Twisting somersault

Abstract

We give a dynamical system analysis of the twisting somersaults using a reduction to a timedependent Euler equation for nonrigid body dynamics. The central idea is that after reduction the twisting motion is apparent in a body frame, while the somersaulting (rotation about the fixed angular momentum vector in space) is recovered by a combination of dynamic phase and geometric phase. In the simplest "kick-model" the number of somersaults m and the number of twists n are obtained through a rational rotation number W = m/n of a (rigid) Euler top. Using the full model with shape changes that take a realistic time we then derive the master twisting-somersault formula: an exact formula that relates the airborne time of the diver, the time spent in various stages of the dive, the numbers m and n, the energy in the stages, and the angular momentum by extending a geometric phase formula due to Cabrera [J. Geom. Phys., 57(2007), pp. 1405-1420]. Numerical simulations for various dives agree perfectly with this formula where realistic parameters are taken from actual observations.