Geometric mechanics examples simulation and theory

几何力学通过包含几何学的思想，与分析方法很好地平衡，增强了传统的力学方法。虽然这个想法的根源可以追溯到力学的创始人，如Jacobi，但在过去的几十年里，随着许多新想法和新联系的注入，这个想法又复苏了。例如，一个基本的事实是，通常被认为属于纯拓扑的标准霍普夫振动，已经出现在刚体力学中（追溯到欧拉和拉格朗日）。在另一个例子中，当前几何力学的重要工具之一是一定的坐标，它由辛几何的各种线性分裂和达布定理推导而来。这些特征在几何力学的现代背景下是很自然的。对称的节能系统构成了这个主题的组织中心。在这些理想模型中，没有摩擦，通常存在多个时间尺度，并且长时间动力学非常微妙。与对称群对齐的解是相当有趣的，这被称为相对平衡。这些解的物理形式取决于对称性，并且可以从圆周运动，螺旋运动和螺旋旋转运动变化。这些系统的模拟最好通过保留系统结构的数值算法来完成。因此，系统的离散（在时间上，或在空间和时间上）版本，以及连续系统的结构保持离散化是一个重要的正在进行的研究课题。控制理论研究非保守系统，是几何力学的主要应用领域之一。虽然力学学科已经建立了几个世纪，但完整的微分几何设置相对较晚，并且在数学上很复杂。在今天的几何力学中，我们找到了传统方法无法解决的动态相关问题的新颖答案，并且我们可以系统地构建新的数值算法，这些算法具有传统算法无法达到的性能。

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.