Integrability and Chaos in Figure Skating

TL;DR

This paper models a figure skater as a three-dimensional non-holonomic system, analyzing conditions for integrability and chaos, revealing new constants of motion and complex behaviors relevant to real skating performance.

Contribution

It introduces a novel three-dimensional model of a figure skater, proving integrability conditions and identifying new constants of motion, while exploring chaotic dynamics in non-integrable scenarios.

Findings

The system is integrable when the center of mass projection aligns with the contact point.

Non-integrable cases exhibit chaotic behavior and sensitivity to initial conditions.

Transition from integrability to chaos demonstrates complex dynamical features.

Abstract

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.