American Institute of Mathematical Sciences

Advanced
  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

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.  doi: 10.1007/BF02183641.  Google Scholar

[2]

M. Robnik and M. V. Berry, Classical billiards in magnetic fields,, J. Phys. A, 18 (1985), 1361.   Google Scholar

[3]

M. Bialy, Convex billiards and a theorem by E. Hopf,, Math. Z., 214 (1993), 147.   Google Scholar

[4]

M. Bialy, Hopf rigidity for convex billiards on the hemisphere and hyperbolic plane,, \arXiv{1205.3873}., ().   Google Scholar

[5]

M. L. Bialy, Rigidity for periodic magnetic fields,, Ergodic Theory Dynam. Systems, 20 (2000), 1619.   Google Scholar

[6]

Chavel, Isaac, "Riemannian Geometry,", A Modern Introduction. Cambridge Studies in Advanced Mathematics, 98 (2006).   Google Scholar

[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.  Google Scholar

[8]

Gutkin, Boris Hyperbolic magnetic billiards on surfaces of constant curvature,, Comm. Math. Phys., 217 (2001), 33.   Google Scholar

[9]

E. Gutkin, Billiard dynamics: A survey with the emphasis on open problems,, Regul. Chaotic Dyn., 8 (2003), 1.   Google Scholar

[10]

E. Gutkin and S. Tabachnikov, Billiards in Finsler and Minkowski geometries,, J. Geom. Phys., 40 (2002), 277.   Google Scholar

[11]

A. Knauf, Closed orbits and converse KAM theory,, Nonlinearity, 3 (1990), 961.   Google Scholar

[12]

S. Tabachnikov, "Billiards,", Panor. Synth., 1 (1995).   Google Scholar

[13]

S. Tabachnikov, Remarks on magnetic flows and magnetic billiards,, Finsler metrics and a magnetic analog of Hilbert's fourth problem. in, (2004), 233.   Google Scholar

[14]

T. Tasnadi, The behavior of nearby trajectoriies in magnetic billiards,, J. Math. Phys., 37 (1996), 5577.   Google Scholar

[15]

A. Veselov, Confocal surfaces and integrable billiards on the sphere and in the Lobachevsky space,, J. Geom. Phys., 7 (1990), 81.   Google Scholar

[16]

M. Wojtkowski, Two applications of Jacobi fields to the billiard ball problem,, J. Differential Geom., 40 (1994), 155.   Google Scholar

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.  doi: 10.1007/BF02183641.  Google Scholar

[2]

M. Robnik and M. V. Berry, Classical billiards in magnetic fields,, J. Phys. A, 18 (1985), 1361.   Google Scholar

[3]

M. Bialy, Convex billiards and a theorem by E. Hopf,, Math. Z., 214 (1993), 147.   Google Scholar

[4]

M. Bialy, Hopf rigidity for convex billiards on the hemisphere and hyperbolic plane,, \arXiv{1205.3873}., ().   Google Scholar

[5]

M. L. Bialy, Rigidity for periodic magnetic fields,, Ergodic Theory Dynam. Systems, 20 (2000), 1619.   Google Scholar

[6]

Chavel, Isaac, "Riemannian Geometry,", A Modern Introduction. Cambridge Studies in Advanced Mathematics, 98 (2006).   Google Scholar

[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.  Google Scholar

[8]

Gutkin, Boris Hyperbolic magnetic billiards on surfaces of constant curvature,, Comm. Math. Phys., 217 (2001), 33.   Google Scholar

[9]

E. Gutkin, Billiard dynamics: A survey with the emphasis on open problems,, Regul. Chaotic Dyn., 8 (2003), 1.   Google Scholar

[10]

E. Gutkin and S. Tabachnikov, Billiards in Finsler and Minkowski geometries,, J. Geom. Phys., 40 (2002), 277.   Google Scholar

[11]

A. Knauf, Closed orbits and converse KAM theory,, Nonlinearity, 3 (1990), 961.   Google Scholar

[12]

S. Tabachnikov, "Billiards,", Panor. Synth., 1 (1995).   Google Scholar

[13]

S. Tabachnikov, Remarks on magnetic flows and magnetic billiards,, Finsler metrics and a magnetic analog of Hilbert's fourth problem. in, (2004), 233.   Google Scholar

[14]

T. Tasnadi, The behavior of nearby trajectoriies in magnetic billiards,, J. Math. Phys., 37 (1996), 5577.   Google Scholar

[15]

A. Veselov, Confocal surfaces and integrable billiards on the sphere and in the Lobachevsky space,, J. Geom. Phys., 7 (1990), 81.   Google Scholar

[16]

M. Wojtkowski, Two applications of Jacobi fields to the billiard ball problem,, J. Differential Geom., 40 (1994), 155.   Google Scholar

[1]

Misha Bialy. Hopf rigidity for convex billiards on the hemisphere and hyperbolic plane. Discrete & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 2011, 29 (3) : 953-978. doi: 10.3934/dcds.2011.29.953
[3]

Federica Dragoni. Metric Hopf-Lax formula with semicontinuous data. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 713-729. doi: 10.3934/dcds.2007.17.713
[4]

Hanming Zhou. Lens rigidity with partial data in the presence of a magnetic field. Inverse Problems & Imaging, 2018, 12 (6) : 1365-1387. doi: 10.3934/ipi.2018057
[5]

Silvia Caprino, Guido Cavallaro, Carlo Marchioro. A Vlasov-Poisson plasma with unbounded mass and velocities confined in a cylinder by a magnetic mirror. Kinetic & Related Models, 2016, 9 (4) : 657-686. doi: 10.3934/krm.2016011
[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 & 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 & 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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 2011, 29 (3) : 893-908. doi: 10.3934/dcds.2011.29.893
[12]

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
[13]

Mickaël Kourganoff. Uniform hyperbolicity in nonflat billiards. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1145-1160. doi: 10.3934/dcds.2018048
[14]

Timothy Chumley, Renato Feres. Entropy production in random billiards. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020319
[15]

Yves Coudène. The Hopf argument. Journal of Modern Dynamics, 2007, 1 (1) : 147-153. doi: 10.3934/jmd.2007.1.147
[16]

Sonja Cox, Arnulf Jentzen, Ryan Kurniawan, Primož Pušnik. On the mild Itô formula in Banach spaces. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2217-2243. doi: 10.3934/dcdsb.2018232
[17]

Peter Seibt. A period formula for torus automorphisms. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 1029-1048. doi: 10.3934/dcds.2003.9.1029
[18]

Hans F. Weinberger, Xiao-Qiang Zhao. An extension of the formula for spreading speeds. Mathematical Biosciences & Engineering, 2010, 7 (1) : 187-194. doi: 10.3934/mbe.2010.7.187
[19]

Gamaliel Blé, Carlos Cabrera. A generalization of Douady's formula. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6183-6188. doi: 10.3934/dcds.2017267
[20]

Hong-Kun Zhang. Free path of billiards with flat points. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4445-4466. doi: 10.3934/dcds.2012.32.4445

2019 Impact Factor: 0.5

Tools

Metrics

  • PDF downloads (38)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences