Geometric mechanics examples simulation and theory

几何力学增强了传统的方法来力学的概念，从几何，很好地平衡与分析方法。虽然这个想法的根源可以追溯到力学的创始人，如雅可比，但在过去的几十年里，随着许多新思想和新联系的注入，它又重新兴起。例如，一个基本事实是，通常被认为属于纯拓扑的标准霍普夫纤维化，已经发生在刚体力学中（回到欧拉和拉格朗日）。在另一个例子中，几何力学的当前重要工具之一是某些坐标，其源自辛几何的各种线性分裂以及达布定理。这些特征在现代几何力学中是很自然的。对称的、节能的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，通常存在多个时间尺度，并且长时间动力学非常微妙。 相当感兴趣的是与对称群一致的解，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变化到螺旋运动和螺旋旋转运动。对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，离散（在时间，或在空间和时间）版本的系统，以及结构保持离散化的连续系统，是一个重要的正在进行的研究课题。 控制理论是几何力学的主要应用领域之一。虽然力学的学科已经建立了几个世纪，但全微分几何设置是相对较新的，并且在数学上是复杂的。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接近的性能。

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.