Stability and bifurcations of relative equilibiria in simple mechanical systems

许多机械系统可以建模为哈密顿系统，其中动力学由函数（哈密顿量）和相空间的几何（泊松或辛）决定。理解几何结构的含义是分析运动的核心，特别是当存在对称性时。机械系统的一个重要子类是简单机械系统，其中运动由动力学项和势项决定。机械系统中连续对称性的存在对动力学有着深远的影响。在这样的系统中，相对均衡是最简单的非平凡解。它们是沿对称群轨道解匀速运动的解。例如，牛顿三体问题的相对平衡是这样一种运动：每一个物体都围绕质心在一个圆形轨道上均匀地运动，而三个质点形成的形状不变。与非对称系统的平衡类似，相对平衡是更复杂动力学的组织中心，它们可以用作微扰理论的基点。我的研究计划有两个主要目标。首先是为研究对称哈密顿系统，特别是简单机械系统的相对平衡点附近的动力学提供了建设性的理论。“建构性理论”在这里是指可以应用于具体实例的理论。在对称性的奇异点附近需要特别注意，在那里系统参数的轻微扰动可能引起行为的根本变化。例如，这种情况出现在对称破缺现象中，即相对平衡的对称性发生了变化。许多传统的方法可能需要修改。本提案的第二个主要目标是开发与分子物理和化学（分子相对平衡的分岔），天体力学（N-体问题中相对平衡的非线性稳定性，N>2）和天体动力学（小行星对动力学）相关的应用。

Many mechanical systems can be modeled as Hamiltonian systems, in which the dynamics is determined by a function (the Hamiltonian) and the geometry of the phase space (Poisson or symplectic). Understanding the implications of the geometric structure is central to the analysis of the motion, especially when symmetries are present. An important subclass of mechanical systems is that of simple mechanical systems, where the motion is determined by a kinetic term and a potential term. The presence of continuous symmetries in a mechanical system has profound implications for the dynamics. In such systems, relative equilibria are the simplest non-trivial solutions. They are solutions that move with constant velocity along the symmetry group orbit solution. For instance, a relative equilibrium of the Newtonian 3-body problem is a motion in which each body moves uniformly in a circular orbit around the centre of mass while the shape formed by the 3 mass points is unchanged. Similarly to equilibria for non-symmetric systems, relative equilibria are organizing centres of more complicated dynamics and they can be used as base points for perturbation theory. My research proposal has two main goals. The first is to add to the constructive theory needed to study the dynamics near relative equilibria for symmetric Hamiltonian systems, and in particular, simple mechanical systems. ``Constructive theory" here means a theory which can be applied to concrete examples. Special attention is required near singular points of the symmetry, where a slight perturbation of the system parameters might induce radical changes in behavior. For instance, this situation appears in symmetry breaking phenomena, where the symmetry of the relative equilibrium changes.  Many traditional methods may have to be modified. The second main goal of this proposal is to develop applications with relevance to molecular physics and chemistry (bifurcations from relative equilibria for molecules), celestial mechanics (nonlinear stability of relative equilibria in the N-body problem, N>2) and astrodynamics (dynamics of asteroid pairs).