Mathematical Sciences: Hamiltonian Dynamical Systems

数学科学：哈密顿动力系统

• 批准号: 9626627
• 负责人: Kenneth Meyer
• 金额: $ 8.5万
• 依托单位: University of Cincinnati Main Campus
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-08-01 至 2000-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9626627&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Hamiltonian; Dynamical; Systems

Abstract Meyer Dr. Meyer (with Dr. McCord) will study the topology of the integral manifolds of the regularized spatial three-body problem. He wishes to describe the bifurcations by computing the homology of these manifolds for various values of the parameters. He will try to clarify the existence of Arnold diffusion in the three-body problem using reduction, symplectic scaling, normalization, and the theory of normally hyperbolic manifolds. Dr. Meyer (with Dr. Schmidt) will study the existence of KAM tori near elliptic orbits of the three-body problem. He (with Ms Howison, a graduate student) will establish the existence of various periodic solutions of the spatial restricted three-body problem. He will also try to establish the existence of elliptical comet like orbits in the full three-body problem. Dr. Meyer studies the differential equations the describe the motion of the planets and satellites in our solar system, the N-body problem. He also studies other types of equations that describe mechanical systems. Since these equations are not solvable exactly, he studies questions about special types of solutions. For example, when does the system of equations have periodic solutions or even quasi-periodic solutions? He will attempt to establish several new families of periodic solutions and several new families of quasi-periodic solutions for the N-body problem. He has developed several new methods to attack these problems. The existence of these quasi-periodic solutions give some partial understanding of the stability of the solar system. He will also study some global questions about these systems. What global constraints on the possible configuration of the bodies come from the conservation of energy and momentum?

摘要 Meyer Meyer 博士（与 McCord 博士）将研究正则化空间三体问题的积分流形的拓扑。 他希望通过计算这些流形对于不同参数值的同源性来描述分岔。 他将尝试使用归约、辛标度、归一化和正双曲流形理论来阐明三体问题中阿诺德扩散的存在性。 Meyer 博士（与 Schmidt 博士）将研究三体问题的椭圆轨道附近 KAM tori 的存在性。 他（与研究生 Howison 女士）将建立空间受限三体问题的各种周期解的存在性。 他还将尝试在完整的三体问题中证明椭圆形彗星轨道的存在。 迈耶博士研究描述太阳系中行星和卫星运动的微分方程，即 N 体问题。 他还研究了描述机械系统的其他类型的方程。由于这些方程无法精确求解，因此他研究了有关特殊类型解的问题。 例如，方程组什么时候有周期解甚至准周期解？ 他将尝试为 N 体问题建立几个新的周期解族和几个新的准周期解族。 他开发了几种新方法来解决这些问题。 这些准周期解的存在使人们对太阳系的稳定性有了一些部分的了解。 他还将研究有关这些系统的一些全球性问题。 能量和动量守恒对物体可能的配置有哪些全局限制？