标题： 关于辛数值方法精度的坐标系依赖性 显示英文标题

标题： On the coordinate system-dependence of the accuracy of symplectic numerical methods

摘要： 辛数值方法已成为在各种领域，包括天体力学、分子动力学和机器人学中准确模拟哈密顿系统的一种广泛使用的选项。 尽管它们的特性在数学上已被充分理解，但通常较少关注坐标选择如何影响数值结果精度的实际方面，尽管其后果可能在计算上具有重要意义。 本文旨在通过系统地概述坐标变换如何影响使用辛方法进行的模拟结果来填补这一空白。 我们给出了辛方法的修正哈密顿量在坐标变换下非不变性的推导，以及针对辛欧拉方法对应循环坐标的第一个积分非保持性的充分条件。 我们还考虑了寻找可以补偿阶数的坐标变换的可能性，以提高数值方法的精度。 文中展示了各种数值示例。 显示英文摘要

摘要： Symplectic numerical methods have become a widely-used choice for the accurate simulation of Hamiltonian systems in various fields, including celestial mechanics, molecular dynamics and robotics. Even though their characteristics are well-understood mathematically, relatively little attention has been paid in general to the practical aspect of how the choice of coordinates affects the accuracy of the numerical results, even though the consequences can be computationally significant. The present article aims to fill this gap by giving a systematic overview of how coordinate transformations can influence the results of simulations performed using symplectic methods. We give a derivation for the non-invariance of the modified Hamiltonian of symplectic methods under coordinate transformations, as well as a sufficient condition for the non-preservation of a first integral corresponding to a cyclic coordinate for the symplectic Euler method. We also consider the possibility of finding order-compensating coordinate transformations that improve the order of accuracy of a numerical method. Various numerical examples are presented throughout.