Using the Geometric Phase to Optimise Planar Somersaults

TL;DR

This paper models planar somersaults using dynamic and geometric phases, validates the model with athlete footage, and proposes an optimized shape change strategy to maximize rotation by balancing both phases.

Contribution

It introduces a novel model incorporating geometric phase into somersault dynamics and demonstrates how shape change strategies can optimize rotation.

Findings

Reversing shape change sections can increase geometric phase and total rotation.

The geometric phase, though small, significantly influences optimal shape change strategies.

Validation with elite athlete footage supports the model's accuracy.

Abstract

We derive the equations of motion for the planar somersault, which consist of two additive terms. The first is the dynamic phase that is proportional to the angular momentum, and the second is the geometric phase that is independent of angular momentum and depends solely on the details of the shape change. Next, we import digitised footage of an elite athlete performing 3.5 forward somersaults off the 3m springboard, and use the data to validate our model. We show that reversing and reordering certain sections of the digitised dive can maximise the geometric phase without affecting the dynamic phase, thereby increasing the overall rotation achieved. Finally, we propose a theoretical planar somersault consisting of four shape changing states, where the optimisation lies in finding the shape change strategy that maximises the overall rotation of the dive. This is achieved by balancing the rotational contributions from the dynamic and geometric phases, in which we show the geometric phase plays a small but important role in the optimisation process.