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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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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description 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.
doi_str_mv 10.1007/s10883-022-09608-y
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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
title Estimating Elliptic Billiard Invariants with Spatial Integrals
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