Virtual billiards in pseudo–euclidean spaces: Discrete hamiltonian and contact integrability

Božidar Jovanović; Vladimir Jovanović

doi:10.3934/dcds.2017224

Article Contents

2017, Volume 37, Issue 10: 5163-5190. Doi: 10.3934/dcds.2017224

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Virtual billiards in pseudo–euclidean spaces: Discrete hamiltonian and contact integrability

Božidar Jovanović^1, , and
Vladimir Jovanović^2,

1.
Mathematical Institute SANU, Serbian Academy of Sciences and Arts, Kneza Mihaila 36, 11000 Belgrade, Serbia
2.
Faculty of Sciences, University of Banja Luka, Mladena Stojanovića 2, 51000 Banja Luka, Bosnia and Herzegovina

* Corresponding author
* Corresponding author

Received: September 30, 2016

Revised: April 30, 2017

Published: October 2017

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Abstract

The aim of the paper is to unify the efforts in the study of integrable billiards within quadrics in flat and curved spaces and to explore further the interplay of symplectic and contact integrability. As a starting point in this direction, we consider virtual billiard dynamics within quadrics in pseudo-Euclidean spaces. In contrast to the usual billiards, the incoming velocity and the velocity after the billiard reflection can be at opposite sides of the tangent plane at the reflection point. In the symmetric case we prove noncommutative integrability of the system and give a geometrical interpretation of integrals, an analog of the classical Chasles and Poncelet theorems and we show that the virtual billiard dynamics provides a natural framework in the study of billiards within quadrics in projective spaces, in particular of billiards within ellipsoids on the sphere ${\mathbb{S}^{n - 1}}$ and the Lobachevsky space $\mathbb H^{n-1}$.

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Mathematics Subject Classification: 70H06, 37J35, 37J55, 51M05.

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Figure 1. A segment of a virtual billiard trajectory within hyperbola ($a_1>0, a_2<0$) in the Euclidean space $\mathbb E^{2, 0}$. The caustic is an ellipse

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Figure 2. Families of pseudo-confocal quadrics for $a_1>0, a_2<0$ in $\mathbb E^{1, 1}$ (with $\alpha_1=-a_2>\alpha_2=a_1$) and $\mathbb E^{2, 0}$, respectively

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Figure 3. Family of pseudo-confocal quadrics for $a_1>0, a_2<0$ in $\mathbb E^{1, 1}$, where $\alpha_1=a_1>\alpha_2=-a_2$

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Figure 4. The segments of time-like and space-like billiard trajectories for $a_1>0, a_2<0$, $\alpha_1=-a_2>\alpha_2=a_1$ in $\mathbb E^{1, 1}$. The caustics are hyperbolas

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Figure 5. The segment of a space-like billiard trajectory for $a_1>0, a_2<0$, $\alpha_1=a_1>\alpha_2=-a_2$ in $\mathbb E^{1, 1}$. The caustic is an ellipse

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