Variational approach and invariant tori for the bouncing convex body problem

• Yong-Ping Wang ^[1] ; Yuan Yue ^[1] ; Xiao-Ming Zhang ^[1] ; Deng-Hui Li ^[2]
  1. [1] Southwest Jiaotong University

     Southwest Jiaotong University

     China

     
  2. [2] Suzhou University of Technology
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 25, Nº 3, 2026
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • In this paper, we investigate a Hamiltonian system which describes the motion of a convex body with a smooth boundary, falling under gravity and undergoing completely elastic impacts with a periodically moving horizontal plate. Firstly, by establishing a variational framework, we define the generation function of the system. Then via a Lagrangian version of the twist mapping theorem, we prove the existence of invariant tori of the system, provided that the convex body is closed to a disk whose geometric center coincides with mass center, and the vibration of the plate is sufficiently small.

    Finally, we discuss the equivalence between the bouncing convex body problem and the billiard ball problem in a cylinder.

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