Validated Computational Methods in Global Analysis and Applications to Celestial Mechanics

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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.