Geometric Analysis of Mechanical Systems with Symmetry

摘要提案：DMS-9802106主要研究者：Jerrold E. Marsden提出的这一建议旨在从以下几个方面推进对称性力学系统约化理论的基础研究：首先，在Hamilton系统和Hamilton系统中发展群扩张的阶段约化理论（余切，辛和泊松上下文）和拉格朗日（变分语境）观点。第二，拉格朗日约化的几何和动力学，特别是Lagrange-Poincare方程和Lagrange-d 'Alembert-Poincare方程。此外，几何相位理论的系统减少使用不变量和Kahler结构，以及使用爆破技术在奇异减少，和多辛几何。机械系统的工程分析，如机器人运动和控制装置，利用几何力学中的proposers prosticaltechniques。这种数学基础将越来越重要的复杂性，这样的工程系统变得更加复杂，并表现出更多的异国情调的动态行为，并需要更微妙的控制策略。本建议中的主题是进行此类研究的关键工具，并已证明在卫星稳定和水下航行器动力学等各种系统中具有很大的实用性。它们也被用于力学中的非线性动力系统的数值模拟包，包括碰撞、流体和弹性系统以及卫星使命规划。

AbstractProposal: DMS-9802106Principal Investigator: Jerrold E. MarsdenThis proposal intends to advance funamental research in reductiontheory for mechanical systems with symmetry in the following ways.First, the theory of reduction by stages for group extensions will bedeveloped in both the Hamiltonian (cotangent, symplectic and Poissoncontexts) and the Lagrangian (variational context) points of view.Second, the geometry and dynamics of Lagrangian reduction, especiallythe Lagrange-Poincare equations and the Lagrange-d'Alembert-Poincareequations will be investigated. In addition, the theory of geometricphases for systems reduced using invariants and Kahler structures aswell as the use of blowing up techniques in singular reduction, andmultisymplectic geometry will be advanced.The engineering analysis of mechanical systems such as roboticlocomotion and control devices makes use of the proposers mathematicaltechniques in geometric mechanics. This mathematical foundation willbe of increased importance as the sophistication of such engineeringsystems get more complex and exhibit more exotic dynamical behaviorsand require more delicate control strategies. The topics in thisproposal are critical enabling tools for such research and have provento be of great utility in a variety of systems such as satellitestabilization and underwater vehicle dynamics. They are also being putto fundamental use in numerical simulation packages for nonlineardynamical systems in mechanics, including collision, fluid, andelastic systems as well as in satellite mission planning.