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On resonant energy sets for Hamiltonian systems with reflections

Krzysztof Frączek

TL;DR

The paper addresses resonances in a quasi-integrable Hamiltonian system of two uncoupled oscillators with potentials $V_1,V_2$ constrained by elastic reflections on rectilinear polygons. It combines a translation-surface approach with analytic and Fourier techniques to reduce the dynamics to billiards on polygonal tables $P_{E,\theta}$ and to extract end-behavior independence of turning-time functions. The main contributions are a sharp trichotomy for the resonance set $\mathcal{E}(P,V_1,V_2)$, a complete characterization of when abundant resonances occur (precisely when $V_1,V_2\in\mathcal{SP}$ with a rational ratio of curvatures), and a general non-resonance criterion showing that, outside the special class, resonances are rare or finite. The work highlights a Bertrand-type analogue: abundant resonances arise only for isochronous, specially structured potentials, linking geometric and analytic properties to resonance phenomena with potential implications for KAM-type persistence in quasi-integrable regimes.

Abstract

We study two uncoupled oscillators, horizontal and vertical, residing in rectilinear polygons (with only vertical and horizontal sides) and impacting elastically from their boundary. The main purpose of the article is to analyze the occurrence of resonance in such systems, depending on the shape of the analytical potentials that determine the oscillators. We define resonant energy levels; roughly speaking, these are levels for which the resonance phenomenon occurs more often than rarely. We focus on unimodal analytic potentials with the minimum at zero. The most important result of the work describes the size of the set of resonance levels in the form of the following trichotomy: it is mostly empty or is one-element or is large, i.e. non-empty and open. In this latter case, we show that an abundance of resonant orbits occurs only when the potentials are of a special type; we denote this family by $\mathcal{SP}$. This result can be regarded as a distant analogue of the classical Bertrand's theorem (1873), which characterizes centrally symmetric potentials in the presence of an abundance of periodic orbits.

On resonant energy sets for Hamiltonian systems with reflections

TL;DR

The paper addresses resonances in a quasi-integrable Hamiltonian system of two uncoupled oscillators with potentials constrained by elastic reflections on rectilinear polygons. It combines a translation-surface approach with analytic and Fourier techniques to reduce the dynamics to billiards on polygonal tables and to extract end-behavior independence of turning-time functions. The main contributions are a sharp trichotomy for the resonance set , a complete characterization of when abundant resonances occur (precisely when with a rational ratio of curvatures), and a general non-resonance criterion showing that, outside the special class, resonances are rare or finite. The work highlights a Bertrand-type analogue: abundant resonances arise only for isochronous, specially structured potentials, linking geometric and analytic properties to resonance phenomena with potential implications for KAM-type persistence in quasi-integrable regimes.

Abstract

We study two uncoupled oscillators, horizontal and vertical, residing in rectilinear polygons (with only vertical and horizontal sides) and impacting elastically from their boundary. The main purpose of the article is to analyze the occurrence of resonance in such systems, depending on the shape of the analytical potentials that determine the oscillators. We define resonant energy levels; roughly speaking, these are levels for which the resonance phenomenon occurs more often than rarely. We focus on unimodal analytic potentials with the minimum at zero. The most important result of the work describes the size of the set of resonance levels in the form of the following trichotomy: it is mostly empty or is one-element or is large, i.e. non-empty and open. In this latter case, we show that an abundance of resonant orbits occurs only when the potentials are of a special type; we denote this family by . This result can be regarded as a distant analogue of the classical Bertrand's theorem (1873), which characterizes centrally symmetric potentials in the presence of an abundance of periodic orbits.
Paper Structure (13 sections, 21 theorems, 190 equations, 3 figures)

This paper contains 13 sections, 21 theorems, 190 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Structure and main tools of the paper
  4. From oscillations in dimension two to billiards on polygons
  5. Properties of functions $a$, $b$, $a_{\xi}$ and $b_{\xi}$
  6. Potentials of degree two
  7. General criterion for the absence of resonance
  8. Short introduction to translation surfaces
  9. Partitions of translation surfaces into polygons
  10. From billiards to translation surfaces
  11. On geometrically quasi-periodic maps
  12. Proofs of the main results
  13. Examples of resonant energy levels for potentials outside $\mathcal{SP}$.

Key Result

Theorem 1.3

Let $V_1,V_2:{\mathbb{R}}\to{\mathbb{R}}_{\geq 0}$ be two $\mathcal{UM}$-potentials and let $m_1:=\deg(V_1,0)$ and $m_2:=\deg(V_2,0)$. Suppose that Then $\mathcal{E}(P,V_1,V_2)$ is empty for any polygon $P\in \mathcal{RP}$.

Figures (3)

  • Figure 1: The maps $a$, $\bar{a}$, $a_\xi$, $\bar{a}_\xi$ for non-even $V_1$.
  • Figure 2: Internal and marginal sides of $P_{E,\theta}$.
  • Figure 3: The billiard table $P$ and the translation surface $(M,\omega)$ - connected case.

Theorems & Definitions (63)

  • Definition 1
  • Definition 2
  • Remark 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 2.1
  • Remark 2.2
  • Remark 3.1
  • ...and 53 more