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

大部分的工作涉及到哈密顿系统的周期轨道的存在性和数量。此外，研究人员将研究在次黎曼几何和控制理论中产生的Sard定理的无限维版本的可能有效性。 最后，研究人员建议计算某些等变辛配边环相关的非阿贝尔版本的Duistermaat-Heckman公式和几何量化。周期性轨道研究的激励性例子是磁场中粒子的动力学，此后称为磁动力学。黎曼曲面上的单周流就是这种动力学的一个例子。这个流程是至关重要的工作，因为它是基本的构建块构造哈密顿系统，有界能量表面上没有周期轨道。已知这种不存在现象的最小自由度为4。研究人员将构建这个数字等于3的示例。磁动力学也是一个例子，激励阿诺尔德的apturtures关于下界的数量周期轨道。在磁动力学的情况下，这个猜想的一些例子仍然是开放的。调查人员计划回答其中的一些问题。平面三体方程经辛约化后，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有助于理解“三体问题”中的一些悬而未决的问题。在这方面的主要目标是证明存在一个封闭的轨迹表示任何给定的自由同伦类减少空间（三流形）减去碰撞。 研究人员将研究各种系统可能的运动。他们将主要集中在两种类型的系统上，一种涉及在磁场中移动的电荷，与控制聚变的问题密切相关，另一种涉及根据牛顿定律移动的行星或卫星。我们的主要兴趣是在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它的起点。一个熟悉的例子是每天早上太阳的升起。周期轨道被认为是理解系统行为的关键，尤其是复杂的混沌系统。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。研究人员计划寻找更多没有周期轨道的系统，同时寻找保证一定数量周期轨道存在的简单条件。这将是对物理系统运动的普遍理解和控制的重要一步。

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.