Eighty New Invariants of N-Periodics in the Elliptic Billiard (2004.12497v11)

Abstract: We introduce several-dozen experimentally-found invariants of Poncelet N-periodics in the confocal ellipse pair (Elliptic Billiard). Recall this family is fully defined by two integrals of motion (linear and angular momentum), so any "new" invariants are dependent upon them. Nevertheless, proving them may require sophisticated methods. We reference some two-dozen proofs already contributed. We hope this article will motivate contributions for those still lacking proof.

• The paper experimentally identifies over eighty new mathematical invariants associated with N-periodic trajectories in elliptic billiards.
• Key findings include novel length, area, and angular invariants, along with consistent properties found in associated pedal and antipedal polygons.
• These findings provide experimental evidence supporting the integrable nature of elliptic billiards and offer a significant basis for future mathematical research and formal proofs.

Overview of "Eighty New Invariants in the Elliptic Billiard"

The paper "Eighty New Invariants in the Elliptic Billiard" by Dan Reznik, Ronaldo Garcia, and Jair Koiller explores the fascinating mathematical properties of the elliptic billiard, a special case of Poncelet's Porism. Specifically, the authors introduce and experimentally identify a substantial number of new invariants—eighty in total—pertaining to Poncelet N-periodic trajectories within confocal ellipse pairs. This work builds upon the existing understanding that such trajectories in the elliptic billiard are governed by a two-integral system related to linear and angular momentum. Despite being dependent on these known integrals, unveiling and proving these new invariants often demand sophisticated mathematical techniques.

The introduction and thorough experimental analysis of these new invariants may prove foundational for further research into integrable systems, providing the community with a richer set of tools to understand elliptic billiards. Key quantities explored include distances, areas, and angular relationships of N-periodic trajectories and associated polygons like pedal and antipedal polygons, along with their central properties like curvature and invariant centroids.

Key Contributions and Findings

1. Invariant Properties: The authors document over eighty invariants associated with N-periodic trajectories and derived geometrical structures. These include novel length, area, and angular invariants, some of which depend on the parity of N. For example, the ratio of outer to inner polygon areas is invariant for odd N.
2. Invariant Areas and Lengths: Through exhaustive experimentation, the authors find products and ratios of certain polygonal areas, such as pedal polygons relative to N-periodics, to be invariant. These invariants appear consistently across numerous scenarios, suggesting a deep underlying geometric property.
3. Pedal and Antipedal Polygons: Both pedal and antipedal polygons exhibit invariant properties that the authors meticulously document. These findings expand the mathematical landscape surrounding elliptic billiards by revealing symmetrical and invariant behavior in these complex geometric configurations.
4. Impact of Symmetry and Integrability: The findings underscore the integrable nature of the elliptic billiard system, supporting conjectures about the existence of only one such system in planar billiards. Importantly, this research suggests that new methodologies may enrich the understanding of dynamical systems that present integrable characteristics.
5. Potential for Further Mathematical Simplification: Although many observed invariants lack formal proofs, the paper offers a significant opportunity for future work. The mathematical community may find engagement with the tools and new invariants an advantageous path towards better understanding the relationship between periodic orbits and their geometric frames.

Implications and Future Developments

From a theoretical perspective, the breadth of invariants uncovered in this work could offer new insights into the symmetry and structure inherent in integrable systems. Practically, the experimental and computational techniques developed and used to identify these invariants provide a foundation for further exploration and discovery in mathematical billiards and related disciplines. The invariants may have broader applications in optimization problems due to their connection with conserved quantities, allowing them to be considered within other analytical contexts.

The motivation for continuing this line of work includes establishing formal proofs for the conjectured invariants and investigating their applications in other domains governed by similar mathematical structures. This research could inspire further analysis into other integrable and near-integrable systems, offering potential advances in mathematical physics, geometry, and computational methods.

In conclusion, this paper represents a substantial contribution to the field of elliptic billiards through the experimental discovery of new invariants. By challenging existing frameworks and proposing numerous mathematically consistent invariants, the authors invite the community to explore, prove, and apply these results in diverse scientific areas.