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

该研究包括两部分：车辆运动模型和轮胎轨迹几何模型，有界区域弹性反射模型和几何光学模型。在第一部分中，调查者将研究车辆运动学的各种具体问题。直接应用包括追逐问题、多拖车拖拉机的控制和预防驾驶危险。同样的数学方法也适用于其他看似无关的应用问题的研究，包括浮体的稳定性和约瑟夫森效应的建模(1973年诺贝尔奖)，这些问题在量子计算机的量子力学电路设计中很重要。在第二部分中，研究人员将研究射线光学的基本问题和具有弹性碰撞的机械系统的模型，例如理想气体。尽管光线光学只提供了更精确的波动光学的近似值，但这种近似值对于许多应用都足够精确，包括激光整形、捕获光线和存储太阳能、光污染控制和不可见性。现代技术使制造具有特殊反射和折射性能的材料成为可能，并创造出形状复杂的近乎理想的镜子，从而实现玻璃、金属和塑料的几何光学设计。大多数提出的问题既承认理论研究，也承认计算机实验研究，在许多情况下，后者是迈向前者的第一步。这位研究人员将积极邀请本科生和研究生参与他的研究计划。建议的研究与完全可积系统理论、连续和离散系统理论有很强的联系，它依赖于自20世纪60年代发现孤子以来在这一理论中发展起来的各种方法，特别是离散微分几何理论。例如，描述在所有位置平衡漂浮的圆柱体的问题与描述细丝方程的孤子密切相关，丝方程是一个完全可积的系统，模拟流体和气体涡旋的运动，这个漂浮问题的所有已知解的横截面都是扣环(加压弹性体)，这也解决了19世纪末和20世纪初广泛研究的变分问题。一般来说，完全可积系统的理论使我们有可能找到模型中出现的微分方程式和差分方程式的显式解；这些解通常是以椭圆函数的形式给出的。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.