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Geometric Origin of the Tennis Racket Effect
P Mardešić1, G J Gutierrez Guillen2, L Van Damme3
1Institut de Mathématiques de Bourgogne-UMR 5584 CNRS, Université de Bourgogne-Franche Comté, 9 avenue Alain Savary, BP 47870, 21078 Dijon, France.
Physical Review Letters
|August 27, 2020
Summary
The tennis racket effect, a geometric phenomenon in rigid body rotation, is explained using complex phase space and Riemann surfaces. This study reveals its robustness and provides bounds for the twist defect in rotating objects.
Area of Science:
- Physics
- Mathematics
- Geometric Mechanics
Background:
- The tennis racket effect is a counterintuitive phenomenon observed during the free rotation of a three-dimensional rigid body.
- This effect, also known as the tennis racket paradox, involves an unexpected flip during rotation.
Purpose of the Study:
- To provide a mathematical explanation for the tennis racket effect using concepts from complex phase space.
- To demonstrate the connection between the tennis racket effect, Riemann surfaces, and the Picard-Lefschetz formula.
- To analyze the conditions for a perfect twist and establish bounds for the twist defect in rigid body rotation.
Main Methods:
- Analysis of rigid body rotation in a complex phase space.
- Application of Riemann surface theory and the Picard-Lefschetz formula.
- Derivation of upper and lower bounds for the twist defect.
Main Results:
- The tennis racket effect originates from a pole of a Riemann surface.
- A perfect twist is achieved in the limit of an ideal asymmetric object.
- Robustness of the effect is demonstrated through derived bounds for the twist defect.
- A similar mathematical approach explains the Dzhanibekov effect and the monster flip.
Conclusions:
- The study offers a novel geometric and mathematical framework for understanding the tennis racket effect and related phenomena.
- The findings highlight the importance of complex phase space analysis in explaining physical rotation dynamics.
- The derived bounds provide insights into the stability and predictability of such rotational behaviors.
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