Mathematical Sciences: Some Questions in Nonlinear Differential Equations

有限数量粒子的运动模型， 不存在摩擦力时，通常通过 称为哈密顿系统的微分方程。 期间 在过去的十年里，人们一直在集中研究 这类系统的周期解的存在性 工作 由这个奖项赞助的部分是一个更普遍的产物， 变分方法的发展，特别是极小极大和莫尔斯 理论方法及其在非线性 微分方程 它说明了有效性， 当抽象数学研究的力量 是由外部刺激驱动的 这项工作的一个焦点是关于哈密顿量本身， 包含位置的两个（向量）变量的函数 以及粒子的速度信息。 传统上， 假设哈密顿算子是连续可微的。 然而，许多有趣的问题涉及奇异的Hamitonians。 天体力学的经典n体问题是一个 example. 使用minimax参数，可以证明 有无穷多个不同周期的周期解 的 目前的工作将寻求澄清是否有多个 周期解可以以相同的周期存在。 此外，本发明还提供了一种方法， 随着奇异哈密顿算子的出现， 碰撞轨道。 将在确定 在这些条件下，人们可以区分常规和 碰撞轨道 还将努力寻找定期 具有规定能量的溶液。 从周期解出发，研究将开始 研究下一个最简单的类， 随着时间的增加， 无限期 这些被称为连接轨道，因为它们 加入时代。 一些结果可用于所谓的 超二次哈密顿系统，但该地区仍在其 婴儿期。

Models for the motion of a finite number of particlies where no frictional forces are present are usually given by means of differential equations called Hamiltonian systems. During the past decade there has been intensive research focusing on the existence of periodic solutions of such systems. The work sponsored by this award is in part an outgrowth of a more general development of variational methods, expecially minimax and Morse theoretic methods, and their application to nonlinear differential equations. It illustrates the effectiveness which can be achieved when the power of abstract mathematical research is driven by external stimuli. One focus of this work concerns the Hamiltonian itself, a function of two (vector) variables which contains the position and velocity information about the particles. Traditionally one assumes that the Hamiltonians are continuously differentiable. Many interesting problems, however, involve singular Hamitonians. The classical n-body problem of celestial mechanics is an example. Using minimax arguments, one can show that there are infinitely many periodic solutions with distinct periods. The present work will seek to clarify whether or not multiple periodic solutions can exist with the same period. In addition, with the advent of singular Hamiltonians, the possibility arises for collision orbits. Work will be done in determining conditions where one can distinguish between regular and collision orbits. Efforts will also be made to find periodic solutions with prescribed energy. Branching away from periodic solutions, research will begin on investigations into the next simplest class, those orbits which tend to equilibrium solutions as times increases indefinitely. These are known as connecting orbits, for they join period ones. Some results are available for so-called superquadratic Hamiltonian systems, but the area is still in its infancy.