基尔霍夫动力学的若干探索
Some explorations of Kirchhoff dynamics
- University of Nevada, Reno（内华达大学雷诺分校）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
以理想流体中刚体的基尔霍夫动力学问题为对象，研究椭球体的可积与非可积运动，揭示其解子流形的连通性、分岔及混沌运动性质。
AI中文摘要：
理想流体中刚体的经典问题是一个具有三个守恒量的六维动力学系统。尽管该问题历史悠久，且作为许多流固问题的简化模型具有重要意义，但仍存在基础问题，包括三维解子流形的形状与连通性，以及其上混沌运动的性质。我们以特定椭球体为研究对象展开探索，从可积运动入手——稳态直线平移、旋转、周期性平面翻滚与扑动。这些状态的稳定性取决于线动量、角动量或能量的相对大小。包含Floquet分析的低维扰动系统线性稳定性分析，与全系统的直接积分结果一致。我们记录了导致多种规则及混沌运动的不稳定性，包括翻转与旋转，其轨迹似乎沿可积状态间的连通路径延伸。对这些连通性的间接观察，为理解该简单系统丰富动力学背后的结构提供了洞见。对于线动量与角动量垂直的子流形，我们进一步研究解如何组合以填充可积状态不同稳定区域内的动量空间，进而揭示连通性的额外分岔，以及看似新的可积解。
英文摘要：
The classical problem of a rigid body in an ideal fluid is a six-dimensional dynamical system with three conserved quantities. Despite its long history and its importance as a reduced model of many fluid-structure problems, basic questions remain, including the shapes and connectivities of the three-dimensional submanifolds of solutions, and the nature of chaotic motions on these. We explore the problem with a particular ellipsoidal body, beginning with integrable motions--- steady linear translation and rotation and periodic planar tumbling and fluttering. The stability of these states depends on the relative magnitudes of linear and angular momentum or energy. Linear stability, including Floquet, analysis of lower-dimensional perturbed systems is consistent with direct integration of the full system. We document instabilities leading to a variety of regular and chaotic motions, including flipping and twirling, whose trajectories appear to follow paths near connections between the integrable states. Such indirect observation of these connections provides insight into the structure underlying the rich dynamics of this simple system. For the submanifold where linear and angular momentum are perpendicular, we further examine how solutions fit together to fill momentum space in different zones of stability of the integrable states, and thereby uncover additional bifurcations in connectivity and what appear to be new integrable solutions.
