Nonlinear Dynamics with Applications to Physical Systems

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The project involves three directions. The first one deals with so-called twist maps -- a basic building block in studying almost every kind of moving or stationary physical system where friction is negligible. As an illustrating example, a twist map can arise from the equation governing the distribution of electrons in models of crystal lattices. Prior work by the investigator suggests an unexpected mathematical effect; interpreted for the case of a crystal lattice, this general mathematical result amounts to an unexpectedly low electric resistance. The mechanism of this effect is not understood, and the investigator works to develop a mathematical theory explaining it. The second part of the project deals in particular with the recently discovered "ponderomotive magnetism." Recall that any change in an electric field gives rise to a magnetic field, as has been known since Faraday. Surprisingly, a superficially similar effect takes place in mechanics (without electric fields present), as was discovered only recently by the investigator and collaborators. For example, particles in the gravitational field of a spinning (non-spherical) asteroid behave as if they were charged and in the presence of a magnetic field -- with neither charge nor the field actually present. This is a mysterious and fundamental mathematical phenomenon. A goal of this project is to develop a geometrical explanation of this effect using tools of differential geometry. The third part of the project explores the recently discovered connection between Hill's equation (ubiquitous in innumerable problems in physics and engineering) on the one hand, and the "tire track" problem on the other. The main value of this topic lies in unifying and simplifying two seemingly distinct areas of mathematics, resulting in richer understanding of both, and perhaps in new discoveries.The project involves three directions unified by use of geometry and analysis with physical motivation. The first direction involves area-preserving twist maps of the cylinder; such maps arise in many settings, e.g. the Frenkel-Kontorova (F-K) model of a crystal lattice. The investigator studies whether the robustness of periodic orbits with respect to certain perturbations (e.g., addition of a tilt to the periodic potential in the F-K example) increases with the number of harmonics in the map. Two entirely different (and slightly simpler) objects are known to show an analogous effect: circle mappings and Mathieu-type Hill's equations. If successful, the project would add a new class of systems to the latter two classes. The second direction of research includes the goal to explain geometrically the recently discovered "ponderomotive magnetism," in which rapid time-periodic oscillation of a conservative force field gives rise to a Faraday-like effect discovered by the investigator and collaborators through a normal-form computation: particles in such a field behave as if they possessed electric charge in the presence of a magnetic field, though neither an electric charge nor a magnetic field is present. One of the goals is to explain the geometry underlying this effect. The third direction explores questions raised by the connection between Hill's equation (arising in many areas of mathematics and applications) on the one hand and the geometrical "tire track" problem on the other.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.