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Collision Systems, with Applications to Geodesic Networks and Point Billiards

A mix of human-written and AI-generated textHuman understanding: some partsmath.DS — Dynamical Systems

Contributed by Alec Shelley ↗, Henry Bosch

Version 1 / Oct 29, 2026 / CC BY 4.0

Abstract

A collision system is a finite set of point particles with masses, moving freely with velocities given by the gradient of a convex energy function of the momentum, whose collisions conserve total momentum and total energy and are otherwise unconstrained. Stationary geodesic networks near a line of multiplicity nn, point billiards with momentum- and energy-conserving collisions, and relativistic point particles are collision systems. We prove that the number of collisions of a collision system is at most exponential in its total mass, with an explicit base, whenever the energy is close enough to the Newtonian one, and that it can be exponential for every energy on a line. For geodesic networks this replaces the previously reported bounds, of order n2n^2 from below and nn2n^{n^2} from above, by exponential bounds on both sides; for point billiards it gives explicit exponential bounds. The exponential growth rates remain open.

Provenance statement

This work was generated using a combination of human mathematics and AI. Previous suboptimal upper bounds and lower bounds for the geodesic net were discovered by hand by Henry Bosch, and I discovered upper and lower bounds for billiard systems in two dimensions. These human results were the inspiration for developing a unified theory of "collision systems," with the hope that the proofs of all the above bounds could arise from this common framework. We used Danus, Codex, and Claude to prove these results.

Tools used

OpenAI
Codex
OpenAI
ChatGPT
Frenzymath
Danus
Anthropic
Claude Fable
Anthropic
Claude Opus

References

  1. W. K. Allard and F. J. Almgren, Jr., The structure of stationary one dimensional varifolds with positive density , Invent. Math. 34 (1976), 83--97.
  2. D. Burago, S. Ferleger, and A. Kononenko, Uniform estimates on the number of collisions in semi-dispersing billiards , Ann. of Math. (2) 147 (1998), 695--708.
  3. D. Burago and S. Ivanov, Examples of exponentially many collisions in a hard ball system , Ergodic Theory Dynam. Systems 41 (2021), no. 9, 2754--2769; arXiv:1809.02800.arXiv
  4. H. Bosch, A partial ϵ\epsilon-regularity theorem for integral stationary geodesic networks in Rn\mathbb{R}^n , arXiv:2607.23872v1 (2026).arXiv
  5. K. Burdzy, An improved upper bound on the number of billiard ball collisions , Comm. Math. Phys. 391 (2022), no. 1, 107--117; arXiv:2107.10362.arXiv
  6. O. Chodosh and C. Mantoulidis, The pp-widths of a surface , Publ. Math. Inst. Hautes Études Sci. 137 (2023), 245--342.
  7. J. Liu, G. Gao, Z. Sun, B. Wu, S. Liu, J. Jiang, H. Ju, L. Chen, R. Cheng, X. Zhang, and B. Dong, Danus: Orchestrating mathematical reasoning agents with fact-graph memory , arXiv:2607.06447 (2026).arXiv
  8. I. Frolov, D. Gorodkov, D. Korshunov, and A. Nabutovsky, Geodesic nets on the Euclidean plane and closed geodesic nets on Riemannian surfaces , arXiv:2606.14080v2 (2026).arXiv
  9. J. Féjoz, A. Knauf, and R. Montgomery, Lagrangian relations and linear point billiards , Nonlinearity 30 (2017), no. 4, 1326--1355.
  10. S. Gasiorek, Counting collisions in an NN-billiard system using angles between collision subspaces , SIGMA 16 (2020), 119, 13 pp., doi:10.3842/SIGMA.2020.119 (https://doi.org/10.3842/SIGMA.2020.119).DOI
Show all 18 referencesShow fewer references
  1. G. A. Gal'perin, Systems of locally interacting and repelling particles that are moving in space (Russian), Trudy Moskov. Mat. Obshch. 43 (1981), 142--196.
  2. G. Galperin, Playing pool with π\pi (the number π\pi from a billiard point of view) , Regul. Chaotic Dyn. 8 (2003), 375--394.
  3. A. Nabutovsky and F. Parsch, Geodesic nets: some examples and open problems , arXiv:1904.00483 (2019).arXiv
  4. F. Parsch and H. Zhang, Towards constructing geodesic nets with four boundary vertices and an increasing number of balanced vertices , arXiv:2511.07810 (2025).arXiv
  5. H. Qiu, Example of exponentially many collisions of identical hard disks in the plane , arXiv:2610.14282 (2026).arXiv
  6. Ya. G. Sinai, Billiard trajectories in a polyhedral angle (Russian), Uspekhi Mat. Nauk 33 (1978), no. 1(199), 229--230.
  7. S. Tabachnikov, Geometry and billiards , Student Mathematical Library 30 , Amer. Math. Soc., Providence, RI, 2005.
  8. L. N. Vaserstein, On systems of particles with finite-range and/or repulsive interactions , Comm. Math. Phys. 69 (1979), 31--56.

Version history

  1. v1Submitted by Alec ShelleyInitial depositCurrentOct 28, 2026