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Estimating Elliptic Billiard Invariants with Spatial Integrals

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Abstract

We give closed-form expressions for recently discovered invariants of Poncelet N-periodic trajectories in the elliptic billiard with spatial integrals evaluated over the boundary of the elliptic billiard. The integrand is weighed by a measure equal to the density of rays hitting a given boundary point. We find that aperiodic averages are smooth and monotonic on caustic eccentricity, and are consistent with the invariant values predicted at the discrete caustic parameters which admit a given N-periodic family. Our results are verified by numerical simulation.

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Notes

  1. In our case, we find the caustic using an optimization procedure described in [18].

  2. A billiard N-periodic is fully specified by its two integrals of motion, so any “new” invariants are ultimately dependent on basic conservations.

References

  1. Akopyan A, Schwartz R, Tabachnikov S. Billiards in ellipses revisited. Eur J Math. 2020.

  2. Bialy M, Tabachnikov S. Dan Reznik’s identities and more. Eur J Math. 2020.

  3. Birkhoff GD. What is the ergodic theorem? Amer Math Monthly. 1942;49(4):222–6.

    Article  MathSciNet  MATH  Google Scholar 

  4. del Centina A. Poncelet’s porism: a long story of renewed discoveries I. Arch Hist Exact Sci 2016;70(2):1–122.

    Article  MathSciNet  MATH  Google Scholar 

  5. Chavez-Caliz A. More about areas and centers of Poncelet polygons. Arnold Math J. 2020.

  6. Connes A, Zagier D. A property of parallelograms inscribed in ellipses. Amer Math Monthly. 2007;114(10):909–14 people.mpim-bonn.mpg.de/zagier/files/amm/114/fulltext.pdf.

    Article  MathSciNet  MATH  Google Scholar 

  7. Dragović V, Radnović M. Poncelet porisms and beyond: Integrable billiards, hyperelliptic jacobians and pencils of quadrics frontiers in mathematics. Basel: Springer; 2011.

    Book  MATH  Google Scholar 

  8. Fierobe C. Complex caustics of the elliptic billiard. Arnold Math J 2021;7:1–30.

    Article  MathSciNet  MATH  Google Scholar 

  9. Flatto L. Poncelet’s theorem. Providence, RI: American Mathematical Society; 2009.

    MATH  Google Scholar 

  10. Glutsyuk A. On curves with Poritsky property. arXiv:1901.01881. 2019.

  11. Gradshteyn IS, Ryzhik IM. Table of integrals, series, and products, 4th ed. New York: Academic Press; 1965.

    MATH  Google Scholar 

  12. Kaloshin V, Sorrentino A. On the integrability of Birkhoff billiards. Phil Trans R Soc, A(376). 2018.

  13. Kolodziej R. The rotation number of some transformation related to billiards in an ellipse. Studia Math 1985;81(3):293–302.

    Article  MathSciNet  MATH  Google Scholar 

  14. Lazutkin VF. The existence of caustics for a billiard problem in a convex domain. Izv Akad Nauk SSSR Ser Mat 1973;3(1):186–216.

    MathSciNet  Google Scholar 

  15. Lynch P. Integrable elliptic billiards and ballyards, Eur J Phys, 41(1). 2019.

  16. Poritsky H. The billiard ball problem on a table with a convex boundary – an illustrative dynamical problem. Ann Math 1950;2(51):446–470.

    Article  MathSciNet  MATH  Google Scholar 

  17. Reznik D, Garcia R, Koiller J. Can the elliptic billiard still surprise us? Math Intelligencer 2020;42:6–17.

    Article  MathSciNet  MATH  Google Scholar 

  18. Reznik D, Garcia R, Koiller J. Fifty new invariants of N-periodics in the elliptic billiard. Arnold. Math J 2021;7:341–355.

    MathSciNet  MATH  Google Scholar 

  19. Tabachnikov S. Geometry and Billiards, vol. 30 of Student Mathematical Library. Providence, RI: American Mathematical Society Mathematics Advanced Study Semesters, University Park, PA; 2005.

    Google Scholar 

  20. Zhang J. Suspension of the billiard maps in the Lazutkin’s coordinate. Discrete and Continuous Dynamical Systems 2017;37(4):2227–2242.

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We would like to thank Sergei Tabachnikov, Arseniy Akopyan, Hellmuth Stachel, and Sergey Galkin for invaluable discussions.

The second author is fellow of CNPq and coordinator of Project PRONEX/ CNPq/ FAPEG 2017 10 26 7000 508.

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Authors and Affiliations

  1. Univ. Federal de Goiás, Goiânia, Brazil

    Ronaldo Garcia

  2. Univ. Federal de Juiz de Fora, Juiz de Fora, Brazil

    Jair Koiller

  3. Data Science Consulting Ltd., Rio de Janeiro, Brazil

    Dan Reznik

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  1. Ronaldo Garcia
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  2. Jair Koiller
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Correspondence to Dan Reznik.

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Garcia, R., Koiller, J. & Reznik, D. Estimating Elliptic Billiard Invariants with Spatial Integrals. J Dyn Control Syst 29, 757–767 (2023). https://doi.org/10.1007/s10883-022-09608-y

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