Geometric Analysis of Mechanical Systems with Symmetry

具有对称性的机械系统的几何分析

基本信息

批准号：
9802106
负责人：
Jerrold Marsden
金额：
$ 8.01万
依托单位：
California Institute of Technology
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
1998
资助国家：
美国
起止时间：
1998-07-15 至 2001-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=9802106&HistoricalAwards=false
关键词：
Geometric Analysis Mechanical Systems Symmetry

项目摘要

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.

摘要提案：DMS-9802106 首席研究员：Jerrold E. Marsden该提案旨在通过以下方式推进对称性机械系统还原理论的基础研究。首先，群扩张的阶段还原理论将在哈密顿量（余切、辛和泊松上下文）和拉格朗日（变分上下文）的观点。二、几何将研究拉格朗日约简的动力学，特别是拉格朗日-庞加莱方程和拉格朗日-达朗贝尔-庞加莱方程。此外，还将推进使用不变量和卡勒结构简化系统的几何相理论以及在奇异简化和多重辛几何中使用爆炸技术。机器人运动和控制装置等机械系统的工程分析利用了该提议者几何力学中的数学技术。随着此类工程系统的复杂性变得更加复杂并表现出更奇特的动态行为并且需要更精细的控制策略，这种数学基础将变得越来越重要。本提案中的主题是此类研究的关键支持工具，并已被证明在卫星稳定和水下航行器动力学等各种系统中具有巨大的实用性。它们还广泛应用于力学中非线性动力系统的数值模拟软件包，包括碰撞、流体和弹性系统以及卫星任务规划。