Constrained geometric mechanics: theory and applications

本文主要讨论几何方法在弹性结构约束动力学中的应用。在第一部分中，我们将应用几何力学的方法，以及相应的约束，涉及点方向的系统，非完整约束，并以接触的弹性杆为例。这个问题将使用最近发展起来的几何拉格朗日-达朗贝尔方法来解决。该提案将探索该理论提供的有趣和新的方向，例如由接触引起的杆的高度复杂动力学，对离散杆的推广，对称约简，稳定和非稳定精确解等。在第二部分中，我们将探索具有链接分支的树状结构，或者，从几何的角度评估一组链接的树状分支。这些问题将用具有完整约束的迭代半直积群的几何力学方法来处理。我们将研究PI最近为这些物体导出的几何结构的约束，如守恒定律和泊松括号的影响。树状结构动力学问题也将用于纳米机械传感器和宏观树状能量收集装置的开发。上面概述的数学问题在这两种设备的设计中起着至关重要的作用。该传感器将与科罗拉多州立大学PI的合作者一起开发，并将利用EUV脉冲激光直接可视化纳米级运动。这种树状的能量收集器是与PI在日本的合作者（京都大学和滋贺县）共同开发的。PI在动力学和建模方面的专业知识，NINT校园的纳米制造设施，以及与CSU和日本的EUV激光小组的合作，使该项目有可能成功。该项目将重点培养研究生和博士后开发现代数学工具，并随后将这些工具应用于现代实际问题。虽然该项目与实验密切相关，但仅为理论部分寻求支持。

This proposal focuses on the applications of geometric methods to constrained dynamics of elastic structures. In the first part, we shall apply the methods of geometric mechanics, and corresponding constraints, to systems involving point wise, non-holonomic constraints, with a particular example of elastic rods in contact. This problem will be tackled using the recently developed methods of geometric Lagrange-d'Alambert's methods. The proposal will explore interesting and new directions provided by this theory, such as highly complex dynamics of rods caused by the contact, generalizations to discrete rods, symmetry reductions, steady and unsteady exact solutions, and others. In the second part, we will explore tree-like structures with linked branches, or, alternatively, an array of linked tree-like branches, evaluated from the point of view of geometry. These problems will be treated with methods geometric mechanics of iterative semidirect product groups with holonomic constraints. We shall study the effect of constraints on the geometric structures recently derived by the PI for such objects, such as conservation laws and Poisson brackets. The problem of tree-like structure dynamics will also be used in the development of a nanomechanical sensor and a macroscale tree-like energy harvesting device. Mathematical questions outlined above play a crucial role in the design of both devices. The sensor will be developed together with the PI's collaborators at CSU and will utilize a direct visualization of nanoscale motion using EUV pulsed laser. The tree-like energy harvester is being developed with the PI's collaborators in Japan (U of Kyoto and Shiga Prefecture). The PI's expertise in the dynamics and modeling, nano-fabrication facilities at NINT on campus, and collaboration with the EUV laser group at CSU and Japan makes this project likely to succeed. The project will focus heavily on the training of graduate students and postdocs in the development of tools of modern mathematics and subsequent application of these tools to modern, practical problems. While the project is closely involved with experiments, support is sought only for the theory part.