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Sergei Tabachnikov

University Park, PA 16802, USA
Department of Mathematics, Pennsylvania State University


Tabachnikov S.
Two Variations on the Periscope Theorem
2020, vol. 25, no. 1, pp.  11-17
A (multidimensional) spherical periscope is a system of two ideal mirrors that reflect every ray of light emanating from some point back to this point. A spherical periscope defines a local diffeomorphism of the space of rays through this point, and we describe such diffeomorphisms. We also solve a similar problem for (multidimensional) reversed periscopes, the systems of two mirrors that reverse the direction of a parallel beam of light.
Keywords: periscope, optical reflection, projectively gradient vector field
Citation: Tabachnikov S.,  Two Variations on the Periscope Theorem, Regular and Chaotic Dynamics, 2020, vol. 25, no. 1, pp. 11-17
Khesin B. A., Tabachnikov S.
Contact complete integrability
2010, vol. 15, no. 4-5, pp.  504-520
Complete integrability in a symplectic setting means the existence of a Lagrangian foliation leaf-wise preserved by the dynamics. In the paper we describe complete integrability in a contact set-up as a more subtle structure: a flag of two foliations, Legendrian and co-Legendrian, and a holonomy-invariant transverse measure of the former in the latter. This turns out to be equivalent to the existence of a canonical $\mathbb{R} \times \mathbb{R}^{n−1}$ structure on the leaves of the co-Legendrian foliation. Further, the above structure implies the existence of n commuting contact fields preserving a special contact 1-form, thus providing the geometric framework and establishing equivalence with previously known definitions of contact integrability. We also show that contact completely integrable systems are solvable in quadratures.
We present an example of contact complete integrability: the billiard system inside an ellipsoid in pseudo-Euclidean space, restricted to the space of oriented null geodesics. We describe a surprising acceleration mechanism for closed light-like billiard trajectories.
Keywords: complete integrability, contact structure, Legendrian foliation, pseudo-Euclidean geometry, billiard map
Citation: Khesin B. A., Tabachnikov S.,  Contact complete integrability, Regular and Chaotic Dynamics, 2010, vol. 15, no. 4-5, pp. 504-520
Dogru F., Tabachnikov S.
On Polygonal Dual Billiard in the Hyperbolic Plane
2003, vol. 8, no. 1, pp.  67-81
We study the polygonal dual billiard map in the hyperbolic plane. We show that for a class of convex polygons called large all orbits of the dual billiard map escape to infinity. We also analyse the dynamics of the dual billiard map when the dual billiard table is a regular polygon with all right angles.
Citation: Dogru F., Tabachnikov S.,  On Polygonal Dual Billiard in the Hyperbolic Plane, Regular and Chaotic Dynamics, 2003, vol. 8, no. 1, pp. 67-81

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