Estimating Elliptic Billiard Invariants with Spatial Integrals

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 poi...

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Bibliographic Details
Published in:Journal of dynamical and control systems 2023-09, Vol.29 (3), p.757-767
Main Authors: Garcia, Ronaldo, Koiller, Jair, Reznik, Dan
Format: Article
Language:English
Subjects:
Calculus of Variations and Optimal Control
Optimization
Control
Dynamical Systems
Dynamical Systems and Ergodic Theory
Integrals
Invariants
Mathematics
Mathematics and Statistics
Systems Theory
Vibration
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Summary: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.
ISSN:1079-2724
1573-8698
DOI:10.1007/s10883-022-09608-y