On Totally integrable magnetic billiards on constant curvature surface

Previous Article
Locally decodable codes and the failure of cotype for projective tensor products
ERA-MS Home
This Volume
Next Article
Simple loops on 2-bridge spheres in Heckoid orbifolds for 2-bridge links

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

On Totally integrable magnetic billiards on constant curvature surface

Misha Bialy ^1,

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

Received August 2012 Published November 2012

Abstract
Related Papers

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.

Keywords: Magnetic Billiards, Hopf rigidity., Mirror formula.

Mathematics Subject Classification: 37J40; Secondary: 37J3.

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), 81-126. doi: 10.1007/BF02183641.
[2]	M. Robnik and M. V. Berry, Classical billiards in magnetic fields, J. Phys. A, 18 (1985), 1361-1378.
[3]	M. Bialy, Convex billiards and a theorem by E. Hopf, Math. Z., 214 (1993), 147-154.
[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), 1619-1626.
[6]	Chavel, Isaac, "Riemannian Geometry," A Modern Introduction. Cambridge Studies in Advanced Mathematics, 98. Cambridge University Press, Cambridge, 2006.
[7]	B. Gutkin, U. Smilansky and E. Gutkin, Hyperbolic billiards on surfaces of constant curvature, Comm. Math. Phys., 208 (1999), 65-90. doi: 10.1007/s002200050748.
[8]	Gutkin, Boris Hyperbolic magnetic billiards on surfaces of constant curvature, Comm. Math. Phys., 217 (2001), 33-53.
[9]	E. Gutkin, Billiard dynamics: A survey with the emphasis on open problems, Regul. Chaotic Dyn., 8 (2003), 1-13.
[10]	E. Gutkin and S. Tabachnikov, Billiards in Finsler and Minkowski geometries, J. Geom. Phys., 40 (2002), 277-301.
[11]	A. Knauf, Closed orbits and converse KAM theory, Nonlinearity, 3 (1990), 961-973.
[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 "Modern Dynamical Systems and Applications" 233-250, Cambridge Univ. Press, Cambridge, 2004.
[14]	T. Tasnadi, The behavior of nearby trajectoriies in magnetic billiards, J. Math. Phys., 37 (1996), 5577-5598.
[15]	A. Veselov, Confocal surfaces and integrable billiards on the sphere and in the Lobachevsky space, J. Geom. Phys., 7 (1990), 81-107.
[16]	M. Wojtkowski, Two applications of Jacobi fields to the billiard ball problem, J. Differential Geom., 40 (1994), 155-164.

show all references

References:

[1]	N. Berglund and H. Kunz, Integrability and ergodicity of classical billiards in a magnetic field, J. Statist. Phys., 83 (1996), 81-126. doi: 10.1007/BF02183641.
[2]	M. Robnik and M. V. Berry, Classical billiards in magnetic fields, J. Phys. A, 18 (1985), 1361-1378.
[3]	M. Bialy, Convex billiards and a theorem by E. Hopf, Math. Z., 214 (1993), 147-154.
[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), 1619-1626.
[6]	Chavel, Isaac, "Riemannian Geometry," A Modern Introduction. Cambridge Studies in Advanced Mathematics, 98. Cambridge University Press, Cambridge, 2006.
[7]	B. Gutkin, U. Smilansky and E. Gutkin, Hyperbolic billiards on surfaces of constant curvature, Comm. Math. Phys., 208 (1999), 65-90. doi: 10.1007/s002200050748.
[8]	Gutkin, Boris Hyperbolic magnetic billiards on surfaces of constant curvature, Comm. Math. Phys., 217 (2001), 33-53.
[9]	E. Gutkin, Billiard dynamics: A survey with the emphasis on open problems, Regul. Chaotic Dyn., 8 (2003), 1-13.
[10]	E. Gutkin and S. Tabachnikov, Billiards in Finsler and Minkowski geometries, J. Geom. Phys., 40 (2002), 277-301.
[11]	A. Knauf, Closed orbits and converse KAM theory, Nonlinearity, 3 (1990), 961-973.
[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 "Modern Dynamical Systems and Applications" 233-250, Cambridge Univ. Press, Cambridge, 2004.
[14]	T. Tasnadi, The behavior of nearby trajectoriies in magnetic billiards, J. Math. Phys., 37 (1996), 5577-5598.
[15]	A. Veselov, Confocal surfaces and integrable billiards on the sphere and in the Lobachevsky space, J. Geom. Phys., 7 (1990), 81-107.
[16]	M. Wojtkowski, Two applications of Jacobi fields to the billiard ball problem, J. Differential Geom., 40 (1994), 155-164.

