Spectral Variational Integrators And Geometric Control Of Rigid Bodies

This paper is mainly focusing on the construction of high-order computational geometric mechanical methods for the simulation of realistic dynamic and control systems.Spectral methods are popular choice for constructing numerical approximations for smooth problems,as they can achieve geometric rates of convergence and have a relatively small memory footprint.In this paper,we introduce a general framework to convert a spectral-collocation method into a structure-preserving numerical method for Hamiltonian systems.The structure-preserving numerical method mentioned above include Galerkin spectral variational integrators,spectral collocation variational integrators,Lie group spectral methods.The constructions of these methods are mainly based on the discrete geometric mechanics and the proposed methods are derived from effective combination strategies(Galerkin and shooting)of spectral methods and variational integrators.To construct high-order variational integrators,we discretize the original variational principles of the Lagrangian and Hamiltonian systems rather than the resulting Euler–Lagrange and Hamilton's equations.For the construction of Lie group spectral methods,we first transfer the dynamic or control systems on Lie group to its corresponding Lie algebra.As Lie algebra space is linear,we can use the spectral collocation methods to deal with the resulting equations on Lie algebra,then map the solution back to the original Lie group by canonical coordinate map.We also compare the proposed high-order computational geometric mechanical methods with some other conventional structure-preserving or high-order methods in terms of their ability to reproduce accurate trajectories in configuration and phase space,their ability to conserve momentum and energy,as well as the relative computational efficiency of these methods when applied to some classical Hamiltonian systems.It can be observed from the numerical results that all numerical methods introduced in this paper preserve the geometric structures and physic properties very well and inherit both bonus from the original geometric numerical methods and spectral methods at the same time,including the convergence rate and computational efficiency.It would be more realistic and reliable to use the proposed methods for the long-time simulations of dynamic or control systems.What's more,all the numerical approximation methods introduced in this paper can be naturally extended in the simulation of dissipative systems and optimal control problems.