Twisting Somersault

Abstract

A complete description of twisting somersaults is given using a reduction to a time-dependent Euler equation for non-rigid 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 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. This rotation number is obtained by a slight modification of Montgomery's formula [9] for how much the rigid body has rotated. 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 [3]. Numerical simulations for various dives agree perfectly with this formula where realistic parameters are taken from actual observations.

16 pages, 6 figures, work from PhD thesis of William Tong

References in corpus (1)

• A Generalized Montgomery Phase Formula for Rotating Self Deforming Bodies

Cited by in corpus (4)

• Euler top with a rotor: classical analogies of spin squeezing and quantum phase transitions in a generalized Lipkin-Meshkov-Glick model
• A New Twisting Somersault - 513XD
• Cosymplectic geometry, reductions, and energy-momentum methods with applications
• Using the Geometric Phase to Optimise Planar Somersaults