Twisting Somersault

Holger R. Dullin; William Tong

doi:10.1137/15m1055097

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arXiv · 1510.08046

Twisting Somersault

Holger R. Dullin·William Tong

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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.

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Holger R. Dullin, William Tong. 2015-10-24. Twisting Somersault. https://doi.org/10.1137/15m1055097

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