Geometric Analysis of Mechanical Systems with Symmetry

摘要提案：dms -9802106首席研究员：Jerrold E. marsden本提案旨在通过以下方式推进具有对称性的机械系统的约化理论基础研究。首先，群扩展的阶段约简理论将在哈密顿（余切、辛和泊松）和拉格朗日（变分）的观点中得到发展。其次，研究了拉格朗日约简的几何和动力学，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，还将介绍用不变量和Kahler结构约简系统的几何相理论，以及在奇异约简和多辛几何中的爆破技术的应用。机械系统（如机器人运动和控制装置）的工程分析利用了几何力学中的数学技术。随着这些工程系统变得越来越复杂，表现出更多的奇异动力学行为，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使能工具，并且在诸如卫星建立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。

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.