Integrability and Chaos in Figure Skating
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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 non-trivial 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.
KeywordsNon-holonomic dynamics Integrable systems Mechanics of sports
Mathematics Subject Classification70H06 70H33 70H45 70K50 70K55
We are grateful to P. Balseiro, A. M. Bloch, F. Fasso, H. Dullin, I. Gabitov, L. Garcia-Naranjo, D. D. Holm, T. Ohsawa, P. Olver, T. S. Ratiu, S. Venkataramani and D. Volchenkov for enlightening scientific discussions. We want to especially thank Prof. D. V. Zenkov for his interest, availability and patience in answering our questions. We would like to thank Dr. S. M. Rogers for careful reading of the manuscript and finding errors and inaccuracies. We are also grateful to M. Hall and J. Hocher for teaching us the intricacies of skating techniques and the differences between hockey and figure skating. We are thankful to M. Hall for providing skating expertise and C. Hansen’s graphics processing for Fig. 1. The research of VP was partially supported by the University of Alberta and NSERC Discovery grant, which also partially supported VG through NSERC USRA program.
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