
|
error of splitting method in terms of commutator of the |
| 832 |
|
operators(\ref{introEquation:exponentialOperator}) associated with |
| 833 |
|
the sub-flow. For operators $hX$ and $hY$ which are associate to |
| 834 |
< |
$\varphi_1(t)$ and $\varphi_2(t$ respectively , we have |
| 834 |
> |
$\varphi_1(t)$ and $\varphi_2(t)$ respectively , we have |
| 835 |
|
\begin{equation} |
| 836 |
|
\exp (hX + hY) = \exp (hZ) |
| 837 |
|
\end{equation} |
| 846 |
|
\] |
| 847 |
|
Applying Baker-Campbell-Hausdorff formula to Sprang splitting, we |
| 848 |
|
can obtain |
| 849 |
< |
\begin{eqnarray*} |
| 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{eqnarray*} |
| 854 |
> |
\end{equation} |
| 855 |
|
Since \[ [X,Y] + [Y,X] = 0\] and \[ [X,X] = 0\], the dominant local |
| 856 |
|
error of Spring splitting is proportional to $h^3$. The same |
| 857 |
|
procedure can be applied to general splitting, of the form |
| 1238 |
|
where $I_{ii}$ is the diagonal element of the inertia tensor. This |
| 1239 |
|
constrained Hamiltonian equation subjects to a holonomic constraint, |
| 1240 |
|
\begin{equation} |
| 1241 |
< |
Q^T Q = 1$, \label{introEquation:orthogonalConstraint} |
| 1241 |
> |
Q^T Q = 1, \label{introEquation:orthogonalConstraint} |
| 1242 |
|
\end{equation} |
| 1243 |
|
which is used to ensure rotation matrix's orthogonality. |
| 1244 |
|
Differentiating \ref{introEquation:orthogonalConstraint} and using |
