Validated Computational Methods in Global Analysis and Applications to Celestial Mechanics

天体力学是数学物理学的一个分支，研究行星、小行星和彗星的运动，并促进安全高效的太空任务的设计。 虽然天体力学的知识基础属于古代，但由于二十世纪太空探索的出现，该领域比以往任何时候都更加重要。 动力系统理论的现代发展提供了深刻的见解，到目前为止，一些太空任务已经将启发性的数学转化为成功的实践。 这些发展的关键是理解某些称为不变流形的标志性物体，并研究它们之间的联系。 在实际应用中，发现这些标志的唯一方法是通过数值计算。 由于这些计算的全局性质，涉及误差和准确性的问题既重要又微妙。 在本项目中，这些问题得到了非常仔细的处理，最终形成了一个描述所有离散化和截断误差的数学严格框架。 研究结果为科学计算的实践者提供了量化计算误差的严格数学工具，同时为数学家提供了证明定理的新方法。 研究人员将这个新框架应用于天体力学中那些无法进行早期分析的问题。 这样做需要理论的分析和计算方面的实质性进步。 由此产生的基础设施将免费提供，并可用于研究其他复杂的数学模型。研究人员研究计划的中心主题是非线性系统全局分析的数值和分析方法的统一。 该项目建立在研究者早期研究成功的基础上，展示了其可扩展性和并行性。 该项目的重点是引力 N 体问题的全局分析，该领域有许多长期悬而未决的理论问题和许多实际应用的机会。 其中一个项目以数字方式研究压缩相空间上的碰撞动力学，而另一个项目则回答了有关从拉格朗日多边形中心配置分叉的周期轨道全局分支的问题。 该项目的一个更具计算性的方面开发了一种新的集群计算方法和不变流形图集的归档，为空间任务设计提供了对动力学和新可能性的更深入的理解。 自始至终，这项工作都遵循这样一个原则：平衡、周期轨道以及它们之间的异宿/同宿连接等基本不变集是理解复杂动力学的基本构件。 由此产生的计算机辅助分析可用于研究其他问题，例如涉及测地线流动、紧凑流形上的机械系统、化学和电动力学的问题。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。

Celestial mechanics is the branch of mathematical physics that studies the motion of planets, asteroids, and comets, and that facilitates design of safe and efficient space missions. While the intellectual foundations of celestial mechanics belong to antiquity, the field is more relevant than ever thanks to the advent of space exploration in the twentieth century. Modern developments in dynamical systems theory provide deep insights, and by now several space missions have transformed inspired mathematics into successful practice. The key to these developments is to understand certain landmark objects known as invariant manifolds, and to study connections between them. In realistic applications the only way to discover these landmarks is through numerical computations. Due to the global nature of these computations, questions involving errors and accuracy are both important and subtle. These issues are treated with great care in the present project, ultimately leading to a mathematically rigorous framework describing all discretization and truncation errors. The results provide practitioners of scientific computing with mathematically rigorous tools for quantifying computational errors, while providing mathematicians with new methods for proving theorems. The investigator applies this new framework to problems in celestial mechanics that have resisted earlier analysis. Doing so requires substantial advancement of both analytical and computational aspects of the theory. The resulting infrastructure will be made freely available, and can be used to study other complex mathematical models.A central theme of the investigator's research program is the unification of numerical and analytical methods for global analysis of nonlinear systems. This project builds on the success of the investigator's earlier research, demonstrating its scalability and parallelizability. The focus of the project is global analysis in gravitational N-body problems, an area with many longstanding unanswered theoretical questions and many opportunities for practical application. One project numerically studies collision dynamics on a compactified phase space, while another answers questions about global branches of periodic orbits bifurcating from the polygonal central configurations of Lagrange. A more computational aspect of the project develops a new approach to cluster computing and archiving of invariant manifold atlases, providing deeper understanding of the dynamics and new possibilities for space mission design. Throughout, the work is guided by the principle that basic invariant sets like equilibria, periodic orbits, and heteroclinic/homoclinic connections between them are the fundamental building blocks for understanding complicated dynamics. The resulting computer-assisted analysis is useful for studying other problems such as those involving geodesic flows, mechanical systems on compact manifolds, chemistry, and electrodynamics.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.