Topics in Kinematics and Geometrical Optics: Tire Track Geometry and Billiard Models

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The proposed research consists of two parts: models of vehicle motion and tire track geometry, and models of elastic reflection in bounded regions and geometric optics. In the first part, the investigator will study a variety of concrete problems of vehicle kinematics. Direct applications involve pursuit problems, control of tractors with many trailers, and preventing driving hazard. The same mathematical methods apply to the study of other, seemingly unrelated, applied problems, including stability of floating bodies and modeling of the Josephson effect (Nobel Prize in 1973), important in the design of quantum-mechanical circuits for quantum computers. In the second part, the investigator will study fundamental problems of ray optics and models of mechanical systems with elastic collision, such as the ideal gas. Although ray optics provides only an approximation to a more precise wave optics, this approximation is accurate enough for many applications, including laser beam shaping, trapping rays of light and storing solar energy, control of light pollution, and invisibility. Modern technology makes it possible to manufacture materials with unusual reflecting and refracting properties and to create nearly ideal mirrors of complicated shape, thus realizing geometrical optical designs in glass, metal, and plastic. Most of the suggested problems admit both theoretical and computer experimental study, in many cases the latter being the first step toward the former. The investigator will actively involve undergraduate and graduate students in his research program.The proposed research has strong connections with the theory of completely integrable systems, continuous and discrete, and it relies on a variety of methods developed in this theory since the discovery of solitons in the 1960s and, in particular, on the theory of discrete differential geometry. For example, the problem of describing cylindrical bodies that float in equilibrium in all positions is intimately related with the description of solitons of the filament equation, a completely integrable system modeling the motion of fluid and gas vortices, and the cross-sections of all known solutions to this flotation problem are buckled rings (pressurized elastica), that also solve a variational problem extensively studied in the late 19th and early 20th centuries. In general, the theory of completely integrable systems makes it possible to find explicit solutions to the differential and difference equations that arise in the models; often these solutions are given in terms of elliptic functions.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.