Twisting somersault


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 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. Numerical simulations for various dives agree perfectly with this formula where realistic parameters are taken from actual observations.
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Schlagworte: Mathematik Modellierung Wasserspringen Physik System Drehung Längenachsendrehung Analyse Theorie dynamisch
Notationen: Naturwissenschaften und Technik technische Sportarten
Veröffentlicht in: arXiv e-prints Abstract Service
Veröffentlicht: 2015
Seiten: 1-16
Dokumentenarten: Artikel
Sprache: Englisch
Level: hoch