Abstract
We give a dynamical system analysis of the twisting somersaults using a reduction to a time-dependent 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.
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Title of paper: Twisting Somersault
Authors: Holger R. Dullin and William Tong
File: M105509_01.mp4
Type: animation
Contents: Animation of a twisting somersault 5136D, 3/2 somersault, 6/2 twists as described by Theorem 6.
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Information & Authors
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Published In
SIAM Journal on Applied Dynamical Systems
Pages: 1806 - 1822
DOI: 10.1137/15M1055097
ISSN (online): 1536-0040
Copyright
© 2016, Society for Industrial and Applied Mathematics.
History
Submitted: 4 January 2016
Accepted: 27 July 2016
Published online: 4 October 2016
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Australian Research Council : LP100200245
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