Geometric mechanics examples simulation and theory

几何力学增强了传统的力学方法，吸收了几何学的思想，与分析方法很好地平衡了。虽然这个想法的根源可以追溯到像雅各比这样的力学创始人，但在过去的几十年里，随着许多新想法和联系的注入，这种想法又卷土重来。例如，通常被认为属于纯拓扑学的标准Hopf纤维运动已经出现在刚体力学中(追溯到欧拉和拉格朗日)，这是一个基本事实。在另一个例子中，当前几何力学的重要工具之一是某些坐标，它是由辛几何的各种线性分裂和达布定理一起得出的。这些特征在几何力学的现代背景下是很自然的。对称的能量守恒系统形成了主体的组织中心。在这些理想化的模型中，没有摩擦力，通常存在多个时间尺度，并且长时间动力学非常微妙。有一些相当感兴趣的解是与对称群对齐的解，称为相对平衡。这些解的物理形式取决于对称性，可以从圆周运动到螺旋运动和螺旋旋转运动。对这些系统的模拟最好是通过保留系统结构的数值算法来完成。因此，系统的离散(时间、空间和时间)形式以及连续系统的保结构离散是一个重要的研究课题。控制理论产生了非保守系统，是几何力学的主要应用领域之一。虽然力学这门学科已经建立了几个世纪，但完全微分几何环境是相对较新的，在数学上也很复杂。在今天的几何力学中，我们发现了更多传统方法无法解决的动力学相关问题的新答案，我们可以系统地构造具有传统算法无法达到的性能的新的数值算法。

Geometric Mechanics enhances the traditional approach to mechanics by the inclusion of ideas from geometry, nicely balanced with analytical methods. While this idea has its roots going back to the founders of mechanics, such as Jacobi, there has been a resurgence in the past few decades, with the infusion of many new ideas and links. For instance, it is a basic fact that the standard Hopf fibration, usually thought of as belonging to pure topology, already occurs in rigid body mechanics (going back to Euler and Lagrange). In another instance, one of the current important tools of Geometric Mechanics is certain coordinates, derived from symplectic geometry's various linear splittings together with the Darboux theorem. These sorts of features are natural in the modern setting of Geometric Mechanics. The symmetric, energy-conserving systems form an organizing center for the subject.  In these idealized models, friction is absent, multiple time scales are usually present, and the long time dynamics is very delicate.  Of considerable interest are solutions which are aligned with the symmetry group, which are called relative equilibria.  The physical form of these solutions depends on the symmetry, and can vary from circular motions, to screw motions and screw-spinning motions. Simulation of these systems is best done by numerical algorithms that preserve the systems' structures. Consequently, discrete (in time, or in space and time) versions of the systems, as well as structure preserving discretizations of the continuous systems, are an important ongoing research topic.  Control theory results in nonconservative systems, and is one of the main application areas of Geometric Mechanics. While the subject of mechanics has been established for centuries, the full differential geometric setting is relatively recent, and mathematically sophisticated. In Geometric Mechanics today, we find novel answers to dynamically relevant questions that more traditional approaches cannot ask, and we can systematically construct new numerical algorithms which have performance that traditional algorithms cannot approach.