Mathematical Sciences: Some Questions in Nonlinear Differential Equations

在没有摩擦力的情况下，有限个粒子的运动模型通常由称为哈密顿系统的微分方程式给出。在过去的十年里，人们对这类系统的周期解的存在性进行了深入的研究。该奖项所赞助的工作在某种程度上是变分方法，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中的应用的更广泛发展的产物。它说明了当抽象数学研究的力量由外部刺激驱动时所能达到的效果。这项工作的一个焦点是哈密顿量本身，它是包含关于粒子的位置和速度信息的两个(矢量)变量的函数。传统上，人们假设哈密顿是连续可微的。然而，许多有趣的问题都涉及到奇特的哈米顿人。天体力学的经典n体问题就是一个例子。利用极小极大变元，可以证明存在无穷多个具有不同周期的周期解。目前的工作将试图澄清在同一时期内是否可以存在多个周期解。此外，随着奇异哈密顿量的出现，碰撞轨道的可能性也随之增加。将在确定人们可以区分规则轨道和碰撞轨道的条件方面进行工作。还将努力寻找具有规定能量的周期性解决方案。除了周期解之外，研究将开始研究下一类最简单的轨道，即随着时间的无限增加而趋于平衡解的轨道。这些轨道被称为连接轨道，因为它们与周期轨道相连。对于所谓的超二次哈密顿系统，已有一些结果，但这一领域仍处于起步阶段。

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.