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Elastica, filament equation, and bicycle models
Speaker: Sergei Tabachnikov


Abstract: Elastica are solutions to a classic variational problem of Bernoulli: to describe the curves of fixed lengths that extremize the bending energy. I will describe several problems where these ubiquitous curves appear.
One is the study of the filament equation, a completely integrable evolution on curves that models the propagation of vortices in liquid or gas. Another is a simple model of a bicycle, presented as a directed segment of a fixed length that can move so that the velocity of the rear end is always aligned with the segment.  A bicycle path is a motion of the segment, subject to this nonholonomic constraint, and the length of the path, by definition, is the length of the front track. This defines a problem of sub-Riemannian geometry, and  the respective geodesics are closely related to classical elastica and its "relatives". Another bicycle problem where elastica appears is as follows: given the front and rear bicycle tracks, can one determine which way the bicycle went?


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Date : Tue, Jul 9

Time: 11:30

Place: SR C