---
title: "The Integrability of a knife-edge Billiard in a Disk"
authors:
  - "Alejandro Bravo-Doddoli"
  - "Huaidian Hou"
  - "William Clark"
  - "Anthony M. Bloch"
year: 2026
venue: "arXiv:math.DS"
arxiv: "2609.12963"
doi: null
source: "https://arxiv.org/abs/2609.12963"
topics:
  - "curiosity"
  - "maths-foundations"
  - "dynamical-systems"
  - "billiards"
seed_rank: 1830
seed_batch: "curiosity-2026-09-15"
reviewed: "2026-09-15"
pool: "maths-foundations"
relevance_score: 9
lineage: nonholonomic-billiards
cites:
  - title: "The Integrability of a knife-edge Billiard in a Disk"
    url: "https://arxiv.org/abs/2609.12963"
    year: 2026
    arxiv: "2609.12963"
    doi: null
see:
  - "1620-analytic-rigidity-and-symbolic-dynamics-for-two-centre-billiards"
  - "1583-symplectic-tiling-billiards-on-complete-affine-tori"
---

# The Integrability of a knife-edge Billiard in a Disk

## One-sentence takeaway

A knife-edge (nonholonomic) billiard in a disk is integrable: rotational symmetry reduces the impact map to an explicit integrable return.

## Why it's lovely

Why you might love this: ordinary billiards bounce; knife-edge billiards skate — a nonholonomic constraint that usually destroys integrability. Inside a disk the rotational symmetry is enough to prove the impact map integrable anyway. Tiny mechanical story, clean dynamical punchline, sits next to Broadside's two-centre and tiling-billiard cards.

## Key ideas

- Nonholonomic knife-edge constraint inside a circular table.
- Impact-space coordinates adapted to rotational symmetry.
- Post-impact return map shown integrable.
- Adds a constrained sibling to symplectic/tiling billiard shelf (1583, 1620).

## Caveats

Disk geometry is special; does not claim integrability for general tables or rough boundaries. Classical mechanics / DS paper, not a graphics technique.

## Links

- arXiv abs: https://arxiv.org/abs/2609.12963
- PDF: https://arxiv.org/pdf/2609.12963.pdf