Hamiltonian Dynamical Systems and N-Body Problem

哈密​​顿动力系统和 N 体问题

• 批准号: 0071494
• 负责人: Kenneth Meyer
• 金额: --
• 依托单位: University of Cincinnati Main Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-06-01 至 2004-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0071494&HistoricalAwards=false
• 关键词: Hamiltonian; Dynamical; Systems; Body; Problem

DMS-0071494Dr. Meyer will work on a variety of problems involving the $N$-body problem,Hamiltonian systems and dynamical systems. Building on his recent work with Dr. McCord on the integral manifolds of the spatial three body problem he will study the geometry and homology of the {\it regularized} (compactified)integral manifolds of the spatial three-body problem. These are eight-dimensionalalgebraicsets which vary with the masses, energy and angular momentum of the particles. These computations should refute Birkhoff conjecture, yield information on theexistence of cross sections, and answer the Birkhoff question on cross sections in the negative asthey did in the non-regularized problem. As another application of the richtopology found in the integral manifolds of the three-body problem, he will usevariational arguments to establish new classes of periodic solutions. Dr. Meyerwill also work on (i) simplifying the proof of the existence of Xia diffusionin the three-body problem; (ii) the existence of comet-like periodic orbits in the N-body problem; (iii) establishing the stability of anequilibrium point of a Hamiltonian system in the case of 1:1 resonance; and(iv) the evolution and bifurcation of invariant manifolds of a Hamiltoniansystem which depend on a parameter. Dr. Meyer will work on a number of problems in the qualitative theory of theequations which describe the motion of mechanical systems and celestial bodies, namelyHamiltonian systems. One of the important equations of this type is the N-bodyproblem which describes the motion of N masses -- the planets. Systems of this type typically conserveenergy and momentum and these conservation laws place global restrictions onthe possible motion of the bodies. Dr. Meyer has an ongoing research programstudying the ramifications of these conserved quantities on the nature of the motion. Dr. Meyer will alsoinvestigate the existence of regular and chaotic motions in these systems andhow these types of motions depend of various physical parameters. One type ofregular motion he will establish is periodic motions in the N-body problem and one type of chaotic motion he willtry to establish is known as Xia diffusion. Dr. Meyer will also study theevolution of the stable manifold of equilibrium solutions i.e. the evolution ofthe set of solutions which tend to an equilibrium (the stable manifold).

DMS-0071494DR。 Meyer将解决涉及$ n $ body问题，哈密顿系统和动态系统的各种问题。基于他最近与麦考德博士在空间三个身体问题的整体歧管上的工作，他将研究{\ it正则化}（压缩）空间三体问题的整体歧管的几何形状和同源性。 这些是八维级甲壳虫，它们随颗粒的质量，能量和角动量而变化。这些计算应驳斥Birkhoff的猜想，产生有关横截面的存在的信息，并回答伯克霍夫在负ashey中的横截面问题，他们在非规范化问题中所做的。作为在三体问题的整体歧管中发现的Richtopology的另一个应用，他将使用不同的论点来建立新的定期解决方案。 Meyerwill博士还致力于（i）简化Xia扩散的证据，即三体问题； （ii）在N体问题中存在类似彗星的周期轨道； （iii）在1：1共振的情况下，建立哈密顿系统的静态点的稳定性； （iv）依赖参数的哈密顿系统的不变流形的进化和分叉。 Meyer博士将在描述机械系统和天体的运动的定性理论中处理许多问题，Namelyhamiltonian Systems。 这种类型的重要方程之一是N-BodyPromblem，它描述了N质量的运动 - 行星的运动。 这种类型的系统通常会节省势头和动量，这些保护法对身体可能的运动施加了全球限制。 迈耶博士正在进行的研究计划，研究了这些保守数量在运动性质上的影响。 Meyer博士还将对这些系统中的规则和混乱运动的存在进行评估，并将这些类型的运动取决于各种物理参数。 他将建立的一种规则运动是N体问题中的周期性动作，而他要确定的一种混乱运动被称为Xia扩散。迈耶博士还将研究平衡溶液稳定歧管的进化，即倾向于平衡的溶液集的演变（稳定的歧管）。