Geometry of escape and transition dynamics in the presence of dissipative and gyroscopic forces in two degree of freedom systems
Geometry of escape and transition dynamics in the presence of dissipative and gyroscopic forces in two degree of freedom systems
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DOI：
10.1016/j.cnsns.2019.105033
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发表时间：
2019-07
期刊：
影响因子：
--
通讯作者：
Jun-Hao Zhong;S. Ross
中科院分区：
文献类型：
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作者：
Jun-Hao Zhong;S. Ross
势阱的逃逸可能发生在不同的物理系统中，例如船舶倾覆、天体力学中的共振跃迁、拱门和贝壳的动态突跳，以及化学反应中的分子重新配置。一自由度系统的逃逸标准和路线已经在理论上得到了很好的研究，与实验具有合理的一致性。轨迹只能从一维势能面的山顶过渡。当系统具有更高的自由度时，情况变得更加复杂，因为系统状态有多种途径可以通过鞍型平衡，特别是指数1鞍型来逃脱。本文总结了一些众所周知的具有两个自由度的物理系统中跨越鞍座的逃逸几何形状，并建立了逃逸标准，在称为管动力学的概念框架下提供了方法和结果。根据考虑阻尼和/或陀螺效应时辛本征空间中的鞍投影和焦点投影是否耦合或解耦合，将这些问题分为两类。为简单起见，仅分析鞍点周围的线性化系统，但结果可推广到非线性系统。我们将过渡区域 T h 定义为给定初始能量 h 的初始条件区域，该区域从鞍座的一侧过渡到另一侧。我们发现，在保守系统中，过渡区域的边界∂ T h 是圆柱体，而在耗散系统中，∂ T h 是椭球体。
Escape from a potential well can occur in different physical systems, such as capsize of ships, resonance transitions in celestial mechanics, and dynamic snap-through of arches and shells, as well as molecular reconfigurations in chemical reactions. The criteria and routes of escape in one-degree of freedom systems have been well studied theoretically with reasonable agreement with experiment. The trajectory can only transit from the hilltop of the one-dimensional potential energy surface. The situation becomes more complicated when the system has higher degrees of freedom since the system state has multiple routes to escape through an equilibrium of saddle-type, specifically, an index-1 saddle. This paper summarizes the geometry of escape across a saddle in some widely known physical systems with two degrees of freedom and establishes the criteria of escape providing both a methodology and results under the conceptual framework known as tube dynamics. These problems are classified into two categories based on whether the saddle projection and focus projection in the symplectic eigenspace are coupled or uncoupled when damping and/or gyroscopic effects are considered. For simplicity, only the linearized system around the saddle points is analyzed, but the results generalize to the nonlinear system. We define a transition region, T h, as the region of initial conditions of a given initial energy h which transit from one side of a saddle to the other. We find that in conservative systems, the boundary of the transition region,∂ T h, is a cylinder, while in dissipative systems,∂ T h is an ellipsoid.