Semi-Annular Spiral Billiard with Periodic Orbits

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This Demonstration shows spiral-shaped billiard tables patched together using annular pieces. Starting at zero radians on the positive axis, and repeating every radians counter-clockwise around the spiral, periodic orbits occur where tangents to a table's inner and outer curved boundaries are parallel for the same spiral angle. To construct a table, users can choose the number of loops around the spiral and the rate at which the width of the spiral increases.

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The more interesting cases (when "connect table ends" is "yes") form the spiral into a doubly connected region with the table's central and peripheral straight ends "seamed" together. In the doubly connected cases, all non-periodic trajectories are chaotic and time-irreversible. Since limited precision rounding errors produce information loss, when trajectories cross from a longer to shorter straight table edge (at the "seam"), backtracking calculations of earlier bounce positions contain minute errors. A table's two straight table edges form the table seam. All non-periodic trajectories cross the table seam.

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Contributed by: William C. Hill (March 2012)
Open content licensed under CC BY-NC-SA


Snapshots


Details

The trajectory calculations are based on figure 1 on page 539 of [1].

Reference

[1] J. S. Espinoza Ortiz and R. Egydio de Carvalho, "Energy Spectrum and Eigenfunctions through the Quantum Section Method," Brazilian Journal of Physics, 31(4), 2001 pp. 538–545.



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