# Long Time Simulation Analysis of Geometry Dynamics Model under Iteration
## 4 The Solution of the Dynamics of a Planer Pendulum
\right] \tag{50}\]  
Using \(\omega_{k},\mathbf{\Gamma}_{k}\) and \(\Delta\omega_{k},\Delta\mathbf{\Gamma}_{k}\) as parameters, the discretion of \(\omega\) is Equation (44), and the discretion of \(\mathbf{\Gamma}\) is Equation (52). \[\dot{\mathbf{\Gamma}}_{k+\frac{1}{2}}=\frac{\Delta\mathbf{\Gamma}_{k}}{h},\mathbf{\Gamma} _{k+\frac{1}{2}}=\mathbf{\Gamma}_{k}+\frac{\Delta\mathbf{\Gamma}_{k}}{2} \tag{51}\]  
Substitute Equation (52) into Equation (48). The dynamics equation of system is Equation (53). \[\left\{ \begin{array}{l}2J\Delta\omega_{k}+hmgle_{1}^{T}(2\mathbf{\Gamma}_{k}+ \Delta\mathbf{\Gamma}_{k})=0\\ 4\Delta\mathbf{\Gamma}_{k}+h(2\omega_{k}+\Delta\omega_{k})\mathbf{S}_{1}(2\mathbf{\Gamma} _{k}+\Delta\mathbf{\Gamma}_{k})=0\end{array} \right. \tag{52}\]  
The Jacobi matrix of Equation (53) is a \(3\times 3\) matrix:  
\[\left[ \begin{array}{cc}2J&hmgle_{1}^{T}\\ hS_{1}(2\mathbf{\Gamma}_{k}+\Delta\mathbf{\Gamma}_{k})&4\mathbf{I}+h(2\omega_{k}+\Delta \omega_{k})\mathbf{S}_{1}\end{array} \right] \tag{53}\]  
If \(\mathbf{R}\) is discrete by Equations (35) and (36), and \(\omega\) is discrete as in Equation (45), then substitute them into Equation (39) to get the dynamics equation of Equation (55). \[\left\{\begin{array}{l}2J\Delta\omega_{k}+hmgle_{2}^{T}\mathbf{R}_{k}\mathbf{\Gamma} _{k}\mathbf{e}_{1}+hmgle_{2}^{T}\mathbf{R}_{k}\mathbf{e}_{1}=0\\ 2\mathbf{F}_{k}\mathbf{e}_{1}-h\mathbf{F}_{k}\mathbf{S}\  
In Equation (57), \(e_{12}=[1;1]\). The above method changes \(R\) into vector \(\Gamma\). Although the numerical result is correct, the complete character of \(R\) is broken, which may lead to constraint loss of the system. In order to reflect the complete structure of \(R\), let \(R^{T}e_{1}=\Gamma_{a},R^{T}e_{2}=\Gamma_{b}\); then, Equation (39) can be written as Equation (58). \[\left\{ \begin{array}{l}\dot{J}\dot{\omega}+mgle_{1}^{T}\Gamma_{b}=0\\ \dot{\Gamma}_{a}=-\mathcal{S}(\mathbf{\omega})\Gamma_{a}\\ \dot{\Gamma}_{b}=-\mathcal{S}(\mathbf{\omega})\Gamma_{b}\end{array} \right. \tag{57}\]  
The discrete regulars of \(\Gamma\) and \(\omega\) are Equation (49) and Equation (42), respectively. With these regulars, the discrete dynamics equation is Equation (59). \[\left\{ \begin{array}{l}2/(\omega_{k+1}-\omega_{k})+hmgle_{1}^{T}(\Gamma_{b,k+1}+\Gamma_{b,k})=0\\ 4(\Gamma_{a,k+1}-\Gamma_{a,k})+h(\omega_{k+1}+\omega_{k})\mathcal{S}_{1}( \Gamma_{a,k+1}+\Gamma_{a,k})=0\\ 4(\Gamma_{b,k+1}-\Gamma_{b,k})+h(\omega_{k+1}+\omega_{k})\mathcal{S}_{1}( \Gamma_{b,k+1}+\Gamma_{b,k})=0\end{array} \right. \tag{58}\]  
It is necessary to notice that the significance of parameters in Equation (59) is different from those in Equation (56). If the equation is discrete with \(\Delta\omega_{k},\Delta\Gamma_{a,k},\Delta\Gamma_{b,k}\), the result is Equation (60). \[\left\{ \begin{array}{l}2/\Delta\omega_{k}+2 hmgle_{1}^{T}\Gamma_{b,k}+hmgle_{ 1}^{T}\Delta\Gamma_{b,k}=0\\ 4\Delta\Gamma_{a,k}+h(2\omega_{k}+\Delta\omega_{k})\mathcal{S}_{1}(2\Gamma_{a,k}+\Delta\Gamma_{a,k})=0\\ 4\Delta\Gamma_{b,k}+h(2\omega_{k}+\Delta\omega_{k})\mathcal{S}_{1}(2\Gamma_{b,k}+\Delta\Gamma_{b,k})=0\end{array} \right. \tag{59}\]  
If the parameters are \(\omega_{k+\frac{1}{2}}\Gamma_{a,k+\frac{1}{2}}\Gamma_{b,k+\frac{1}{2}}\), Equations (59) and (60) have the same Jacobi matrix as Equation (61). \[\left[ \begin{array}{ccc}2J&\mathbf{0}_{1\times 2}&hmgle_{1}^{T}\\ 2h\mathcal{S}_{1}\Gamma_{a,k+\frac{1}{2}}&4\Gamma+2h\omega_{k+\frac{1}{2}} \mathcal{S}_{1}&\mathbf{0}_{2\times 2}\\ 2h\mathcal{S}_{1}\Gamma_{b,k+\frac{1}{2}}&\mathbf{0}_{2\times 2}&4\Gamma+2h\omega_{k+ \frac{1}{2}}\mathcal{S}_{1}\end{array} \right] \tag{60}\]  
The above dynamics equations have Lie group expression. Next, the dynamics equation with Lie algebra is discussed. Firstly, the dynamics equation is changed to be the exponential type:  
\[\left\{ \begin{array}{l}J\dot{\omega}+mgle_{2}^{T}\mathcal{E}^{\mathcal{S}( \theta)}e_{1}=0\\ \dot{\theta}=\omega\end{array} \right. \tag{61}\]  
The discrete dynamics equation is shown in Equations (63) and (64).