Mathematical Sciences: Some Questions in Nonlinear Differential Equations

有限数量的没有摩擦力的粒子的运动模型通常是用称为哈密顿系统的微分方程给出的。在过去的十年中，人们对这类系统周期解的存在性进行了大量的研究。该奖项资助的工作部分是变分方法，特别是极大极小和莫尔斯理论方法，及其在非线性微分方程中的应用的更一般发展的结果。它说明了当抽象数学研究的力量由外部刺激驱动时可以实现的有效性。这项工作的一个重点是哈密顿量本身，它是包含粒子位置和速度信息的两个（矢量）变量的函数。传统上认为哈密顿函数是连续可微的。然而，许多有趣的问题都涉及到单一的哈密顿量。天体力学中的经典n体问题就是一个例子。利用极大极小论证，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多个周期解是否可以存在于同一周期。此外，随着奇异哈密顿量的出现，出现碰撞轨道的可能性。将在确定能够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。从周期解的分支出发，研究将开始对下一个最简单的类别进行调查，这些轨道随着时间的无限增加而趋向于平衡解。它们被称为连接轨道，因为它们的周期为1。对于所谓的超二次哈密顿系统，一些结果是可用的，但该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.