# Long Time Simulation Analysis of Geometry Dynamics Model under Iteration
## 1 Introduction  
Computational geometric mechanics can keep the geometric characteristics of a system and have higher stability than traditional methods. In addition, the geometry-based model are more concise than trigonometric functions, which has incomparable advantages for spatial rigid-flexible coupling multi-body systems. Geometric methods are also used in the fields of informatics and information geometry [1], which a provides unified mathematical basis for the integration of mechanics and other disciplines. Therefore, the exploration of geometric representations, numerical calculations and physical significance will promote the engineering applications of geometric mechanics. Geometric mechanics exploration has two directions: symplectic geometry and Lie group methods. The symplectic geometry method is used to solve differential algebraic equations of dynamic systems which combine dynamic analysis and differential equation solving. The discrete Lie group variational integrator is another geometry method which can conserve the energy, momentum and geometric structure. It is a recursive equation which can be used in the optimal control of a space dynamics system. In recent years, [PERSON] et al. [2] studied the indirect symplectic method for solving optimal control problems of differential algebraic equations, which improved calculation efficiency, and [PERSON] et al. [3] established the discrete Singer formula for optimal control of compressed bodies based on the Lagrange-D'Alembert principle and discrete variational principle, which reduced the number of drives required for optimal control. [PERSON], [PERSON] et al. [4] studied a symplectic optimal control method for robot trajectory control and obtained a symplectic discrete format for differential algebraic equations based on the discrete variational principle. [PERSON] et al. [5] studied the relationship between symmetry rupture and energy dissipation in dynamics based on multi-symplectic theory. [PERSON] et al. [6] explored structure conservation based on local structure conservation of the multi-symplectic method and establisheda dynamic model of the flexible damping plate-spring mass system. These show the symplectic geometry method has developed towards the direction of multi-symplectic theory, mainly based on the conservation of local symplectic structure. Differently from symplectic method, Lie group method seeks global conservation of geometry structure of system. Recent research achievements of Lie group are as follows: [PERSON] et al. [7] studied the polysymplectic Galerkin Lie group of geometrically accurate beams. [PERSON] et al. [8] explored flexible continuous manipulator modeling using a Lie group variational integrator. [PERSON] et al. [9] studied the dynamic model of a crane system based on Lie group theory. [PERSON] et al. [10] studied the application of spinor and Lie group theory in a recursive algorithm for a topologically tree-based multi-body system. [PERSON] et al. [11] studied the forward dynamics of fixed-wing aircraft on the Lie group and realized the attitude non-singular reconstruction on SO(3). [PERSON] et al. [12] studied the momentum and energy conservation Lie group integral method for rigid body rotational dynamics. [PERSON] et al. [13] studied the finite-time optimal control of dynamic systems on Lie groups. [PERSON] et al. [14; 15; 16] studied a Lie group variational integrator for full rigid body systems. [PERSON] et al. [17] studied a Lie group variational integrator for geometrically accurate beam dynamics. In recent years, the moving frame method has provided a new idea for solving dynamics equations based on Lie group theory. The main results are: [PERSON] et al. [18] studied Hamel form and variational integrators on a sphere. [PERSON] et al. [19] studied the Hamel form under classical field theory and used the variational integrator in sledge model [20]. He also studied the minimum time optimal control based on a Hamel product molecule [21] and the Hamel form of infinite dimensional mechanical system [22].