Twisting Somersault

TL;DR

This paper provides a comprehensive mathematical model for twisting somersaults, linking body dynamics, geometric phases, and rotation numbers to predict dive outcomes with high accuracy.

Contribution

It introduces an exact formula connecting dive parameters, energy, and angular momentum, extending previous geometric phase theories to realistic, shape-changing models.

Findings

Numerical simulations match the exact formula with realistic parameters.

The model accurately predicts the number of twists and somersaults in dives.

The approach unifies rigid body dynamics with shape changes in a novel way.

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.