Abstract
Abstract Lie group methods are an excellent choice for simulating differential equations evolving on Lie groups or homogeneous manifolds, as they can preserve the underlying geometric structures of the corresponding manifolds. Spectral methods are a popular choice for constructing numerical approximations for smooth problems, as they can converge geometrically. In this paper, we focus on developing numerical methods for the simulation of geometric dynamics and control of rigid body systems. Practical algorithms, which combine the advantages of Lie group methods and spectral methods, are given and they are tested both in a geometric dynamic system and a geometric control system.
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