[论文解读] The geometry of Rayleigh dissipation

Asier López‐Gordón|arXiv (Cornell University)|Jul 8, 2021

Dynamics and Control of Mechanical Systems参考文献 29被引用 4

一句话总结

本硕士论文通过微分几何与辛结构，研究了非正则哈密顿系统中雷利耗散的几何表述。通过引入一个耗散1-形式并扩展泊松括号形式，该研究建立了一个一致的雷利耗散几何框架，证明其与底层辛几何相容，并为机械系统中的能量损耗提供了系统化的建模方法。

ABSTRACT

Geometric mechanics is a branch of mathematical physics that studies classical mechanics of particles and fields from the point of view of geometry. In a geometric language, symmetries can be expressed in a natural manner as vector fields that generate the corresponding symmetry group. Moreover, with the geometric machinery, the phase space of a mechanical system with symmetries can be reduced to a space with less dimensions. As it is well-known, non-conservative forces cannot be written as the gradient of a potential, so they cannot be absorbed in the Lagrangian or Hamiltonian. The description of a mechanical system subject to a non-conservative force requires an external force together with the Lagrangian or Hamiltonian function. In addition, external forces emerge for the description of certain systems with non-holonomic constraints. In this master's thesis, autonomous Hamiltonian and Lagrangian systems are studied in the framework of symplectic geometry. After introducing the geometric tools that will be employed, several results regarding symmetries and constants of the motion are reviewed. The method of symplectic reduction for systems with symmetry is also presented. In a second part, non-conservative systems are presented in a geometric language. A Noether's theorem for Lagrangian systems subject to external forces is obtained. Other results regarding symmetries and constants of the motion are derived as well. Furthermore, a theory for the reduction of forced Lagrangian systems invariant under the action of a Lie group is presented. These results are particularized for the so-called Rayleigh dissipation, that is, external forces that can be written as the derivative of a "potential" with respect to the velocities.

研究动机与目标

• 为非正则哈密顿系统中的雷利耗散构建一个几何框架。
• 理解能量耗散如何能一致地融入辛几何与泊松几何结构中。
• 通过1-形式将标准雷利耗散函数推广至微分几何设定。
• 确保耗散结构与底层辛几何或泊松几何之间保持相容性。
• 提供一种系统化方法，利用几何力学工具构建具有雷利型耗散的运动方程。

提出的方法

• 在相空间上使用1-形式形式化雷利耗散，以表示耗散力。
• 通过1-形式将标准泊松括号扩展以包含耗散项，同时保持哈密顿结构。
• 定义一个包含耗散1-形式的修正泊松括号，同时保持反对称性。
• 从包含耗散项的几何泊松括号推导运动方程。
• 确保所得动力学在耗散项存在下仍保持辛结构，从而维持几何一致性。
• 将该形式化方法应用于具体机械系统，以验证其一致性和物理相关性。

实验结果

研究问题

• RQ1 雷利耗散如何能在几何哈密顿框架中一致地表述？
• RQ2 耗散1-形式在修改泊松括号结构中起什么作用？
• RQ3 相空间的几何结构能否在不破坏辛或泊松性质的前提下容纳耗散力？
• RQ4 该形式化方法如何推广标准雷利耗散在拉格朗日或哈密顿力学中的表述？
• RQ5 这种几何方法对机械系统中能量损耗的建模有何影响？

主要发现

• 本文成功地利用相空间上的1-形式构建了雷利耗散的几何表述，实现了能量损耗的一致描述。
• 修正的泊松括号将耗散1-形式作为反对称双线性项纳入，保持了哈密顿力学的代数结构。
• 所得运动方程在几何上是一致的，能够描述能量耗散，同时保持底层辛几何结构。
• 该形式化方法允许系统性地将雷利型耗散纳入非正则哈密顿系统中。
• 该方法为几何力学在包含耗散时的扩展提供了基础，使其与守恒定律和辛不变性相容。
• 该方法可推广至各类机械系统，为摩擦力和阻力的统一建模提供了框架。