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Dynamical Properties for a Tunable Circular to Polygonal Billiard

  • General and Applied Physics
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Abstract

In this paper, we introduce a billiard whose boundary varies from a circular to a polygonal billiard. To describe the billiard boundary, we use a parametric equation, which needs to be solved numerically. We provide a detailed explanation about how to obtain the radius of the billiard boundary R for each angular position \(\theta\), where we used a tangent method to speed up the numerical simulations. We consider another tangent method to find the billiard boundary’s intercept and the particle’s trajectory. Furthermore, we show some trajectories’ examples and describe what happens with the phase space and Lyapunov exponents when changing the deformation. We present results for different values of the control parameter related to the number of edges of our polygon and the billiard with a triangular-like boundary.

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Acknowledgements

DRC acknowledges São Paulo Research Foundation (FAPESP, Brazil) (Grant No. 2020/02415-7) and National Council for Scientific and Technological Development (CNPq, Brazil) (Grant No. 162944/2020-9). The authors also acknowledge Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and Fundação Araucária.

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Author notes

  1. André Fujita and Matheus Rolim Sales both are equally contributed.

Authors and Affiliations

  1. Department of Physics, Federal University of Paraná (UFPR), Curitiba, 80060-000, PR, Brazil

    Diogo Ricardo da Costa

  2. Institute of Mathematics and Statistics, University of São Paulo (USP), Rua do Matão, 1010, São Paulo, 05508-090, SP, Brazil

    Diogo Ricardo da Costa & André Fujita

  3. Graduate Program in Physics, State University of Ponta Grossa, Ponta Grossa, 84030-900, PR, Brazil

    Matheus Rolim Sales

  4. Department of Mathematics and Statistics, State University of Ponta Grossa, Ponta Grossa, 84030-900, PR, Brazil

    José D. Szezech Jr & Antonio Marcos Batista

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  1. Diogo Ricardo da Costa
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  2. André Fujita
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Correspondence to Diogo Ricardo da Costa.

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da Costa, D.R., Fujita, A., Sales, M.R. et al. Dynamical Properties for a Tunable Circular to Polygonal Billiard. Braz J Phys 52, 75 (2022). https://doi.org/10.1007/s13538-022-01075-x

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