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

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  • 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 $\mathbb H^{n-1}$.

    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

    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

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

    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

    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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