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首页> 外文期刊>Discrete and continuous dynamical systems >THE GEOMETRIC DISCRETISATION OF THE SUSLOV PROBLEM: A CASE STUDY OF CONSISTENCY FOR NONHOLONOMIC INTEGRATORS
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THE GEOMETRIC DISCRETISATION OF THE SUSLOV PROBLEM: A CASE STUDY OF CONSISTENCY FOR NONHOLONOMIC INTEGRATORS

机译：Suslov问题的几何离散：以非完整积分器的一致性为例

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摘要

Geometric integrators for nonholonomic systems were introduced by Cortes and Martinez in [4] by proposing a discrete Lagrange-D'Alembert principle. Their approach is based on the definition of a discrete Lagrangian L_d and a discrete constraint space D_d. There is no recipe to construct these objects and the performance of the integrator is sensitive to their choice. Cortes and Martinez [4] claim that choosing L_d and D_d in a consistent manner with respect to a finite difference map is necessary to guarantee an approximation of the continuous flow within a desired order of accuracy. Although this statement is given without proof, similar versions of it have appeared recently in the literature. We evaluate the importance of the consistency condition by comparing the performance of two different geometric integrators for the nonholonomic Suslov problem, only one of which corresponds to a consistent choice of L_d and D_d. We prove that both integrators produce approximations of the same order, and, moreover, that the non-consistent discretisation outperforms the other in numerical experiments and in terms of energy preservation. Our results indicate that the consistency of a discretisation might not be the most relevant feature to consider in the construction of nonholonomic geometric integrators.
机译：通过提出离散Lagrange-D'Alembert原理，Cortes和Martinez在[4]中介绍了非完整系统的几何积分器。他们的方法基于离散拉格朗日L_d和离散约束空间D_d的定义。没有构造这些对象的方法，并且积分器的性能对其选择很敏感。科尔特斯和马丁内斯[4]声称，对于有限差分图，以一致的方式选择L_d和D_d对于保证连续流在所需精度范围内的近似是必要的。尽管此陈述没有证据，但最近在文献中出现了类似的说法。我们通过比较两个不同几何积分器对非完整Suslov问题的性能来评估一致性条件的重要性，其中只有一个对应于L_d和D_d的一致选择。我们证明了这两个积分器都可以产生相同阶数的近似值，此外，在数值实验和能量保存方面，不一致的离散化效果要好于另一个。我们的结果表明，离散化的一致性可能不是构造非完整几何积分器时要考虑的最相关的特征。

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