Dynamics of carving runs in alpine skiing. I. The basic centrifugal pendulum

We studied perfect carving turns of alpine skiing using the simple model of an inverted pendulum which is subject to the gravity force and the force mimicking the centrifugal force emerging in the turns. Depending on the turn speed the model describes two different regimes. In the subcritical regime, there exist three equilibrium positions of the pendulum where the total torque applied to the pendulum vanishes-the marginally stable vertical position and two unstable tilted positions on both sides of the vertical. The tilted equilibria correspond to the ski turns executed in perfect balance. The vertical equilibrium corresponds to gliding down the fall line without turns. In the supercritical regime, the tilted equilibria disappear. In addition to the equilibria, the model allows fall-rise solutions, where the pendulum (skier) rises from the ground on one side and hits the ground on the other side, and solutions describing oscillations about the vertical equilibrium. These oscillations correspond to the so-called dynamic skiing where the skier never settles to a balanced position in the turn. Analysis of the available data on World Cup races shows that elite racers ski mostly in the supercritical regime.