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

AbstractAward：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。这项研究的目的是进一步证明关于在这种系统中存在无能量拓扑不稳定性的猜想，即所谓的“阿诺德扩散”，通过共振产生影响。第一个目标是获得一个适当的本地描述的各种geometricobjects的扰动原因出现。这些物体可以用它们的旋转矢量来表征。 首先，它们是低维的部分双曲（须状）环面和拉格朗日流形（须状）渐近于这些环面。一般来说，所有产生须状环面的旋转向量的集合形成第一类。在它的“间隙”中的旋转数描述了所谓的“Aubry-Mather集”。然而，后一个术语在两个自由度之外有些模糊。 将上述提到的所有类型的许多对象表示成一个单一的全局几何框架，作为相空间上的图谱，这是第二个目标，也是主要目标。关于“Aubry-Mather集”，这将需要通过拉格朗日力学的变分方法和几何隐函数定理，如KAM，Hamiltonian approach.The建议的研究重点是不稳定性作为复杂力学系统的一般特征，无论是太阳系，还是多原子分子得到的结果相互解释。为了描述这种情况，我们需要知道系统的当前状态，以及它未来演化的方程。通常情况下，由于其复杂性，人们不能得到这些方程的解与初始条件纳入。否则，这是一个例外的情况，这个函数被称为可积的。这将是太阳系，如果只考虑每个单独的行星与太阳的相互作用，特别是忽略行星之间的相互引力，等等。然而，运动定律是确切的，一个系统的初始状态必须测量，这会导致误差，无论多么小。不稳定性假说，又称“阿诺德扩散”，认为两个极小不同的初始条件可能导致两个性质上不同的无限情景。特别是，这被认为是正确的小扰动的积分系统，例如，如果一个考虑的影响，月球或木星上的轨道运动的地球。几何力学提供了建立这类系统模型的数学环境。这项研究的技术动力来自分子化学和生物学，在那里，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.