American Institute of Mathematical Sciences

2012, 19: 112-119. doi: 10.3934/era.2012.19.112

On Totally integrable magnetic billiards on constant curvature surface

 1 School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University

Received  August 2012 Published  November 2012

We consider billiard ball motion in a convex domain of a constant curvature surface influenced by the constant magnetic field. We prove that if the billiard map is totally integrable then the boundary curve is necessarily a circle. This result shows that the so-called Hopf rigidity phenomenon which was recently obtained for classical billiards on constant curvature surfaces holds true also in the presence of constant magnetic field.
Citation: Misha Bialy. On Totally integrable magnetic billiards on constant curvature surface. Electronic Research Announcements, 2012, 19: 112-119. doi: 10.3934/era.2012.19.112
References:
 [1] N. Berglund and H. Kunz, Integrability and ergodicity of classical billiards in a magnetic field,, J. Statist. Phys., 83 (1996), 81. doi: 10.1007/BF02183641. [2] M. Robnik and M. V. Berry, Classical billiards in magnetic fields,, J. Phys. A, 18 (1985), 1361. [3] M. Bialy, Convex billiards and a theorem by E. Hopf,, Math. Z., 214 (1993), 147. [4] M. Bialy, Hopf rigidity for convex billiards on the hemisphere and hyperbolic plane,, \arXiv{1205.3873}., (). [5] M. L. Bialy, Rigidity for periodic magnetic fields,, Ergodic Theory Dynam. Systems, 20 (2000), 1619. [6] Chavel, Isaac, "Riemannian Geometry,", A Modern Introduction. Cambridge Studies in Advanced Mathematics, 98 (2006). [7] B. Gutkin, U. Smilansky and E. Gutkin, Hyperbolic billiards on surfaces of constant curvature,, Comm. Math. Phys., 208 (1999), 65. doi: 10.1007/s002200050748. [8] Gutkin, Boris Hyperbolic magnetic billiards on surfaces of constant curvature,, Comm. Math. Phys., 217 (2001), 33. [9] E. Gutkin, Billiard dynamics: A survey with the emphasis on open problems,, Regul. Chaotic Dyn., 8 (2003), 1. [10] E. Gutkin and S. Tabachnikov, Billiards in Finsler and Minkowski geometries,, J. Geom. Phys., 40 (2002), 277. [11] A. Knauf, Closed orbits and converse KAM theory,, Nonlinearity, 3 (1990), 961. [12] S. Tabachnikov, "Billiards,", Panor. Synth., 1 (1995). [13] S. Tabachnikov, Remarks on magnetic flows and magnetic billiards,, Finsler metrics and a magnetic analog of Hilbert's fourth problem. in, (2004), 233. [14] T. Tasnadi, The behavior of nearby trajectoriies in magnetic billiards,, J. Math. Phys., 37 (1996), 5577. [15] A. Veselov, Confocal surfaces and integrable billiards on the sphere and in the Lobachevsky space,, J. Geom. Phys., 7 (1990), 81. [16] M. Wojtkowski, Two applications of Jacobi fields to the billiard ball problem,, J. Differential Geom., 40 (1994), 155.

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References:
 [1] N. Berglund and H. Kunz, Integrability and ergodicity of classical billiards in a magnetic field,, J. Statist. Phys., 83 (1996), 81. doi: 10.1007/BF02183641. [2] M. Robnik and M. V. Berry, Classical billiards in magnetic fields,, J. Phys. A, 18 (1985), 1361. [3] M. Bialy, Convex billiards and a theorem by E. Hopf,, Math. Z., 214 (1993), 147. [4] M. Bialy, Hopf rigidity for convex billiards on the hemisphere and hyperbolic plane,, \arXiv{1205.3873}., (). [5] M. L. Bialy, Rigidity for periodic magnetic fields,, Ergodic Theory Dynam. Systems, 20 (2000), 1619. [6] Chavel, Isaac, "Riemannian Geometry,", A Modern Introduction. Cambridge Studies in Advanced Mathematics, 98 (2006). [7] B. Gutkin, U. Smilansky and E. Gutkin, Hyperbolic billiards on surfaces of constant curvature,, Comm. Math. Phys., 208 (1999), 65. doi: 10.1007/s002200050748. [8] Gutkin, Boris Hyperbolic magnetic billiards on surfaces of constant curvature,, Comm. Math. Phys., 217 (2001), 33. [9] E. Gutkin, Billiard dynamics: A survey with the emphasis on open problems,, Regul. Chaotic Dyn., 8 (2003), 1. [10] E. Gutkin and S. Tabachnikov, Billiards in Finsler and Minkowski geometries,, J. Geom. Phys., 40 (2002), 277. [11] A. Knauf, Closed orbits and converse KAM theory,, Nonlinearity, 3 (1990), 961. [12] S. Tabachnikov, "Billiards,", Panor. Synth., 1 (1995). [13] S. Tabachnikov, Remarks on magnetic flows and magnetic billiards,, Finsler metrics and a magnetic analog of Hilbert's fourth problem. in, (2004), 233. [14] T. Tasnadi, The behavior of nearby trajectoriies in magnetic billiards,, J. Math. Phys., 37 (1996), 5577. [15] A. Veselov, Confocal surfaces and integrable billiards on the sphere and in the Lobachevsky space,, J. Geom. Phys., 7 (1990), 81. [16] M. Wojtkowski, Two applications of Jacobi fields to the billiard ball problem,, J. Differential Geom., 40 (1994), 155.
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