Mathematical Sciences: Hamiltonian Dynamical Systems

抽象迈耶 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结构。 他希望通过计算这些流形的各种参数值的同调来描述分叉。 他将试图澄清阿诺德扩散的存在，在三体问题使用减少，辛缩放，规范化，和理论的正常双曲流形。 博士 Meyer（与施密特博士）将研究三体问题椭圆轨道附近KAM环面的存在性。 他（与Howison女士，一名研究生）将建立空间限制三体问题的各种周期解的存在性。 他还将试图建立椭圆彗星轨道的存在，在完整的三体问题。 Meyer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。 他还研究其他类型的方程描述机械系统。由于这些方程是不完全可解的，他研究问题的特殊类型的解决方案。 例如，什么时候方程组有周期解甚至是准周期解？ 他将尝试建立几个新的家庭的周期性解决方案和几个新的家庭的准周期性解决方案的N体问题。 他研究出了几种新方法来解决这些问题。 这些拟周期解的存在性使我们对太阳系的稳定性有了部分的了解。 他还将研究有关这些系统的一些全球性问题。 从能量和动量守恒出发，对物体可能的构型有什么全局约束？

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?