非完整积分器为何有效？
What makes nonholonomic integrators work?
AI总结：
本文针对非完整积分器长期优异性能的原因展开研究，提出可逆性而非离散拉格朗日-达朗贝尔原理是关键，通过分析测试问题结构、构造扰动问题验证，并指出需考虑不可逆系统，为评估积分器提供新视角。
摘要：
非完整系统是具有非源于位置约束的速度约束的力学系统，典型例子为无滑滚动。非完整积分器是专为非完整系统设计的数值方法。数值观察发现，许多非完整积分器应用于各类测试问题时展现出优异的长期行为，其出色性能常被归因于拉格朗日-达朗贝尔原理的某种潜在离散形式。然而，本文提供证据表明可逆性是观察到的行为背后的原因。实际上，我们证明许多标准非完整测试问题具有在可逆可积系统上叶状化的结构。由于大多数非完整积分器保留叶状结构和可逆结构，首次积分的近似守恒是可逆KAM理论的结果。因此，要全面评估非完整积分器，还需考虑不可逆非完整系统。为此，我们构造了可积但不再可逆（相对于标准可逆映射）的扰动测试问题。将文献中的各类非完整积分器应用于这些问题，我们观察到没有方法在所有问题上都表现良好。这进一步表明可逆性是非完整积分器首次积分近似守恒的主要机制。文中还给出了相关开放问题列表。
英文摘要：
A nonholonomic system is a mechanical system with velocity constraints not originating from position constraints; rolling without slipping is the typical example. A nonholonomic integrator is a numerical method specifically designed for nonholonomic systems. It has been observed numerically that many nonholonomic integrators exhibit excellent long-time behaviour when applied to various test problems. The excellent performance is often attributed to some underlying discrete version of the Lagrange--d'Alembert principle. Instead, in this paper, we give evidence that reversibility is behind the observed behaviour. Indeed, we show that many standard nonholonomic test problems have the structure of being foliated over reversible integrable systems. As most nonholonomic integrators preserve the foliation and the reversible structure, near conservation of the first integrals is a consequence of reversible KAM theory. Therefore, to fully evaluate nonholonomic integrators one has to consider also non-reversible nonholonomic systems. To this end we construct perturbed test problems that are integrable but no longer reversible (with respect to the standard reversibility map). Applying various nonholonomic integrators from the literature to these problems we observe that no method performs well on all problems. This further indicates that reversibility is the main mechanism behind near conservation of first integrals for nonholonomic integrators. A list of relevant open problems is given.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
