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-0071494博士。迈耶将致力于解决涉及 $N$ 体问题、哈密尔顿系统和动力系统的各种问题。基于他最近与 McCord 博士在空间三体问题积分流形方面的合作，他将研究空间三体问题的{\it正则化}（紧致）积分流形的几何和同源性。 这些是八维代数集，随着粒子的质量、能量和角动量而变化。这些计算应该反驳伯克霍夫猜想，产生关于横截面存在的信息，并像在非正则化问题中所做的那样，以否定的方式回答关于横截面的伯克霍夫问题。作为三体问题积分流形中丰富拓扑的另一个应用，他将使用变分论证来建立新类别的周期解。 Meyerwill博士还致力于（i）简化三体问题中夏扩散存在性的证明； (ii) N体问题中类彗星周期轨道的存在性； (iii)建立哈密顿系统在1:1共振情况下平衡点的稳定性； (iv) 依赖于参数的哈密顿系统的不变流形的演化和分岔。迈耶博士将研究描述机械系统和天体运动的方程定性理论中的许多问题，即哈密尔顿系统。 此类重要方程之一是 N 体问题，它描述 N 个质量（行星）的运动。 这种类型的系统通常守恒能量和动量，并且这些守恒定律对物体可能的运动施加全局限制。 迈耶博士正在进行一项研究项目，研究这些守恒量对运动性质的影响。 迈耶博士还将研究这些系统中是否存在规则运动和混沌运动，以及这些类型的运动如何依赖于各种物理参数。 他将建立的一种规则运动是 N 体问题中的周期性运动，而他将尝试建立的一种混沌运动称为夏扩散。迈耶博士还将研究平衡解的稳定流形的演化，即趋于平衡的解集（稳定流形）的演化。