Geometric Mechanics

项目负责人：Jerrold E. MarsdenInstitution: California Institute of technology标题：Geometric mechanics摘要这项NSF资助的工作将在几个感兴趣的问题的背景下发展几何力学和约简理论：离散力学的积分算法和约简理论，碰撞算法的辛性质，对称控制拉格朗日和哈密顿系统，协变场论（如电磁和规范场）背景下的约简，使用协变泊松括号和拉格朗日约简，以及使用爆破技术在对称力学系统中的分岔，包括建立与奇异约简的联系。几何技术和分析技术的结合在力学的许多方面继续发挥着重要作用。这包括计算算法的发展。例如，变分积分算法的最新进展产生了最先进的碰撞代码，例如，弹性体和弹性壳之间的碰撞。本提案中的工作进一步发展了导致这些算法的数学框架。这包括在一个精确的意义上证明，算法尊重力学的特殊结构，比如守恒定律。此外，研究对称性对这些算法的影响，并在离散水平上直接分解出这种对称性是这些算法的一个重要方面，目前只在特殊情况下才被理解；本建议将发展和扩展这些技术。对称和几何技术很重要的另一个领域是机械系统的控制（例如如何使用转子来控制航天器的方向）。控制这种机械系统的对称性缩减是将在这项资助下发展的技术之一。

Proposal #0204474 PI: Jerrold E. MarsdenInstitution: California Institute of TechnologyTitle: Geometric MechanicsABSTRACTWork under this NSF grant will develop geometric mechanics and reduction theory in the context of several problems of interest: integration algorithms and reduction theory for discrete mechanics, the symplectic nature of collision algorithms, controlled Lagrangian and Hamiltonian systems with symmetry, reduction in the context of covariant field theory (such as electromagnetism and gauge fields) using covariant Poisson brackets as well as Lagrangian reduction, and bifurcations in mechanical systems with symmetry using the blowing-up technique, including the establishment of connections with singular reduction. The combination of geometric and analytical techniques continues to play an important role in many aspects of mechanics. This includes developments in computational algorithms. For instance, recent advances in variational integration algorithms have produced state of the art collision codes, for example, for collisions between elastic bodies and elastic shells. The work in this proposal further develops the mathematical framework that led to these algorithms in the first place. This includes proving, in a precise sense, that the algorithms respect the special structure of mechanics, such as conservation laws. In addition, studying the effect of symmetry on such algorithms and factoring out this symmetry directly on the discrete level is an important aspect of these algorithms which is only understood in special cases at the moment; this proposal will develop and extend these techniques. Another area where symmetry and geometric techniques are important is that of the control of mechanical systems (such as how rotors are used to control spacecraft orientation). Symmetry reduction for the control of such mechanical systems is one of the techniques that will be developed under this grant.