Transverse foliations for two-degree-of-freedom mechanical systems

Naiara V. de Paulo, Seongchan Kim, Pedro A. S. Salomão, Alexsandro Schneider
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Abstract

We investigate the dynamics of a two-degree-of-freedom mechanical system for energies slightly above a critical value. The critical set of the potential function is assumed to contain a finite number of saddle points. As the energy increases across the critical value, a disk-like component of the Hill region gets connected to other components precisely at the saddles. Under certain convexity assumptions on the critical set, we show the existence of a weakly convex foliation in the region of the energy surface where the interesting dynamics takes place. The binding of the foliation is formed by the index-$2$ Lyapunov orbits in the neck region about the rest points and a particular index-$3$ orbit. Among other dynamical implications, the transverse foliation forces the existence of periodic orbits, homoclinics, and heteroclinics to the Lyapunov orbits. We apply the results to the H\'enon-Heiles potential for energies slightly above $1/6$. We also discuss the existence of transverse foliations for decoupled mechanical systems, including the frozen Hill's lunar problem with centrifugal force, the Stark problem, the Euler problem of two centers, and the potential of a chemical reaction.
二自由度机械系统的横向拓扑结构
我们研究了略高于临界值的二自由度机械系统的动力学。假设势函数的临界集包含有限数量的鞍点。当能量越过临界值时,希尔区域的圆盘状分量恰好在鞍点处与其他分量相连。在临界集的某些凸性假设下,我们证明了在发生有趣动力学的能量面区域存在弱凸对折。褶皱的结合是由颈部区域关于静止点的 index-$2$Lyapunov 轨道和一个特定的 index-$3$ 轨道形成的。除其他动力学意义外,横向褶皱迫使周期轨道、同轴和异轴与拉普诺夫轨道同时存在。我们将这些结果应用于略高于1/6$的H\'enon-Heiles势能。我们还讨论了解耦机械系统横向叶的存在,包括具有离心力的冰冻希尔月球问题、斯塔克问题、双中心欧拉问题和化学反应势。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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