Dynamics of mechanical systems with symmetry

对未来的预测和控制是科学技术的中心主题。我们都想知道未来会带来什么。我们还希望控制未来，以便要么消除不想要的状态(例如，防止疾病传播)，要么达到想要的状态(例如，将卫星的相机重定向到目标)。*这一提议涉及机械系统的动力学，即它们如何随着时间的推移而移动或演变。虽然计算机算法可以在一定程度上预测系统的未来演化，但对可能的动力学特征的理论理解可以改进预测，并为控制它的策略奠定基础。许多系统可以一起分析，如果它们表现出相似的特征。对于机械系统，这些特征可以指环境的几何形状(例如，在平面上的运动，而不是在空间中的运动)、物理定律(例如，能量守恒的运动，而不是有摩擦的运动)或对称性(例如，没有标记的完美圆球，从各个方向看都是相同的)。然后，系统被组织到特定的类中，在每个类中，我们试图描述可能的动态行为。特别令人感兴趣的是系统的稳态(其中系统保持“冻结”)和其他特殊轨迹，它们的稳定性，以及当某些物理特征或参数变化时可能发生的变化(“分叉”)。*我的目标是开发有用的方法和技术来捕捉具有对称性的节能机械系统的共同动力学特征。该提案考虑了两个子类：1)无约束力学系统，如分子的n粒子模型，或重力相互作用中的刚体(例如，小行星)；2)受约束的机械系统，如雪橇、滚动球和鳍箭或水下航行器，其建模假设是忽略耗散力(如摩擦力)。在每个子类中，我建议发展研究分叉和稳定性的方法。我将使用微分方程组(描述系统的演化)、微分几何和拓扑学(描述空间)和分叉理论(描述动力学的可能分支)的数学形式和方法。*我的研究增加了动力学系统的基本知识，并在机械工程(机械设备的控制和稳定)、空间科学(例如小行星系统的稳定性、月球-行星系统的自旋轨道耦合)和物理化学(例如多原子分子的旋振态)中有应用。**

Prediction and control of the future are central themes of science and technology. We all want to know what the future will bring. We also want to control the future so that either undesired states are eliminated (e.g., preventing the spreading of a disease), or desired states are attained (e.g., redirecting the camera of a satellite towards a target). *** This proposal concerns the dynamics of mechanical systems, that is, how they move or evolve over time. While the future evolution of a system can to some degree be predicted using computer algorithms, a theoretical understanding of possible dynamical features can improve prediction and also lay the groundwork for strategies to control it.  *** Many systems may be analyzed together, if they display similar characteristics. For mechanical systems, these characteristics may refer to the geometry of the environment (e.g., motion on a flat surface as opposed to motions in space), the physical laws (e.g., motions where the energy is conserved, as opposed to motions with friction), or symmetries (e.g. a perfectly round ball, without markings on it, looks the same from all directions). Systems are then organized in specific classes, and within each one, we attempt to describe the possible dynamical behaviours. Of particular interest are the steady states (in which the system remains "frozen") and other special trajectories, their stability, and the changes ("bifurcations'') that may occur as certain physical features or parameters vary.*** My aim is to develop methods and techniques useful for capturing the common dynamical features of energy conserving mechanical systems with symmetry. The proposal considers two sub-classes: 1) unconstrained mechanical systems such as n-particle models of molecules, or rigid bodies in gravitational interactions (for example, asteroids); and 2) constrained mechanical systems, such as sleighs, rolling spheres, and finned arrows or underwater vehicles modelled under the assumption that dissipative forces (for instance, friction) are neglected. Within each subclass, I propose to develop methods for studying bifurcation and stability. I will use the mathematical formalism and methods of differential equations (to describe the evolution of the system), differential geometry and topology (to describe the space) and bifurcation theory (to describe the possible branches of the dynamics).*** My research adds to the fundamental knowledge of dynamical systems and has applications in mechanical engineering (control and stabilization of mechanical devices), space science (e.g., stability of asteroidal systems, spin-orbit coupling of moon-planet systems) and physical chemistry (e.g., roto-vibrational states of polyatomic molecules). **