带质量弦的摆动阿特伍德机的动力学与不可积性：混沌、周期轨道和共振结构

中文摘要 AI 辅助

在我们之前关于非线性变长度摆系统研究的基础上，本文研究带质量弦的摆动阿特伍德机。与经典模型不同，弦的惯性引入了与构型相关的转动惯量，从而产生修正的哈密顿结构和丰富得多的动力学行为。为了揭示相空间的整体结构，我们将庞加莱截面、分岔图、李雅普诺夫指数图与我们近期开发的数值框架“李雅普诺夫精细图”相结合，该方法可对周期、准周期、混沌及终止运动进行统一可视化，展现出复杂的共振网络和高阶周期结构。我们通过在参数空间、初始条件空间以及固定能量曲面上构建李雅普诺夫图，研究了弦质量、系统参数和能量的影响。在莫拉莱斯-拉米斯理论框架内对刘维尔可积性进行了研究，通过分析沿显式非定常径向解的正则变分方程并应用科瓦西算法，我们证明微分伽罗瓦群一般为SL(2,C)，这对任意非零弦质量下的亚纯刘维尔可积性构成了严格阻碍，因此，经典摆动阿特伍德机的特殊可积情形会因引入弦惯性而被破坏。

英文摘要

Building upon our previous studies on nonlinear variable-length pendulum systems, we investigate the Swinging Atwood Machine with a massive string. In contrast to the classical model, string inertia introduces a configuration-dependent moment of inertia, leading to a modified Hamiltonian structure and substantially richer dynamics. To uncover the global organization of the phase space, we combine Poincaré sections, bifurcation diagrams, and Lyapunov exponent maps with our recently developed numerical framework, ,,Lyapunov Refined Maps". This approach provides a unified visualization of periodic, quasi-periodic, chaotic, and terminating motions, revealing intricate resonance networks and high-order periodic structures. We investigate the influence of the string mass, system parameters, and energy by constructing Lyapunov maps in parameter and initial-condition spaces and on fixed-energy surfaces. Liouville integrability is studied within the Morales--Ramis theory. By analyzing the normal variational equations along explicit non-stationary radial solutions and applying the Kovacic algorithm, we prove that the differential Galois group is generically SL(2,C), providing a rigorous obstruction to meromorphic Liouville integrability for every nonzero string mass. Thus, the exceptional integrable case of the classical Swinging Atwood Machine is destroyed by the inclusion of string inertia.

补充信息

备注

40 pages, 14 figures