Integrability and Control of Figure Skating

Figure skating is a beautiful sport combining elegance, precision, and athleticism. MS2Discovery is hosting a webinar with Professor Vakhtang Putkaradze from the Department of Mathematics and Statistics of the University of Alberta on Dec 2, 2022 at 03:00 PM to understand some of the mechanics and complexity involved in this sport and discuss the integrability and control of figure skating.

Webinar:
https://wilfrid-laurier.zoom.us/j/92935575874?pwd=eTFMbjlxK0JhNFhSdm95Z0tTMDZIZz09

Bio: Prof Vakhtang Putkaradze received his PhD from the University of Copenhagen, Denmark, and held faculty positions in New Mexico, Colorado State University, and, most recently, at the University of Alberta, where he was a Centennial Professor between 2012-2019. From 2019 to 2022, he led the science and tech part of the Transformation Team at ATCO Ltd, first as a Senior Director and then Vice-President. He is currently on a sabbatical in Albuquerque, New Mexico, where he enjoys the sunshine and his collaborations with the national labs on applications of geometric mechanics to neural networks, in particular, efficient computations of Hamiltonian systems using data-based techniques. His main topic of interest is using geometric methods in mechanics and various applications. He has received numerous prizes and awards for research and teaching, including Humboldt Fellowship, Senior JSPS fellowship, CAIMS-Fields industrial math prize.

Abstract: Figure skating is a beautiful sport combining elegance, precision, and athleticism. To understand some of the mechanics and complexity involved in this sport, we derive and analyze a three-dimensional model of a figure skater in continuous contact with the ice (i.e., no jumps). We model the skater as a three-dimensional body moving in space subject to a non-holonomic constraint enforcing movement along the skate's direction and holonomic constraints of continuous contact with ice and pitch constancy of the skate. We derive a surprising result that for a static (i.e., non-articulated) skater, the system is integrable if and only if the projection of the center of mass on the skate's direction coincides with the contact point with ice and some mild (and realistic) assumptions on the directions of inertia's axes. We also consider the case when the projection of the center of mass on the skate's direction does not coincide with the contact point and show that this non-integrable case exhibits apparent chaotic behavior. After a pedagogical introduction to Hamel's approach to mechanics, we extend the study to see how an articulated skater traces trajectories on ice by considering a two-dimensional skater (a Chaplygin's sleigh) with a controlled moving mass. We derive a control procedure by approximating the trajectories using circular arcs. We show that there is a control procedure minimizing the 'relative kinetic energy' of the control mass, which leads to well-posed equations for the control masses. We demonstrate examples of our system tracing actual compulsory figure skating trajectories. We also discuss further extensions of the model and applications to real-life figure skating.

This research is a joint work with M. Rhodes and V. Gzenda.