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Top on a smooth plane.

Maria Przybylska1, Andrzej J Maciejewski2

  • 1Institute of Physics, University of Zielona Góra, Licealna 9, 65-417 Zielona Góra, Poland.

Chaos (Woodbury, N.Y.)
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Summary
This summary is machine-generated.

This study investigates the integrable dynamics of a sliding top. We prove that only two specific cases are integrable, analogous to classical top problems, with or without gravity.

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Area of Science:

  • Rigid Body Dynamics
  • Mathematical Physics
  • Classical Mechanics

Background:

  • The classical top problem is a fundamental model in mechanics.
  • Investigating integrable systems is crucial for understanding complex dynamics.

Purpose of the Study:

  • To determine the conditions for the integrability of a sliding top in a frictionless plane.
  • To analyze the impact of gravitational fields on the system's integrability.

Main Methods:

  • Perturbation theory applied to classical top equations.
  • Analysis of differential Galois groups of variational equations.
  • Application of the Ziglin theorem for non-integrability proofs.

Main Results:

  • Identified two integrable cases for the sliding top, analogous to Euler and Lagrange cases.
  • Proved non-integrability for two other cases using differential Galois groups.
  • Established symmetry as a necessary condition for integrability in the absence of gravity.

Conclusions:

  • The integrability of the sliding top is strictly limited to specific configurations.
  • The presence and absence of gravity significantly influence the system's integrability.
  • Gyrostatic terms do not alter the integrability of the identified cases.