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<p>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.</p>

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Variational approach and invariant tori for the bouncing convex body problem

  • 期刊论文

  • 发表日期：

    2026年5月24日

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  • Yong-Ping Wang,
  • Yuan Yue,
  • Xiao-Ming Zhang,
  • Deng-Hui Li

摘要

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.