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Dynamical and Statistical Methods Applied to Hamiltonian Systems

应用于哈密顿系统的动力学和统计方法

基本信息

批准号：
1814543
负责人：
Pablo Roldan Gonzalez
金额：
$ 42.5万
依托单位：
Yeshiva University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-08-01 至 2023-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1814543&HistoricalAwards=false
关键词：
Dynamical Statistical Methods Applied Hamiltonian

项目摘要

This project is devoted to research on dynamical systems that represent mathematical models of complex physical systems evolving over time. Dynamical systems of interest include mechanical systems (e.g., motion of satellites under the influence of gravity, solar pressure, electromagnetic forces, etc.), astrophysics system (e.g., expansion of the early universe, motion near a Black Hole, and evolution of exoplanetary systems), and high energy physics (e.g., particle accelerators, and plasma confinement devices). The goal of this research is to analyze and predict patterns of complex behavior emerging from the dynamics, and to obtain statistical information on the long-term evolution of the underlying systems. The obtained results are expected to contribute towards advancing scientific discovery and promoting innovative technologies, for example to design more efficient satellite orbits, to optimize performance of particle accelerators, and to understand more about the formation and evolution of the universe and of our solar system. Additionally, this project integrates research with education, and contributes towards preparing a diverse, highly qualified mathematics workforce.At the fundamental level, the mathematical models to be investigated by this project can be framed in the formalism of Hamiltonian dynamics. A quintessential question is to understand the long-term behavior of orbits in Hamiltonian systems. This question is investigated in the context of nearly-integrable Hamiltonian systems (the Arnold diffusion problem), in the context of the N-body problem, and in the context of randomly perturbed Hamiltonian systems. The main objective of the research is to obtain qualitative and quantitative information on typical orbits in such systems. The information of interest concerns the distance traveled, speed and stability of orbits, the measure/Hausdorff dimension of the set of initial conditions for orbits that visit a prescribed sequence of targets, and the statistical distribution of orbits evolving from an initial data set. For this purpose, the investigator and his colleagues develop innovative methodologies that combine geometric, topological, and statistical methods, which can be applied to concrete examples, and can be implemented in high-precision numerical computations and computer assisted proofs.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目致力于研究动态系统，该系统代表复杂物理系统随时间演变的数学模型。感兴趣的动力系统包括机械系统（例如，卫星在重力、太阳压力、电磁力等影响下的运动）、天体物理系统（例如，早期宇宙的膨胀、黑洞附近的运动和系外行星系统的演化）和高能物理（例如，粒子加速器和等离子体约束装置）。本研究的目的是分析和预测动态中出现的复杂行为模式，并获得底层系统长期演变的统计信息。预计获得的结果将有助于推进科学发现和促进创新技术，例如设计更有效的卫星轨道，优化粒子加速器的性能，以及更多地了解宇宙和太阳系的形成和演化。此外，该项目将研究与教育相结合，有助于培养多样化、高素质的数学人才。在基本层面上，这个项目要研究的数学模型可以在哈密顿动力学的形式主义框架。一个典型的问题是理解哈密顿系统中轨道的长期行为。本文在近可积哈密顿系统（阿诺德扩散问题）、n体问题和随机扰动哈密顿系统的背景下研究了这个问题。研究的主要目的是获得这类系统中典型轨道的定性和定量信息。感兴趣的信息涉及轨道的行进距离、速度和稳定性，访问指定目标序列的轨道的初始条件集的测量/豪斯多夫维数，以及从初始数据集演变的轨道的统计分布。为此，研究者和他的同事们开发了结合几何、拓扑和统计方法的创新方法，这些方法可以应用于具体的例子，并且可以在高精度数值计算和计算机辅助证明中实现。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。