Periodic Orbits, Magnetic Fields, and Other Topics in Symplectic Geometry

大部分工作涉及哈密顿系统的周期轨道的存在性和数目。此外，研究人员还将研究出现在次黎曼几何和控制理论中的萨德定理的无限维版本的可能有效性。最后，研究人员建议计算与Duistermaat-Heckman公式的非阿贝尔形式和几何量子化有关的某些等变辛余边环。周期性轨道研究的激励例子是粒子在磁场中的动力学，此后称为磁动力学。黎曼曲面上的圆环流就是这种动力学的一个例子。这一流对这项工作至关重要，因为它是构造在有界能量面上没有周期轨道的哈密顿系统的基本构件。已知这种不存在现象存在的最小自由度是四个。调查人员将构造这个数字等于3的例子。磁动力学也是激发Arnod关于周期轨道数目下限的猜想的例子之一。在磁动力学的情况下，该猜想的许多例子仍然是开放的。调查人员计划回答其中一些问题。平面三体方程经过辛约化后，成为磁动力学的一个实例。(虚构的)“约化”粒子在一个三流形上运动(奇点对应于三重碰撞)，其点代表平面三角形的同余类。对磁动力学的一般性研究应该有助于理解“三体问题”中的一些未决问题。这方面的主要目的是证明关于减去碰撞的约化空间(三流形)表示任意给定的自由同伦类的闭轨的存在性。调查人员将研究各种系统可能发生的运动。他们将主要集中在两种类型的系统上，一种涉及电荷在磁场中运动，与聚变控制问题密切相关，另一种涉及行星或卫星根据牛顿定律运动。我们的主要兴趣是这些系统的周期轨道。周期轨道是一种无限重复的运动，在一段固定的时间后回到起点。一个熟悉的例子是每天早上太阳升起。周期轨道被认为是理解系统，特别是复杂混沌系统行为的关键。最近最令人惊讶的发展之一是发现了没有周期轨道的复杂系统。研究人员计划找到更多没有周期轨道的系统，同时寻找保证存在一定数量周期轨道的简单条件。这将是朝着对物理系统运动的普遍理解和控制迈出的重要一步。

The bulk of the proposed work concerns the existence and number of periodic orbits for Hamiltonian systems. In addition, the investigators will research the possible validity of an infinite-dimensional version of Sard's theorem arising in subRiemannian geometry and control theory. Finally, the investigators propose to compute certain equivariant symplectic cobordism rings relevant to non-Abelian versions of the Duistermaat-Heckman formula and to geometric quantization. The motivating example for the periodic orbit investigations is the dynamics of a particle in magnetic fields, henceforth referred to as magnetodynamics. The horocycle flow on a Riemann surface is an example of such a dynamics. This flow is crucial to the work since it is the basic building block for constructing Hamiltonian systems which have no periodic orbits on bounded energy surfaces. The smallest number of degrees of freedom for which this non-existence phenomenon is known to hold is four. The investigators will construct examples where this number is equal to three. Magnetodynamics was also one of the examples motivating Arnol'd's conjectures concerning lower bounds for the number of periodic orbits. A number of instances of the conjecture are still open in the magnetodynamic case. The investigators plan to answer some of these. The planar three-body equations, after symplectic reduction, becomes an instance of magnetodynamics. The (fictitious) ``reduced'' particle moves on a three-manifold (with a singularity corresponding to triple collisions) whose points represent congruence classes of planar triangles. The general investigations into magnetodynamics should be of help in understanding some of the open problems remaining in ``the three-body problem''. The main goal in this regard is to prove the existence of a closed trajectory representing any given free homotopy class for the reduced space (three-manifold) minus collisions. The investigators will be studying what motions are possible for various systems. They will be concentrating primarily on two types of systems, a type involving charges moving in magnetic fields, closely related to the problem of controlling fusion, and a type involving planets or satellites moving according to Newton's laws. Our main interest is in the periodic orbits for these systems. A periodic orbit is a motion which infinitely repeats itself, coming back to its starting point after some fixed period of time. A familiar example is the rising of the sun every morning. The periodic orbits are thought to be the key to understanding the behavior of systems, especially complicated chaotic systems. One of the most surprising recent developments has been the discovery of complicated systems with no periodic orbits. The investigators plan to find more systems with no periodic orbits and to simultaneously look for simple conditions which would guarantee the existence of a certain number of periodic orbits. This would be a significant step towards a general understanding and control of motions of physical systems.