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

运动学和几何光学主题：轮胎轨迹几何和台球模型

基本信息

批准号：
2005444
负责人：
Serge Tabachnikov
金额：
$ 34.8万
依托单位：
Pennsylvania State Univ University Park
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-06-01 至 2024-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2005444&HistoricalAwards=false
关键词：
Topics Kinematics Geometrical Optics Tire

项目摘要

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.

拟议的研究包括两部分：车辆运动和轮胎轨迹几何模型，以及有界区域和几何光学的弹性反射模型。在第一部分中，研究者将研究车辆运动学的各种具体问题。直接应用涉及追踪问题、对带有许多拖车的拖拉机的控制以及防止驾驶危险。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），这对于量子计算机的量子力学电路的设计非常重要。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模型，例如理想气体。尽管射线光学仅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来说足够准确，包括激光束整形、捕获光线和存储太阳能、控制光污染和隐形。现代技术使得制造具有不寻常的反射和折射特性的材料成为可能，并创造出近乎理想的复杂形状的镜子，从而实现玻璃、金属和塑料的几何光学设计。大多数提出的问题都允许理论和计算机实验研究，在许多情况下，后者是迈向前者的第一步。研究者将积极让本科生和研究生参与他的研究计划。所提出的研究与连续和离散的完全可积系统理论有很强的联系，并且依赖于自 20 世纪 60 年代发现孤子以来该理论中开发的各种方法，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细丝方程的孤子的描述密切相关，这是一个模拟流体和气体涡流运动的完全可积系统，并且该漂浮问题的所有已知解决方案的横截面都是扣环（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几个世纪。一般来说，完全可积系统的理论使得找到模型中出现的微分方程和差分方程的显式解成为可能；这些解决方案通常以椭圆函数的形式给出。该奖项反映了 NSF 的法定使命，并且通过使用基金会的智力优点和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

期刊论文数量（10）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Remarks on Joachimsthal Integral and Poritsky Property

关于 Joachimsthal 积分和 Poritsky 性质的评论

DOI：
10.1007/s40598-021-00180-0
发表时间：
2021
期刊：
Arnold Mathematical Journal
影响因子：
0
作者：
Arnold, Maxim;Tabachnikov, Serge
通讯作者：
Tabachnikov, Serge

Loewner's ``forgotten" theorem

勒纳“被遗忘”定理

DOI：
10.1007/s00283-021-10144-z
发表时间：
2022
期刊：
The mathematical intelligencer
影响因子：
0
作者：
Albers, P.
通讯作者：
Albers, P.

Open Problems on Billiards and Geometric Optics

台球和几何光学的未决问题

DOI：
10.1007/s40598-022-00198-y
发表时间：
2022
期刊：
Arnold Mathematical Journal
影响因子：
0
作者：
Bialy, Misha;Fierobe, Corentin;Glutsyuk, Alexey;Levi, Mark;Plakhov, Alexander;Tabachnikov, Serge
通讯作者：
Tabachnikov, Serge

Billiards in ellipses revisited