Hamel's Formalism and its Applications

这个项目发展了拉格朗日力学的一个表示，在这个表示中，速度分量是相对于一个与位形空间上的任何局部坐标无关的坐标系来测量的。这种方法比拉格朗日的原始形式提供了更多的灵活性，同时保持了系统配置和速度之间的清晰区分。众所周知，这是有限维连续时间力学中有效而强大的工具，但Georg Hamel引入的这种形式尚未完全发展到具有无限多个自由度的系统。该项目旨在系统地研究无限维环境下的形式主义，理解其变分结构，及其在具有非完整约束的无限维系统动力学中的应用，包括受非完整约束的充液刚体的稳定性，以及Lyapunov函数的构造。具有无限个自由度的系统在工程和工业中有许多应用。例子包括与流体相互作用的柔性结构和刚体。该项目发展了建模和模拟这种复杂机械系统行为所需的数学。这项关于系统力学的工作将进一步整合数学和理论工程。该项目的研究成果将有助于多体动力学和复杂机械系统的理论分析、仿真、控制和长期行为研究。

This project develops a representation of Lagrangian mechanics in which the velocity components are measured relative to a frame that is not related to any local coordinates on the configuration space. This approach offers more flexibility than the original formalism of Lagrange, while keeping a clear distinction between system configurations and velocities. Known to be an effective and powerful tool in finite-dimensional continuous-time mechanics, this formalism introduced by Georg Hamel has not yet been fully developed for systems with infinitely-many degrees of freedom. The project is aimed at systematic study of the formalism in the infinite-dimensional setting, understanding its variational structure, its applications to dynamics of infinite-dimensional systems with nonholonomic constraints, including stability of a liquid-filled rigid body subject to nonholonomic constraints, and Lyapunov function construction.Systems with infinite number of degrees of freedom arise in many applications in engineering and industry. Examples include flexible structures and rigid bodies interacting with fluids. This project develops the mathematics required to model and simulate the behavior of such complex mechanical systems. This work on the mechanics of systems will further integrate mathematics and theoretical engineering. The results of the project will be useful in theoretical analysis, simulations, control, and study of the long time behavior of multi-body dynamics and complex mechanical systems.