Mathematical Sciences: Dynamics of Hamiltonian Systems

数学科学：哈密顿系统动力学

• 批准号: 9401740
• 负责人: John Mather
• 金额: $ 13.5万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-01 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9401740&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Dynamics; Hamiltonian; Systems

9401740 Mather Mather intends to continue his work on dynamics of Hamiltonian systems. In the past, much of his work has been on dynamics of area-preserving mappings and autonomous systems in two degrees of freedom. More recently, he has extended part of this work to more degrees of freedom, in his recent paper, "Variational Construction of Connecting Orbits." However, this paper is only a beginning in this direction, in the sense that it should be possible to obtain similar results under much more general hypotheses. Proving the existence of connecting orbits between minimal sets under more general hypotheses will be a major part of Mather's research. As an example of what one might do with such results, consider the restricted three-body problem in the plane, i.e. the mechanical problem with a Newtonian gravitational potential, two massive bodies and one body of zero mass, all in the plane. In the case that the two massive bodies move about one another in circular orbits, Moser showed, in the early 60's, that if the zero mass body moves in an approximately circular orbit a long distance from the massive bodies, then it does not escape to infinity. Thus one has stability. This is part of KAM theory. In contrast, if the massive bodies move about one another in elliptical orbits, no such result is known. Mather's methods might lead to a proof that such a result is false. There are, however, major hurdles to be overcome before this program can be carried out. ***

小行星9401740 马瑟打算继续他的工作动力学的哈密顿系统。 在过去，他的大部分工作都是在两个自由度的区域保持映射和自治系统的动力学。 最近，他在他最近的论文《连接轨道的变分构造》中将这项工作的一部分扩展到了更多的自由度。然而，这篇论文只是这个方向的一个开始，在这个意义上，在更一般的假设下应该有可能获得类似的结果。 在更一般的假设下证明最小集合之间的连接轨道的存在将是马瑟研究的主要部分。 作为一个例子，我们可以用这样的结果做些什么，考虑平面上的限制性三体问题，即具有牛顿引力势的力学问题，两个质量大的物体和一个质量为零的物体，都在平面上。 在两个大质量物体在圆形轨道上相互运动的情况下，莫泽在60年代初表明，如果零质量物体在近似圆形的轨道上运动，距离大质量物体很远，那么它不会逃逸到无限远。 一是稳定性。 这是KAM理论的一部分。 相反，如果大质量天体在椭圆轨道上相互运动，就没有这样的结果了。 马瑟的方法可能会证明这样的结果是错误的。 然而，在实施这一方案之前，还需要克服一些重大障碍。 ***