Seminar 8/2/19: Integrability and chaos in figure skating

Integrability and chaos in figure skating
Prof Vakhtang Putkaradze
University of Alberta, Canada
Friday 8 February 2019
1pm, Blue Room, 4th floor, Main Building, TU Dublin Kevin Street Campus


We derive and analyze a three dimensional model of a figure skater. 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. For a static (non-articulated) skater, we show that the system is integrable if and only if the projection of the center of mass on skate's direction coincides with the contact point with ice and some mild (and realistic) assumptions on the directions of inertia's axes. The integrability is proved by showing the existence of two new constants of motion linear in momenta, providing a new and highly nontrivial example of an integrable non-holonomic mechanical system. We also consider the case when the projection of the center of mass on skate's direction does not coincide with the contact point and show that this non-integrable case exhibits apparent chaotic behavior, by studying the divergence of nearby trajectories. We also demonstrate the intricate behavior during the transition from the integrable to chaotic case. Our model shows many features of real-life skating, especially figure skating, and we conjecture that real-life skaters may intuitively use the discovered mechanical properties of the system for the control of the performance on ice.

Joint work with Vaughn Gzenda (UofA). The work was supported by NSERC Discovery Grant program (VP), USRA (VG) and the University of Alberta. This talk has also been made possible by the awarding of a James M Flaherty Visiting Professorship from the Ireland Canada University Foundation, with the assistance of the Government of Canada/avec l’appui du gouvernement du Canada.