Mathematical Sciences: Existence and Multiplicity Questions for Periodic Solutions of Hamiltonian Systems and Related Topics

数学科学：哈密顿系统周期解的存在性和多重性问题及相关主题

• 批准号: 8803496
• 负责人: Helmut Hofer
• 金额: $ 4.17万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1988
• 资助国家: 美国
• 起止时间: 1988-06-01 至 1990-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8803496&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Existence; Multiplicity; Questions

The theme of this work centers on the structure of solutions of systems of ordinary differential equations. The equations have roots in classical mechanics described by Hamiltonian systems. To understand the orbit structures of these systems, one tries to find invariant subsystems, the simplest of those being periodic solutions. This leads naturally to boundary value problems. Three different questions will be treated in this project. The first is concerned with conditions one can expect to find periodic solutions on a prescribed energy surface. Complete results are available for convex and star-shaped surfaces. Work will be done in expanding known theory to compact hypersurfaces of contact type. The expected criterion for periodic solutions is the vanishing of the first (real) homology group. A newly observed phenomenon of almost existence of periodic solutions will also be pursued. A second direction this work will follow is one of finding symplectic invariants for periodic solutions on compact hypersurfaces in symplectic manifolds and deciding on multiplicity questions. In attempting to find the number of periodic solutions on an energy surface, one is faced with the difficulty that the usual variational approach does not give the unparametrized periodic solutions. For compact convex surfaces, efforts will be made to establish this number - which is thought to be at least half the dimension of the ambient space. The third area of concentration will deal with symplectic geometry. Considerable activity has been stimulated by the solution of one of Arnold's conjectures and the introduction of holomorphic maps of Gromov. Current work will focus on the role of the 2nd homotopy group in the Lagrangian intersection problem.

这部作品的主题是解决方案的结构 常微分方程组。 方程 都源于经典力学中的哈密顿量 系统. 为了了解这些系统的轨道结构， 人们试图找到不变的子系统，其中最简单的 是周期解 这自然会导致边界值 问题 在这个项目中将处理三个不同的问题。 第一个是关于人们可以期望找到的条件 在给定的能量曲面上的周期解。 完成 结果可用于凸形和星形表面。 工作 将已知理论扩展到紧致超曲面 接触类型。 周期解的期望判据 是第一个（真实的）同调群的消失。 一个新 几乎存在周期解的现象 也将被追究。 这项工作的第二个方向是寻找 紧空间上周期解的辛不变量 辛流形中的超曲面和判定 多样性问题。 在试图找到 在能量表面上的周期解，一个是面对 困难的是，通常的变分方法不给 非参数化周期解 对于紧致凸曲面， 将努力建立这个数字-这是认为， 至少是周围空间尺寸的一半。 集中的第三个领域将处理辛 几何 大量的活动受到了 解决阿诺德的一个问题和介绍 Gromov的全纯映射 目前的工作将侧重于 第二个同伦群在拉格朗日相交问题。