反馈积分器
Feedback Integrators
AI总结：
本文提出一种新方法，用于在流形上数值积分动力系统，使轨迹稳定留在流形上并保持系统的首积分。通过将动力学扩展到欧几里得空间并修改动力学，使交集成为局部吸引子。该方法可与任何普通数值积分器结合使用，并通过三个经典问题验证其有效性。
摘要：
本文提出了一种新方法，用于在流形上数值积分动力系统，使得轨迹稳定地留在流形上并保持系统的首积分。该方法的思路是：给定流形上的初始点，将动力学从流形扩展到其欧几里得空间，并修改动力学，使得流形与首积分的水平集的交集成为修改后动力学的唯一局部吸引子。虽然修改后的动力学理论上产生与原始动力学相同的轨迹，但其数值轨迹稳定地留在流形上并保持首积分。该方法的主要优点是，修改后的动力学可以与任何普通的数值积分器（如欧拉或龙格-库塔方法）结合使用。我们通过将该方法应用于三个著名问题：自由刚体、开普勒问题和具有旋转对称性的扰动开普勒问题来演示该方法。我们还进行了模拟研究，以展示该方法的优越性，并与标准投影方法、分裂方法和斯托默-韦尔特方案进行了比较。
英文摘要：
A new method is proposed to numerically integrate a dynamical system on a manifold such that the trajectory stably remains on the manifold and preserves first integrals of the system. The idea is that given an initial point in the manifold we extend the dynamics from the manifold to its ambient Euclidean space and then modify the dynamics outside the intersection of the manifold and the level sets of the first integrals containing the initial point such that the intersection becomes a unique local attractor of the resultant dynamics. While the modified dynamics theoretically produces the same trajectory as the original dynamics, it yields a numerical trajectory that stably remains on the manifold and preserves the first integrals. The big merit of our method is that the modified dynamics can be integrated with any ordinary numerical integrator such as Euler or Runge-Kutta. We illustrate this method by applying it to three famous problems: the free rigid body, the Kepler problem and a perturbed Kepler problem with rotational symmetry. We also carry out simulation studies to demonstrate the excellence of our method and make comparisons with the standard projection method, a splitting method and Störmer-Verlet schemes.
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。
