Dynamical and Statistical Methods Applied to Hamiltonian Systems

该项目致力于研究动态系统，这些动态系统代表随时间演变的复杂物理系统的数学模型。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，以及系外行星系统的演化），以及高能物理学（例如，粒子加速器和等离子体约束装置）。这项研究的目标是分析和预测从动力学中出现的复杂行为模式，并获得底层系统长期演化的统计信息。预计所获得的结果将有助于推进科学发现和促进创新技术，例如设计更高效的卫星轨道、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，并有助于培养多样化，高素质的数学劳动力。在基础层面上，该项目所研究的数学模型可以在哈密顿动力学的形式主义框架内。一个典型的问题是理解哈密顿系统中轨道的长期行为。这个问题是在近可积哈密顿系统（阿诺德扩散问题）的背景下，在N体问题的背景下，并在随机扰动哈密顿系统的背景下进行研究。研究的主要目的是获得定性和定量信息的典型轨道在这样的系统。感兴趣的信息涉及的距离，速度和轨道的稳定性，测量/豪斯多夫维度的轨道访问一个规定的目标序列的初始条件集，从初始数据集的轨道演变的统计分布。 为此，研究者和他的同事们开发了结合联合收割机几何、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NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.