Constrained geometric mechanics: theory and applications

本文主要研究几何方法在弹性结构约束动力学中的应用。在第一部分中，我们将把几何力学的方法和相应的约束应用于涉及点状、非完整约束的系统，并以弹性杆接触的特定例子为例。这个问题将用最近发展起来的几何拉格朗日-达朗贝尔方法来解决。该提案将探索这一理论提供的有趣和新的方向，如由接触引起的杆的高度复杂的动力学，离散杆的推广，对称性约化，稳定和非稳定的精确解，以及其他。在第二部分中，我们将探索具有链接分支的树状结构，或者，从几何角度计算的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。我们将研究约束对这类物体最近由PI导出的几何结构的影响，例如守恒定律和泊松括号。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都起着至关重要的作用。该传感器将在CSU与PI的合作者一起开发，并将使用极紫外光脉冲激光直接可视化纳米级运动。这种树状的能源收割机正在与日本(京都大学和志贺县大学)的PI合作者一起开发。PI在动力学和建模方面的专业知识，NINT在校园内的纳米制造设施，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。该项目将主要侧重于培训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，但仅寻求理论部分的支持。

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.