# Long Time Simulation Analysis of Geometry Dynamics Model under Iteration
## 2 Lie Group and Lie Algebra  
The Lie group and Lie algebra are abstract mathematical concepts and difficult to understand. Thus, in this section, the concepts are explained based on concrete examples. Consider a rotation problem in plane. The coordinate transformation is expressed as in Figure 1. Suppose the initial coordinate is xOy, after a counterclockwise rotation with \(\theta\). The new coordinate is x\({}^{\prime}\)Oy\({}^{\prime}\). The rotation matrix between them is as shown in Equation (1). The physical significance of the first and second column of \(R\) is the projection of Ox\({}^{\prime}\) and Oy\({}^{\prime}\) on Ox and Oy respectively. \(R\) can be written as the exponential type, as in Equation (2). \[R=e^{S(\theta)} \tag{2}\]  
In Equation (2), \(S(\theta)=\left[ \begin{array}{cc}0&-\theta\\ \theta&0\end{array} \right]=\theta\left[ \begin{array}{cc}0&-1\\ 1&0\end{array} \right]\). The derivation of \(R\) is Equation (3). \[\dot{R}=\left[ \begin{array}{cc}-\sin\theta&-\cos\theta\\ \cos\theta&-\sin\theta\end{array} \right]\dot{\theta}=R\mathbf{S}\Big{(}\dot{\theta} \Big{)} \tag{3}\]  
The derivation of \(R\) can be written as Equation (4) according to Equation (2). \[\dot{R}=e^{S(\theta)}S\Big{(}\dot{\theta}\Big{)} \tag{4}\]  
The results agree with the trigonometric functions type. Considering the system in space, the attitude matrices corresponding to the rotation along \(x\), \(y\), and \(z\) axes are in Equations (5)-(7), respectively. \[R_{x}(\theta_{3})=\left[ \begin{array}{ccc}1&0&0\\ 0&\cos\theta_{1}&-\sin\theta_{1}\\ 0&\sin\theta_{1}&\cos\theta_{1}\end{array} \right] \tag{5}\]  
\[R_{y}(\theta_{2})=\left[ \begin{array}{ccc}\cos\theta_{2}&0&\sin\theta_{2}\\ 0&1&0\\ -\sin\theta_{2}&0&\cos\theta_{2}\end{array} \right] \tag{6}\]  
\[R_{z}(\theta_{1})=\left[ \begin{array}{ccc}\cos\theta_{3}&-\sin\theta_{3}&0 \\ \sin\theta_{3}&\cos\theta_{3}&0\\ 0&0&1\end{array} \right] \tag{7}\]  
Derive Equations (5)-(7). The results are in Equation (8). \[\dot{R}_{x}=\omega_{x}R_{x}S(\mathbf{e}_{1});\dot{R}_{y}=\omega_{y}R_{y}S(\mathbf{e}_{ 2});\dot{R}_{z}=\omega_{z}R_{z}S(\mathbf{e}_{3}) \tag{8}\]  
Figure 1: The coordinate transformation in plane. In Equation (8), \(\mathbf{e}_{1}=[1;0;0],\mathbf{e}_{2}=[0;1;0],\mathbf{e}_{3}=[0;0;1]\). \[\mathbf{S}_{\mathbf{e}_{1}}=\left[ \begin{array}{ccc}0&0&0\\ 0&0&-1\\ 0&1&0\end{array} \right]\mathbf{S}_{\mathbf{e}_{2}}=\left[ \begin{array}{ccc}0&0&1\\ 0&0&0\\ -1&0&0\end{array} \right]\mathbf{S}_{\mathbf{e}_{3}}=\left[ \begin{array}{ccc}0&-1&0\\ 1&0&0\\ 0&0&0\end{array} \right]\]  
Equations (5)-(7) can also be written exponentially, as in Equation (9). \[\mathbf{R}_{x}=e^{\theta_{1}\mathbf{S}_{\mathbf{e}_{1}}};\mathbf{R}_{y}=e^{\theta_{2}\mathbf{S}_{ \mathbf{e}_{2}}};\mathbf{R}_{z}=e^{\theta_{3}\mathbf{e}_{3}} \tag{9}\]  
The derivation of Equation (9) is similar to Equation (4). \(\mathbf{R}\) can be seen as the Lie group expression of planer rotation through the concept of kinematics, and \(\theta\) is the Lie algebra which corresponds to the Lie group. If \(\mathbf{R}\) represents the rotation in space, then the Lie algebra which corresponds to \(\mathbf{R}_{x},\mathbf{R}_{y},\mathbf{R}_{z}\) is \(\theta\mathbf{e}_{1},\theta\mathbf{e}_{2},\theta\mathbf{e}_{3}\). According to the above analysis, \(\mathbf{R}\) is a function about \(\theta\), no matter the triangle function or exponent type. The above theory can be used as the basis for modeling the dynamics of a single body or a series of single bodies combined together which rotate in space. In practical engineering, some rigid body has three degrees of rotational freedom, such as a ship, plane, spacecraft or UAV. They have triangle function expression based on the classical multibody theory. For example, with the Cardan expression, the rotation of a rigid body in space is as the series of X-Y-Z, as in Figure 2, and the rotation matrix can be written as (10). \[\mathbf{R}=\mathbf{R}_{x}\mathbf{R}_{y}\mathbf{R}_{z} \tag{10}\]  
Differentiate \(\mathbf{R}\); then, substitute Equation (8) into Equation (10).