Reduction Methods in Hamiltonian Dynamics, Bifurcation Theory, and Lie Theory of Symplectomorphism Groups

项目负责人：Tudor ratia本项目的第一个目标是利用涉及奇异点的辛约简理论及其推广来研究有限维和无限维李群的各种力学系统和伴随轨道。此外，这些技术将与动力系统方法和经典分岔理论方法相结合，应用于对称哈密顿分岔理论。一个拉格朗日的对应化简将被发展，其中辛几何技术被变分原理所取代。这些变分方法的应用产生了地球物理流体动力学中各种近似模型的方程。本项目的第二个目标是通过应用几何和李理论方法来研究非线性波动方程和理解无限维动量映射的凹凸性来理解微分同态群的补全。第三个目标是应用哈密顿动力学方法，结合奇异约简来研究混凝土力学系统的分岔和长期行为，如黎曼椭球体。同样的方法也适用于设计和研究非线性流形上的数值算法。几何力学中的辛方法是解决具体动力学问题中的难题的有力工具，例如潜艇或机械臂运动的控制，棒状和壳状连续体系统模型的细化，或从微小液滴到星系的液体或气体质量形状的研究。解决这些问题的几何基础的发展，以及它们在几个问题上的应用，构成了本基金要做的工作。

AbstractProposal: DMS-9802378Principal Investigator: Tudor RatiuA first goal of this project is to use symplectic reduction theory andits extension involving singularities to the study of variousmechanical systems and coadjoint orbits of finite and infinitedimensional Lie groups. In addition, these techniques will be appliedto symmetric Hamiltonian bifurcation theory by combining them withmethods of dynamical systems and classical bifurcation theory methods.A Lagrangian counterpart of reduction will be developed, wheresymplectic geometry techniques are replaced by variational principles.Applications of these variational methods yield the equations ofvarious approximate models in geophysical fluid dynamics. A secondgoal of this project is to understand completions of diffeomorphismgroups with the scope of applying geometric and Lie theoreticalmethods to the study of nonlinear wave equations and the understandingof the convexity properties of the momentum map in infinitedimensions. A third goal is the application of methods of Hamiltoniandynamics in conjunction with singular reduction to study thebifurcation and the long term behavior of concrete mechanical systemssuch as the Riemann ellipsoids. The same methods are intended for thedesign and study of numerical algorithms on nonlinear manifolds.Symplectic methods in geometric mechanics are a powerful tool toaddress difficult questions in the dynamics of concrete problems suchas the control of submarine or robotic arm motion, the elaboration ofmodels for continuum systems such as rods and shells, or the study ofthe shape of liquid or gaseous masses ranging from tiny drops togalaxies. The development of the geometric foundations to attack theseproblems as well as their application to several problems comprisesthe work to be done on this grant.