Validated Computational Methods in Global Analysis and Applications to Celestial Mechanics

天体力学是数学物理学的分支，研究行星、小行星和彗星的运动，有助于设计安全有效的太空任务。 虽然天体力学的知识基础属于古代，但由于二十世纪世纪太空探索的到来，该领域比以往任何时候都更加相关。 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被称为不变流形的地标性物体，并研究它们之间的联系。 在实际应用中，发现这些地标的唯一方法是通过数值计算。 由于这些计算的全局性，涉及误差和准确性的问题既重要又微妙。 在本项目中，这些问题都得到了非常仔细的处理，最终导致了一个数学上严格的框架来描述所有的离散化和截断误差。 这些结果为科学计算的从业者提供了量化计算错误的数学严格工具，同时为数学家提供了证明定理的新方法。 研究人员将这个新的框架应用于天体力学中那些抵抗早期分析的问题。 这样做需要理论的分析和计算方面的实质性进展。 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。研究人员的研究计划的中心主题是统一的数值和分析方法的非线性系统的全局分析。 该项目建立在研究人员早期研究的成功基础上，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，这是一个有许多长期悬而未决的理论问题和许多实际应用机会的领域。 一个项目数值研究碰撞动力学的紧致相空间，而另一个回答问题的全球分支的周期性轨道分叉的多边形中心配置的拉格朗日。 该项目更多的计算方面开发了一种新的集群计算方法和不变流形地图集的存档，提供了对空间使命设计的动力学和新的可能性的更深入的理解。 在整个过程中，工作是由基本不变集的原则，如平衡，周期轨道，和它们之间的异宿/同宿连接是理解复杂动力学的基本构建模块。 由此产生的计算机辅助分析是有用的研究其他问题，如涉及测地线流，机械系统的紧凑流形，化学和electrodynamics.This奖项反映了NSF的法定使命，并已被认为是值得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的支持。

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.