[论文解读] Geometrical aspects of contact mechanical systems and field theories

Xavier Rivas Guijarro|arXiv (Cornell University)|Apr 25, 2022

Dynamics and Control of Mechanical Systems被引用 4

一句话总结

本论文利用接触结构与k-接触结构，为耗散性力学与场论构建了一个几何框架，推广了辛几何与k-辛几何的形式体系。论文提出了k-接触系统的Skinner–Rusk形式体系，推导出Euler–Lagrange方程与哈密顿方程，并将该理论应用于阻尼谐振子与阻尼电磁波等实例，为力学与场论中的耗散现象建立了统一的几何描述。

ABSTRACT

Many important theories in modern physics can be stated using differential geometry. Symplectic geometry is the natural framework to deal with autonomous Hamiltonian mechanics. This admits several generalizations for nonautonomous systems, both regular and singular. Some of these extensions are the subject of this thesis. Recently there has been a growing interest in studying dissipative mechanical systems from a geometric perspective using contact geometry. In this thesis we review what has been done in this topic and go deeper, studying symmetries and dissipated quantities of contact systems, and developing the Skinner-Rusk formalism for these systems. With regard to classical field theory, we introduce the notion of k-precosymplectic manifold and use it to give a geometric description of singular nonautonomous field theories. We also devise a constraint algorithm for these systems. Field theories with damping are described through a modification of the De Donder-Weyl Hamiltonian field theory. This is achieved by combining contact geometry and k-symplectic structures, resulting in the k-contact formalism. We introduce two notions of dissipation laws, generalizing the concept of dissipated quantity. These developments are also applied to Lagrangian field theory. The Skinner-Rusk formulation for k-contact systems is described in detail and we show how to recover the Lagrangian and Hamiltonian formalisms from it. Throughout the thesis we present several examples in mechanics and field theory. The most remarkable mechanical examples are the damped harmonic oscillator, the motion in a gravitational field with friction, the parachute equation and the damped simple pendulum. In field theory, we study the damped vibrating string, the Burgers' equation, the Klein-Gordon equation and its relation with the telegrapher's equation, and the Maxwell's equations with dissipation.

研究动机与目标

• 通过接触几何将几何力学推广至耗散系统，解决非保守系统缺乏统一几何形式体系的问题。
• 将Skinner–Rusk形式体系推广至k-接触系统，实现耗散存在下拉格朗日与哈密顿描述的统一。
• 为具有耗散的奇异非自治场论开发k-预辛同调约束算法。
• 提出结合接触几何与k-辛结构的k-接触形式体系，用于建模耗散场论。
• 定义并分析接触与k-接触系统中的耗散律与对称性，将守恒量概念推广至耗散情形。

提出的方法

• 利用接触流形与k-接触结构建模耗散性力学系统，推广辛几何以描述保守系统。
• 在扩展的庞特里亚金丛上应用Skinner–Rusk形式体系，结合预接触典范结构，推导动力学方程。
• 引入k-预辛同调几何作为具有耗散的奇异非自治场论的几何框架。
• 为k-预辛同调系统开发约束算法，以处理具有阻尼的奇异场论。
• 结合De Donder–Weyl理论与接触几何，构建场论的k-接触形式体系。
• 定义两种耗散律的概念，将守恒量概念推广至耗散系统。

实验结果

研究问题

• RQ1 如何系统性地应用接触几何来建模具有对称性及守恒/耗散量的耗散性力学系统？
• RQ2 k-接触系统的Skinner–Rusk形式体系的正确几何表述是什么？它如何同时恢复拉格朗日与哈密顿形式体系？
• RQ3 k-预辛同调结构如何描述具有阻尼的奇异非自治场论？
• RQ4 对称性在k-接触系统中起什么作用？它们与耗散量之间有何关系？
• RQ5 k-接触形式体系能否推广至场论？它如何描述如阻尼波动方程或含耗散的麦克斯韦方程等耗散场方程？

主要发现

• k-接触系统的Skinner–Rusk形式体系成功恢复了拉格朗日与哈密顿形式体系，为耗散动力学提供了统一的几何描述。
• 开发并应用了k-预辛同调约束算法于奇异场论，包括仿射与二次型拉格朗日量，实现了对耗散场系统中约束的分析。
• k-接触形式体系成功应用于场论，包括阻尼振动弦、Burgers方程及含耗散的麦克斯韦方程，展示了能量损耗的一致几何建模。
• 引入了两种新的耗散律概念，将守恒量推广至耗散系统，实现了对能量递减不变量的识别。
• 通过详细实例验证了该理论：阻尼谐振子、阻尼摆以及Klein–Gordon/电报方程，展示了该形式体系的预测能力。
• k-接触形式体系为场论中Herglotz型变分原理提供了几何基础，提示了向奇异耗散场论的多接触形式体系发展的可能路径。