Hamel's Formalism and its Applications

这个项目开发了一种拉格朗日力学的表示法，其中速度分量是相对于与构形空间上的任何局部坐标无关的框架来测量的。这种方法比拉格朗日最初的形式主义提供了更多的灵活性，同时保持了系统配置和速度之间的明确区别。这种由Georg Hamel提出的形式主义在有限维连续时间力学中被认为是一个有效而有力的工具，但对于无限多个自由度的系统来说，它还没有完全发展起来。该项目旨在系统地研究无限维环境中的形式主义，了解它的变分结构，它在无限维非完整约束系统动力学中的应用，包括受非完整约束的充满液体的刚体的稳定性，以及Lyapunov函数的构造。无限多个自由度的系统在工程和工业中有许多应用。例如，柔性结构和刚体与流体相互作用。这个项目开发了对这种复杂机械系统的行为进行建模和模拟所需的数学。这项关于系统力学的工作将进一步将数学和理论工程结合起来。该项目的结果将对多体动力学和复杂机械系统的长时间行为的理论分析、仿真、控制和研究有所帮助。

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.