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Geometric Origin of the Tennis Racket Effect.

P Mardešić1, G J Gutierrez Guillen2, L Van Damme3

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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.

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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.