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

本工作的主题集中在常微分方程组解的结构上。这些方程在经典力学中用哈密顿系统来描述。为了理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。这自然导致了边值问题。本项目将处理三个不同的问题。第一个是关于在规定的能量表面上可以期望找到周期解的条件。完整的结果可用于凸面和星形面。工作是将已知的理论扩展到接触型的紧致超曲面。周期解的期望准则是第一（实）同调群的消失。还将探讨一个新观察到的周期解几乎存在的现象。本研究的第二个方向是寻找辛流形中紧超曲面周期解的辛不变量，并决定多重性问题。在试图求能量曲面上周期解的个数时，通常的变分方法不能给出非参数化的周期解。对于紧凑的凸面，将努力建立这个数字-它被认为至少是周围空间的一半。第三个重点领域是辛几何。阿诺德猜想的解和格罗莫夫全纯映射的引入激发了相当大的活动。本文将重点研究第二同伦群在拉格朗日交点问题中的作用。

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.