论文标题
接触机械系统和现场理论的几何方面
Geometrical aspects of contact mechanical systems and field theories
论文作者
论文摘要
现代物理学中的许多重要理论可以使用差别几何形状来说明。符号几何形状是处理自主哈密顿力学的自然框架。这承认了对规则和单数的非自主系统的几种概括。这些扩展中的一些是本文的主题。 最近，使用接触几何形状从几何学角度研究耗散机械系统的兴趣越来越大。在本论文中，我们回顾了该主题中所做的事情，并进行了更深入的研究，研究了对称性和消散数量的接触系统，并为这些系统开发了Skinner-Rusk形式。 关于经典的田地理论，我们介绍了k-蛋白质歧管的概念，并使用它来对单数非自治理论进行几何描述。我们还为这些系统设计了约束算法。 通过修改De Donder-Weyl Hamiltonian田间理论来描述具有阻尼的现场理论。这是通过结合接触几何形状和K-旋转结构来实现的，从而导致K-contact形式主义。我们介绍了两个耗散法的概念，从而概括了耗散数量的概念。这些发展也适用于拉格朗日现场理论。详细描述了K-Contact系统的Skinner-Rusk配方，我们展示了如何从中恢复Lagrangian和Hamiltonian形式。 在整个论文中，我们介绍了力学和现场理论的几个例子。最显着的机械示例是阻尼的谐波振荡器，带有摩擦的引力场中的运动，降落伞方程和阻尼的简单摆。在现场理论中，我们研究了抑制的振动串，汉堡方程，克莱恩 - 戈登方程及其与电报者方程的关系以及随着耗散的麦克斯韦方程。
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.
