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Some remarks on the periodic motions of a bouncing ball

Stefano Marò

Abstract

We consider the vertical motion of a free falling ball bouncing elastically on a racket moving in the vertical direction according to a regular $1$-periodic function $f$. For fixed coprime $p,q$ we study existence, stability in the sense of Lyapunov and multiplicity of $p$ periodic motions making $q$ bounces in a period. If $f$ is real analytic we prove that one periodic motion is unstable and give some information on the set of these motions.

Some remarks on the periodic motions of a bouncing ball

Abstract

We consider the vertical motion of a free falling ball bouncing elastically on a racket moving in the vertical direction according to a regular -periodic function . For fixed coprime we study existence, stability in the sense of Lyapunov and multiplicity of periodic motions making bounces in a period. If is real analytic we prove that one periodic motion is unstable and give some information on the set of these motions.
Paper Structure (4 sections, 12 theorems, 48 equations)

This paper contains 4 sections, 12 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Some results on periodic orbits of exact symplectic twist maps
  3. The bouncing ball map and its properties
  4. Periodic bouncing motions

Key Result

Theorem 2.5

Let $S:{\mathbb A}\rightarrow{\mathbb A}$ be an exact symplectic twist diffeomorphism that preserves and twists the ends infinitely and let $p,q$ be two coprime integers. Then there exist at least two Birkhoff $(p,q)$-periodic orbits for $S$.

Theorems & Definitions (30)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Theorem 2.5
  • Remark 2.6
  • Proposition 2.7
  • proof
  • Corollary 2.8
  • Theorem 2.9
  • ...and 20 more