抛物轨道和倒钩环面附近光滑哈密顿圆作用的存在性

IF 0.8 4区 数学 Q3 MATHEMATICS, APPLIED Regular and Chaotic Dynamics Pub Date : 2021-12-06 DOI:10.1134/S1560354721060101
Elena A. Kudryavtseva, Nikolay N. Martynchuk
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引用次数: 12

摘要

我们证明了一个二自由度可积系统的每一个抛物轨道都存在一个\(C^{\infty}\) -光滑哈密顿圆作用,该作用在微小的可积\(C^{\infty}\)摄动下是持久的。我们从这个结果推导出抛物线轨道的结构稳定性,并证明它们与标准模型都是光滑等效的(在非辛意义上)。作为一个推论,我们得到了类似的结果,对尖角环面。我们的证明是基于证明一个抛物点的邻域的每一个保持运动第一积分的辛形态是一个哈密顿量,其生成函数在公共水平集的连通分量上是光滑的和常数的。
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Existence of a Smooth Hamiltonian Circle Action near Parabolic Orbits and Cuspidal Tori

We show that every parabolic orbit of a two-degree-of-freedom integrable system admits a \(C^{\infty}\)-smooth Hamiltonian circle action, which is persistent under small integrable \(C^{\infty}\) perturbations. We deduce from this result the structural stability of parabolic orbits and show that they are all smoothly equivalent (in the non-symplectic sense) to a standard model. As a corollary, we obtain similar results for cuspidal tori. Our proof is based on showing that every symplectomorphism of a neighbourhood of a parabolic point preserving the first integrals of motion is a Hamiltonian whose generating function is smooth and constant on the connected components of the common level sets.

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来源期刊
CiteScore
2.50
自引率
7.10%
发文量
35
审稿时长
>12 weeks
期刊介绍: Regular and Chaotic Dynamics (RCD) is an international journal publishing original research papers in dynamical systems theory and its applications. Rooted in the Moscow school of mathematics and mechanics, the journal successfully combines classical problems, modern mathematical techniques and breakthroughs in the field. Regular and Chaotic Dynamics welcomes papers that establish original results, characterized by rigorous mathematical settings and proofs, and that also address practical problems. In addition to research papers, the journal publishes review articles, historical and polemical essays, and translations of works by influential scientists of past centuries, previously unavailable in English. Along with regular issues, RCD also publishes special issues devoted to particular topics and events in the world of dynamical systems.
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