Geometric Numerical Discretizations of Gauge Field Theories and Interconnected Systems

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Practical engineered systems, such as electro-mechanical systems in human prosthetics, typically involve components described by coupled, distinct physical theories. This research project develops methods to easily model complicated systems by connecting simpler models together, which allows sophisticated mathematical and modeling tools to be more readily accessible to the engineering community. This will reduce the barrier to constructing high-fidelity mathematical models that can be used for rapid prototyping in the rational design process, leading to better, cheaper, and faster translation of conceptual designs into commercially realizable products. In addition to modeling, the research also involves designing controllers for interconnected systems. This is relevant to the control of distributed networks of autonomous aerial and underwater vehicles, which have applications to search and rescue, ocean exploration, and distributed sensor networks for both disaster monitoring and homeland security. To further facilitate the dissemination of this research to the engineering community, the investigator and his collaborators will develop a two-volume graduate textbook that will be accessible to a broad technical audience.Dirac and multi-Dirac mechanics and geometry provide a unified mathematical framework for describing Lagrangian and Hamiltonian mechanics and field theories, as well as degenerate, interconnected, and nonholonomic systems. Variational integrators yield geometric structure-preserving numerical methods that automatically preserve the symplectic form and momentum maps, and exhibit excellent long-time energy stability. This project studies far reaching generalizations of the variational integrator approach for discretizing covariant gauge theories, such as electromagnetism and general relativity, and interconnected systems, such as electrical circuits and multibody systems. This involves: (i) a systematic framework for constructing and analyzing multi-Dirac variational integrators for Lagrangian partial differential equations; (ii) the connection between discrete covariant and instantaneous formulations of covariant gauge theories, and group-equivariant finite-element spaces using spacetime finite-element exterior calculus and Lorentzian metric-valued geodesic finite-elements; (iii) discrete interconnection theory for Dirac mechanical systems and their associated control theory.