  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.
