Mathematical Sciences: Geometry and Dynamics of Mechanical Systems with Symmetry

数学科学：对称机械系统的几何和动力学

• 批准号: 9633161
• 负责人: Jerrold Marsden
• 金额: $ 14.4万
• 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: 1996
• 资助国家: 美国
• 起止时间: 1996-07-01 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9633161&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Geometry; Dynamics; Mechanical

9633161 Marsden The proposed research lies in the area of geometric mechanics. It has to do with laying a theoretical foundation for various aspects of classical mechanics. Nonholonomic systems with symmetry are to be investigated as an application of the ideas of Lagrangian reduction. Integrable systems and solitons will be pursued with a view towards certain applications in control dynamics. Bifurcations in Hamiltonian systems will be investigated in the context of singular reduction. Geometry of control systems will also be studied. Geometric mechanics aims to formulate theories and problems in classical mechanics in a mathematically rigorous setting. There have been several recent successes of geometric mechanics. The falling cat example, where the cat executes a complete re-orientation without exerting any angular momentum, has now been explained using the idea of a connection - a connection is a mathematical notion originally introduced to study so called principal bundles in differential geometry. The proposed research will further pursue these theoretical ideas as well as consider possible applications in such areas as control theory and robotics.

9633161 Marsden 拟议的研究属于几何力学领域。它与为经典力学的各个方面奠定理论基础有关。具有对称性的非完整系统将作为拉格朗日约简思想的应用进行研究。将追求可积系统和孤子，着眼于控制动力学中的某些应用。哈密​​顿系统中的分岔将在奇异约简的背景下进行研究。控制系统的几何形状也将被研究。 几何力学旨在在严格的数学环境中阐述经典力学的理论和问题。几何力学最近取得了一些成功。猫下落的例子，猫在不施加任何角动量的情况下执行完全重新定向，现在已经使用连接的概念进行了解释 - 连接是一个数学概念，最初是为了研究微分几何中所谓的主丛而引入的。拟议的研究将进一步追求这些理论思想，并考虑在控制理论和机器人技术等领域的可能应用。