Hamiltonian Dynamical Systems and N-Body Problem

dms - 0071494 -博士。迈耶将研究各种各样的问题，包括N体问题、哈密顿系统和动力系统。在他最近与McCord博士一起研究空间三体问题的积分流形的基础上，他将研究空间三体问题的正则化（紧化）积分流形的几何和同调性。这些是随粒子的质量，能量和角动量变化的八维代数集。这些计算应该反驳Birkhoff猜想，给出关于横截面存在的信息，并像在非正则化问题中那样在负方向上回答关于横截面的Birkhoff问题。作为富拓扑在三体问题的积分流形中的另一个应用，他将使用变分论证来建立新的周期解类。迈耶博士还将致力于(1)简化三体问题中夏扩散存在的证明；（ii） n体问题中类彗星周期轨道的存在；(3)建立了1:1共振情况下哈密顿系统平衡点的稳定性；(4)含参数哈密顿系统不变流形的演化与分岔。迈耶博士将研究描述机械系统和天体运动的方程的定性理论中的一些问题，即哈密顿系统。这种类型的一个重要方程是N体问题，它描述了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，这些守恒定律对物体可能的运动施加了全局限制。迈耶博士正在进行一个研究项目研究这些守恒量对运动本质的影响。迈耶博士还将研究这些系统中规则和混沌运动的存在，以及这些运动类型如何依赖于各种物理参数。他将建立的一种规则运动是n体问题中的周期运动，他将试图建立的一种混沌运动是夏扩散。迈耶博士还将研究平衡解的稳定流形的演化，即趋向于平衡的一组解的演化（稳定流形）。

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).