Theoretical Chemistry, Dynamical Systems, and the Geometry of Global Phase Space Dynamics

9704759威金斯这个研究项目致力于开发分析和计算工具，用于研究分子的半经典模型的全球动力学。重点将放在三个或三个以上自由度的分子模型上，分析方法是非线性动力系统的几何理论。特别是，我们将研究相空间的几何结构如何在实验光谱中表现出来，三个或更多自由度分子的局域到简正模的跃迁，能量传输和不变流形在决定这一转变中的作用，以及三个或更多自由度分子中转动-振动相互作用的动力学。此外，我们还将研究具有三个或更多个自由度的哈密顿系统的全局不稳定性的机制，特别是Arnold机制。这项研究涉及研究三个或更多自由度分子的经典和半经典动力学的技术的发展，以及对三个或更多自由度哈密顿系统不稳定性机制的基础研究。在过去的10年里，化学实验技术已经发展到可以获得与分子相互作用和动力学有关的实时动态数据的程度，这导致了被称为“飞秒化学”的研究领域。在某些方面，今天的化学实验情况与20世纪70年代末的流体力学实验情况有很大的相似之处。当时，基于激光的新实验技术被开发出来，产生了来自流体流动的全球、实时的动态数据。这些数据在流体流动的动力系统分析中发挥了关键作用。事实上，人们可以说，这些实验进展是流体力学在整个80年代成为(可能)动力系统理论的主要应用领域的原因。今天，我们处于理论化学的类似情况下，70年代末流体力学和动力系统理论的实验进展的关系。因此，在动力系统和理论化学中进行集中的、跨学科的研究可以在解释这些新的实验数据以及理论化学领域的进一步发展中发挥重要作用。该奖项将由数学科学处、化学处和数学和物理科学局多学科活动办公室提供支助。

9704759 Wiggins This research project is concerned with developing analytical and computational tools for studying the global dynamics of semiclassical models of molecules. The emphasis will be on molecular models with three or more degrees-of-freedom and the analytical approach is that of the geometric theory of nonlinear dynamical systems. In particular, we will investigate how geometrical structures in phase space are manifested in experimental spectra, the local to normal mode transition in molecules with three or more degrees-of-freedom, energy transport and the role of invariant manifolds in determining this, and the dynamics of rotational-vibrational interaction in molecules with three or more degrees-of-freedom. In addition, we will investigate mechanism for global instabilities in Hamiltonian systems with three or more degrees of freedom; in particular, the Arnold mechanism. The research is concerned with the development of techniques to study classical and semiclassical dynamics of molecules with three or more degrees-of-freedom, as well as a fundamental study of mechanisms for instability in Hamiltonian systems with three or more degrees-of-freedom. Over the past 10 years experimental techniques have been developed in chemistry to the point where real time dynamical data related to molecular interactions and dynamics can be obtained, which has resulted in the area of research known as "femtochemistry". In some ways the experimental situation in chemistry today bears a strong resemblance to the experimental situation in fluid mechanics in the late 1970`s. At that time new laser based experimental techniques were developed that resulted in global, real time dynamical data from fluid flows. Such data played a key role in the dynamical systems analyses of fluid flows. In fact, one could make the case that these experimental advances were the reason that fluid mechanics became (probably) the main application area for dynamical sy stems theory throughout the 1980`s. Today we are in an analogous situation for theoretical chemistry vis- -vis the relation of experimental advances in fluid mechanics and dynamical systems theory in the late 1970`s. Consequently, a focused, interdisciplinary research effort in dynamical systems and theoretical chemistry could play an important role in the interpretation of this new experimental data, as well as the further development of the field of theoretical chemistry. This award will be supported by the Division of Mathematical Sciences, the Division of Chemistry, and the Office of Multidisciplinary Activities of the Directorate for Mathematical and Physical Sciences.