Constrained geometric mechanics: theory and applications

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This proposal focuses on the applications of geometric methods to constrained dynamics of elastic structures. In the first part, we shall apply the methods of geometric mechanics, and corresponding constraints, to systems involving point wise, non-holonomic constraints, with a particular example of elastic rods in contact. This problem will be tackled using the recently developed methods of geometric Lagrange-d'Alambert's methods. The proposal will explore interesting and new directions provided by this theory, such as highly complex dynamics of rods caused by the contact, generalizations to discrete rods, symmetry reductions, steady and unsteady exact solutions, and others. In the second part, we will explore tree-like structures with linked branches, or, alternatively, an array of linked tree-like branches, evaluated from the point of view of geometry. These problems will be treated with methods geometric mechanics of iterative semidirect product groups with holonomic constraints. We shall study the effect of constraints on the geometric structures recently derived by the PI for such objects, such as conservation laws and Poisson brackets. The problem of tree-like structure dynamics will also be used in the development of a nanomechanical sensor and a macroscale tree-like energy harvesting device. Mathematical questions outlined above play a crucial role in the design of both devices. The sensor will be developed together with the PI's collaborators at CSU and will utilize a direct visualization of nanoscale motion using EUV pulsed laser. The tree-like energy harvester is being developed with the PI's collaborators in Japan (U of Kyoto and Shiga Prefecture). The PI's expertise in the dynamics and modeling, nano-fabrication facilities at NINT on campus, and collaboration with the EUV laser group at CSU and Japan makes this project likely to succeed. The project will focus heavily on the training of graduate students and postdocs in the development of tools of modern mathematics and subsequent application of these tools to modern, practical problems. While the project is closely involved with experiments, support is sought only for the theory part.