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Magic Billiards: the Case of Elliptical Boundaries

Vladimir Dragović, Milena Radnović

TL;DR

This work defines magic billiards as a general framework where a particle, after boundary contact, is transported to another boundary point and continues motion. Focusing on elliptical boundaries, it identifies symmetry-based, integrable instances that preserve elliptic-coordinate separability and caustics, and derives comprehensive periodicity criteria in algebro-geometric, analytic (Cayley-type), and polynomial forms, complemented by topological descriptions via Fomenko graphs. The analysis extends to elliptic annuli, revealing richer Liouville foliations with specific atoms and illustrating how boundary identifications influence global dynamics. By tying divisor theory on elliptic curves, Pell-type polynomial relations, and Fomenko invariants, the paper advances a unified perspective on billiard integrability and contributes to the broader Fomenko conjecture on billiard realizations of Liouville foliations.

Abstract

In this work, we introduce a novel concept of magic billiards, which can be seen as an umbrella, unifying several well-known generalisations of mathematical billiards. We analyse properties of magic billiards in the case of elliptical boundaries. We provide explicit conditions for periodicity in algebro-geometric, analytic, and polynomial forms. A topological description of those billiards is given using Fomenko graphs.

Magic Billiards: the Case of Elliptical Boundaries

TL;DR

This work defines magic billiards as a general framework where a particle, after boundary contact, is transported to another boundary point and continues motion. Focusing on elliptical boundaries, it identifies symmetry-based, integrable instances that preserve elliptic-coordinate separability and caustics, and derives comprehensive periodicity criteria in algebro-geometric, analytic (Cayley-type), and polynomial forms, complemented by topological descriptions via Fomenko graphs. The analysis extends to elliptic annuli, revealing richer Liouville foliations with specific atoms and illustrating how boundary identifications influence global dynamics. By tying divisor theory on elliptic curves, Pell-type polynomial relations, and Fomenko invariants, the paper advances a unified perspective on billiard integrability and contributes to the broader Fomenko conjecture on billiard realizations of Liouville foliations.

Abstract

In this work, we introduce a novel concept of magic billiards, which can be seen as an umbrella, unifying several well-known generalisations of mathematical billiards. We analyse properties of magic billiards in the case of elliptical boundaries. We provide explicit conditions for periodicity in algebro-geometric, analytic, and polynomial forms. A topological description of those billiards is given using Fomenko graphs.
Paper Structure (8 sections, 14 theorems, 6 equations, 14 figures)

This paper contains 8 sections, 14 theorems, 6 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Definition of magic billiards
  3. Magic billiards within an ellipse
  4. Elliptical billiard with flipping over long axis
  5. Elliptic billiard with flipping over short axis
  6. Elliptic billiard with half-turn around the center
  7. Magic billiards in elliptic annulus
  8. Conclusions and discussion

Key Result

Theorem 3.6

In the billiard table bounded by the ellipse $\mathcal{E}$ given by eq:ellipse, consider the magic billiard with flipping over the long axis. Consider a trajectory of such billiard with the caustic $\mathcal{C}_{\beta}$ from the confocal family eq:confocal. Such a trajectory is $n$-periodic if and o Here $Q_0$, $Q_b$, $Q_{\infty}$ denote the points with coordinates $(0,\sqrt{ab\beta})$, $(b,0)$, $

Figures (14)

  • Figure 1: A standard billiard trajectory (upper left) and three trajectories of the magic billiard with the same initial segment. Upper right: flipping over the long axis; lower left: flipping over the short axis; lower right: half-turn around the center. In each case, the consecutive points of impact with the boundary are numerated.
  • Figure 2: Billiards within ellipse (upper left), half-ellipses (upper right and lower left), and quarter-ellipse (lower right). All trajectories have the same initial conditions. Unfolding the trajectories within the last three domains about the axes give the trajectory within the ellipse.
  • Figure 3: Magic billiard with flipping over the long axis: each time when the particle hits the boundary, it is reflected off the boundary and immediately magically flipped over the longer axis.
  • Figure 4: Theorem \ref{['th:fomenko-magic1']}: Fomenko graph for the magic billiard with flipping over the long axis.
  • Figure 5: The Liouville torus corresponding to an ellipse as caustic of magic billiard with flipping over the long axis is obtained by gluing four annuli along congruent arcs of the same color and texture.
  • ...and 9 more figures

Theorems & Definitions (41)

  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Theorem 3.6
  • ...and 31 more