# Long Time Simulation Analysis of Geometry Dynamics Model under Iteration
## 2 Lie Group and Lie Algebra
The derivation of \(\mathbf{R}\) can be written as Equation (11). \[\dot{\mathbf{R}}=\dot{\mathbf{R}}_{x}\mathbf{R}_{y}\mathbf{R}_{z}+\mathbf{R}_{x}\dot{\mathbf{R}}_{y} \mathbf{R}_{z}+\mathbf{R}_{x}\mathbf{R}_{y}\dot{\mathbf{R}}_{z}=\omega_{x}\mathbf{R}_{x}\mathbf{S}_{x }\mathbf{R}_{y}\mathbf{R}_{z}+\omega_{y}\mathbf{R}_{x}\mathbf{R}_{y}\mathbf{R}_{z}+\omega_{y}\mathbf{R }_{x}\mathbf{R}_{y}\mathbf{R}_{z}\mathbf{S}_{z} \tag{11}\]  
Equation (11) can be rewritten as Equation (12) according to \(\mathbf{R}^{T}\mathbf{S}(\mathbf{a})\mathbf{R}=\mathbf{S}\big{(}\mathbf{R}^{T}\mathbf{a}\big{)}\). \[ \begin{array}{l}\dot{\mathbf{R}}=\mathbf{R}_{x}\mathbf{R}_{y}\mathbf{R}_{z}\big{(}\mathbf{R}_{y} \mathbf{R}_{z}\big{)}^{T}\mathbf{S}\Big{(}\delta_{x}\mathbf{e}_{1}\Big{)}\mathbf{R}_{y}\mathbf{R}_ {z}+\mathbf{R}_{x}\mathbf{R}_{y}\mathbf{R}_{z}\mathbf{R}_{z}^{T}\mathbf{S}\Big{(}\delta_{y}\mathbf{e} _{2}\Big{)}\mathbf{R}_{z}+\mathbf{R}_{x}\mathbf{R}_{y}\mathbf{R}_{z}\mathbf{S}\Big{(}\delta_{z}\mathbf{ e}_{3}\Big{)}\\ =\mathbf{R}\mathbf{S}\Big{(}\mathbf{R}_{z}^{T}\mathbf{R}_{y}^{T}\delta_{x}\mathbf{e}_{1}+\mathbf{R}_{z}^ {T}\delta_{y}\mathbf{e}_{2}+\delta_{z}\mathbf{e}_{3}\Big{)}\end{array}  \tag{12}\]  
From the above equation, the rotation of a rigid body in space can be written as the nonlinear combination of the rotations on different axes Cardan order. Using the physics definition of \(\mathbf{R}\) in plane, the physics definition of each row of \(\mathbf{R}\) in space can be explained as the projection of X\({}^{\prime}\)Y\({}^{\prime}\)Z\({}^{\prime}\) on X, Y, Z. Using \(\mathbf{\omega}\) to represent the absolute angular velocity of a rotation body in space, the differential of \(\mathbf{R}\) can be written as in Equation (13). \[\dot{\mathbf{R}}=\mathbf{R}\mathbf{S}(\mathbf{\omega}) \tag{13}\]  
Figure 2: The Cardan rotation. In Equation (13), \(\mathbf{\omega}=\left[ \begin{array}{cc}\omega_{x}&\omega_{y}&\omega_{z}\end{array}  \right]^{T}\). \(S(\mathbf{\omega})\) is the skew matrix of \(\mathbf{\omega}\), as in Equation (14). \[S(\mathbf{\omega})=\left[ \begin{array}{ccc}0&-\omega_{z}&\omega_{y}\\ \omega_{z}&0&-\omega_{x}\\ -\omega_{y}&\omega_{x}&0\end{array} \right] \tag{14}\]  
According to Equations (12) and (13), the angular velocity can be written as the combination of the rotation matrix and angular velocities across each axis, as in Equation (15). \[\mathbf{\omega}=\mathbf{R}_{z}^{T}\mathbf{R}_{y}^{T}\theta_{x}\mathbf{e}_{1}+\mathbf{R}_{z}^{T} \theta_{y}\mathbf{e}_{2}+\dot{\theta}_{z}\mathbf{e}_{3} \tag{15}\]  
The above derivations are the expressions of the rotation matrix and their differentials of rigid body rotations in plane and in space. As part of dynamics solving, the above equations usually need to transform to the triangle type, which may make the expressions more complex due to the triangle transformation, especially for the rotation in space or multibody systems. \[\|\mathbf{R}\|=1;\mathbf{R}\mathbf{R}^{T}=\mathbf{I};\mathbf{R}^{T}=\mathbf{R}^{-1} \tag{16}\]  
In order to avoid these problems, the overall natures of Lie group and Lie algebra need to be analyzed. From the overall point of view, \(\mathbf{R}\) satisfies the characteristics in Equation (16), no matter the plane or space. Expect the above characteristics, each row of \(\mathbf{R}\) also satisfies similar characteristics, as the moduli are one in each row. \[\|\mathbf{R}_{1}\|=\|\mathbf{R}_{2}\|=\|\mathbf{R}_{3}\|=1 \tag{17}\]  
If each row of \(\mathbf{R}\) is one element, Equation (13) can be divided into three parts, as in Equation (18). \[\mathbf{e}_{3}^{T}\dot{\mathbf{R}}=\mathbf{e}_{3}^{T}\mathbf{R}S(\mathbf{\omega});\mathbf{e}_{2}^{T} \dot{\mathbf{R}}=\mathbf{e}_{2}^{T}\mathbf{R}S(\mathbf{\omega});\mathbf{e}_{1}^{T}\dot{\mathbf{R}}=\bm {e}_{1}^{T}\mathbf{R}S(\mathbf{\omega}) \tag{18}\]  
Suppose \(\mathbf{\Gamma}_{i}=\mathbf{R}^{T}\mathbf{e}_{3}\).