A new method for global analysis of manifolds asymptotic to partially hyperbolic tori in near-integrable Hamiltonian systems

摘要奖项：DMS-0072153 首席研究员：Michael Rudnev 将研究具有多个自由度的近可积哈密顿系统的全局动力学。该研究旨在进一步证明此类系统中存在非一般拓扑不稳定性的猜想，即通过共振实现的“阿诺德扩散”。第一个目标是获得扰动导致出现的各种几何对象的充分局部描述。这些物体可以通过它们的旋转矢量来表征。 首先，它们是低维部分双曲（须状）环面和渐近于这些环面的拉格朗日流形（须状）。一般来说，产生须状环面的所有旋转向量的集合形成第一类别的集合。其中“间隙”中的旋转数描述了所谓的“奥布里-马瑟集”。然而，在双自由度的情况下，后一个术语有些晦涩难懂。 将上述所有类型的许多对象合并到一个全局几何框架中，作为相空间上的图集，是第二个目标，也是主要目标。关于“奥布里-马瑟集”，这将需要对通过拉格朗日力学的变分方法和哈密顿方法的几何隐函数定理（例如 KAM）获得的结果进行相互解释。所提出的研究重点是将不稳定性作为复杂机械系统的一般特征，无论是太阳系还是多原子分子。为了描述这一点，我们需要知道系统的当前状态及其未来演化的方程。通常，由于其复杂性，人们无法在结合初始条件的情况下获得这些方程的解。否则，这是一个例外情况，该系统称为可积的。如果只考虑每个行星与太阳的相互作用，特别是忽略行星之间的相互引力和吸引力等，那么太阳系就是这样的。然而，运动定律是准确已知的，必须测量系统的初始状态，这会导致误差，无论多么小。不稳定的假设，又名“阿诺德扩散”，表明两个无限小的初始条件可能会无限地导致两个性质不同的场景。特别是，对于可积系统的小扰动，例如，这被认为是正确的。如果考虑到月球或木星对地球轨道运动的影响。几何力学提供了对所讨论的系统进行建模的合适数学环境。这项研究的技术动机来自分子化学和生物学，其中不稳定性需要数十亿年才能在宇宙尺度上发展，转化为几分之一秒。

AbstractAward: DMS-0072153Principal Investigator: Michael RudnevGlobal dynamics in near-integrable Hamiltonian systems withseveral degrees of freedom will be studied. The research aims toprogress towards proving the conjecture about the existence of ageneric topological instability in such systems, known as"Arnold's diffusion", effected via resonances. The first goal isto obtain an adequate local description of various geometricobjects that the perturbation causes to appear. These objects canbe characterized by their rotation vectors. First, they arelower-dimensional partially hyperbolic (whiskered) tori and theLagrangian manifolds (whiskers) asymptotic to these tori.Generally, the collection of all the rotation vectors yieldingwhiskered tori form a set of the first category. Rotation numbersin the "gaps" in it describe so-called "Aubry-Mather sets". Yetthe latter term is somewhat obscure beyond the case of twodegrees of freedom. Incorporating many of the above mentionedobjects of all types into a single global geometric frameworkeffected as an atlas over the phase space, is the second, and themain goal. Apropos of "Aubry-Mather sets", this will requiremutual interpretation of the results obtained via variationalmethods of Lagrangian mechanics and geometric implicit functiontheorems, such as KAM, of Hamiltonian approach.The proposed research focuses on instability as a generic featureof complex mechanical systems, be it the Solar system, or apolyatomic molecule. In order to describe such, one needs to knowthe present state of the system, and the equations of its futureevolution. Typically, due to their complexity, one cannot obtainthe solutions of these equations with the initial conditionsincorporated. Otherwise, and this is an exceptional case, thesystem is called integrable. Such would be the Solar system ifone considers only the interaction of each individual planet withthe Sun, disregarding in particular the mutual gravitationalattraction between the planets, etc. Yet the laws of motion areknown exactly, the initial state of a system has to be measured,which leads to errors, however small. The hypothesis ofinstability, alias "Arnold's diffusion," suggests that twoinfinitesimally different initial conditions may result in twoqualitatively different scenarios ad infinitum. In particular,this is believed to be true for small perturbations of integrablesystems, e.g. if one takes into account the influence of the Moonor Jupiter on the orbital motion of the Earth. The suitablemathematical enviroment to model the systems in question isprovided by geometrical mechanics. The technological motivationfor this research comes from molecular chemistry and biologywhere billions of years it would take the instability to developitself on a cosmic scale, translate into fractions of a second.