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Schroedinger’s Equation, Tire Tracks and Rolling Cones in the Minkowski Space

Webinar Presented By:  Professor Mark Levi, Mathematics, Pennsylvania State University

Speaker Bio:  Mark Levi is Professor of Mathematics at the Pennsylvania State University. He received his Ph.D. from Courant Institute under the direction of Jurgen Moser. He works on problems in dynamical systems, differential equations and geometry, often with physical motivations. He writes regular articles in the series “Mathematical curiosities” in SIAM News (with 57 articles to date), and is the author of three books, including “The Mathematical Mechanic” (One of science editors' Top 10 list for Science in 2009).

Abstract:  I will describe a hidden but very close connection between the objects mentioned in the title. The tire track problem is a simple geometrical model suggested by the motion of a bike: a segment of fixed length with one end (“the front wheel”) moves along a prescribed path in the plane, while the other end ( “the rear wheel”) tracks the front end: its velocity is constrained to the direction of the segment (no sideslip). I will describe the equivalence between this problem on the one hand and the stationary Schroedinger’s equation on the other.  It turns out that the Schroedinger potential can be visualized as a path of the “front wheel”; and solutions of the equation can be viewed as paths of the rear wheel. Stationary Schroedinger’s equation, also known as Hill’s equation, is ubiquitous in mathematical,  physical and engineering settings, and this equivalence opens a new “bicycle” interpretation of this basic system. In conclusion I will mention very briefly – this is joint work with Gil Bor - how all this (and some more general linear Hamiltonian systems)  can be viewed as one cone rolling on another stationary cone. The shape of the stationary cone is determined by the potential in the Schroedinger’s equation, or the Hamiltonian in the more general case.

Date and Time:  March 24, 2021 | 3:00 p.m.

Location:  Online

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