| 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 at a 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 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 the chaotic case. Time permitting, we discuss the questions of control and applications to skating robots. Our model shows many features of real-life 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 the ice. |
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