[1]	Misha Bialy. Hopf rigidity for convex billiards on the hemisphere and hyperbolic plane. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 3903-3913. doi: 10.3934/dcds.2013.33.3903
[2]	Aleksander Ćwiszewski, Wojciech Kryszewski. On a generalized Poincaré-Hopf formula in infinite dimensions. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 953-978. doi: 10.3934/dcds.2011.29.953
[3]	Federica Dragoni. Metric Hopf-Lax formula with semicontinuous data. Discrete and Continuous Dynamical Systems, 2007, 17 (4) : 713-729. doi: 10.3934/dcds.2007.17.713
[4]	Silvia Caprino, Guido Cavallaro, Carlo Marchioro. A Vlasov-Poisson plasma with unbounded mass and velocities confined in a cylinder by a magnetic mirror. Kinetic and Related Models, 2016, 9 (4) : 657-686. doi: 10.3934/krm.2016011
[5]	Hanming Zhou. Lens rigidity with partial data in the presence of a magnetic field. Inverse Problems and Imaging, 2018, 12 (6) : 1365-1387. doi: 10.3934/ipi.2018057
[6]	Juan Pablo Rincón-Zapatero. Hopf-Lax formula for variational problems with non-constant discount. Journal of Geometric Mechanics, 2009, 1 (3) : 357-367. doi: 10.3934/jgm.2009.1.357
[7]	Yernat M. Assylbekov, Hanming Zhou. Boundary and scattering rigidity problems in the presence of a magnetic field and a potential. Inverse Problems and Imaging, 2015, 9 (4) : 935-950. doi: 10.3934/ipi.2015.9.935
[8]	Bo Wang, Jiguang Bao. Mirror symmetry for a Hessian over-determined problem and its generalization. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2305-2316. doi: 10.3934/cpaa.2014.13.2305
[9]	W. Patrick Hooper, Richard Evan Schwartz. Billiards in nearly isosceles triangles. Journal of Modern Dynamics, 2009, 3 (2) : 159-231. doi: 10.3934/jmd.2009.3.159
[10]	Serge Tabachnikov. Birkhoff billiards are insecure. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 1035-1040. doi: 10.3934/dcds.2009.23.1035
[11]	Simon Castle, Norbert Peyerimhoff, Karl Friedrich Siburg. Billiards in ideal hyperbolic polygons. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 893-908. doi: 10.3934/dcds.2011.29.893
[12]	Timothy Chumley, Renato Feres. Entropy production in random billiards. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1319-1346. doi: 10.3934/dcds.2020319
[13]	Richard Evan Schwartz. Outer billiards and the pinwheel map. Journal of Modern Dynamics, 2011, 5 (2) : 255-283. doi: 10.3934/jmd.2011.5.255
[14]	Mickaël Kourganoff. Uniform hyperbolicity in nonflat billiards. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1145-1160. doi: 10.3934/dcds.2018048
[15]	Yves Coudène. The Hopf argument. Journal of Modern Dynamics, 2007, 1 (1) : 147-153. doi: 10.3934/jmd.2007.1.147
[16]	Hong-Kun Zhang. Free path of billiards with flat points. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4445-4466. doi: 10.3934/dcds.2012.32.4445
[17]	Giovanni Panti. Billiards on pythagorean triples and their Minkowski functions. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4341-4378. doi: 10.3934/dcds.2020183
[18]	W. Patrick Hooper, Richard Evan Schwartz. Erratum: Billiards in nearly isosceles triangles. Journal of Modern Dynamics, 2014, 8 (1) : 133-137. doi: 10.3934/jmd.2014.8.133
[19]	Richard Evan Schwartz. Unbounded orbits for outer billiards I. Journal of Modern Dynamics, 2007, 1 (3) : 371-424. doi: 10.3934/jmd.2007.1.371
[20]	Ilesanmi Adeboye, Harrison Bray, David Constantine. Entropy rigidity and Hilbert volume. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 1731-1744. doi: 10.3934/dcds.2019075

2020 Impact Factor: 0.929

Tools

Metrics

Other articles
by authors

on AIMS
Misha Bialy
on Google Scholar
Misha Bialy

[Back to Top]