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

摘要建议：DMS-9802378首席研究员：Tudor Ratiu本项目的第一个目标是使用辛约化理论及其扩展，包括奇异性，以研究有限维和无限维李群的各种力学系统和共伴轨道。此外，将这些方法与动力系统方法和经典分歧理论方法相结合，应用于对称Hamilton分歧理论，发展了相应的拉格朗日约化方法，用变分原理代替辛几何方法，应用这些变分方法得到了地球流体动力学中各种近似模型的方程。 该项目的第二个目标是理解与应用几何和Lie理论方法的范围内的非线性波动方程的研究和理解的凸性性质的动量映射在无限维完成的同态群。第三个目标是应用Hamilton方法和奇异约化方法来研究具体力学系统如Riemann椭球的分叉和长期行为。几何力学中的辛方法是解决具体问题的动力学难题的有力工具，如潜艇或机械臂运动的控制，连续系统（如杆和壳）模型的详细说明，或从微小水滴到星系的液体或气体质量形状的研究。发展的几何基础攻击thesepolutions以及他们的应用程序的几个问题successes的工作要做的这项补助金。

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.