American Institute of Mathematical Sciences

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  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), 81-126. doi: 10.1007/BF02183641.  Google Scholar

[2]

M. Robnik and M. V. Berry, Classical billiards in magnetic fields, J. Phys. A, 18 (1985), 1361-1378.  Google Scholar

[3]

M. Bialy, Convex billiards and a theorem by E. Hopf, Math. Z., 214 (1993), 147-154.  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), 1619-1626.  Google Scholar

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

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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.  Google Scholar

[8]

Gutkin, Boris Hyperbolic magnetic billiards on surfaces of constant curvature, Comm. Math. Phys., 217 (2001), 33-53.  Google Scholar

[9]

E. Gutkin, Billiard dynamics: A survey with the emphasis on open problems, Regul. Chaotic Dyn., 8 (2003), 1-13.  Google Scholar

[10]

E. Gutkin and S. Tabachnikov, Billiards in Finsler and Minkowski geometries, J. Geom. Phys., 40 (2002), 277-301.  Google Scholar

[11]

A. Knauf, Closed orbits and converse KAM theory, Nonlinearity, 3 (1990), 961-973.  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 "Modern Dynamical Systems and Applications" 233-250, Cambridge Univ. Press, Cambridge, 2004.  Google Scholar

[14]

T. Tasnadi, The behavior of nearby trajectoriies in magnetic billiards, J. Math. Phys., 37 (1996), 5577-5598.  Google Scholar

[15]

A. Veselov, Confocal surfaces and integrable billiards on the sphere and in the Lobachevsky space, J. Geom. Phys., 7 (1990), 81-107.  Google Scholar

[16]

M. Wojtkowski, Two applications of Jacobi fields to the billiard ball problem, J. Differential Geom., 40 (1994), 155-164.  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), 81-126. doi: 10.1007/BF02183641.  Google Scholar

[2]

M. Robnik and M. V. Berry, Classical billiards in magnetic fields, J. Phys. A, 18 (1985), 1361-1378.  Google Scholar

[3]

M. Bialy, Convex billiards and a theorem by E. Hopf, Math. Z., 214 (1993), 147-154.  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), 1619-1626.  Google Scholar

[6]

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

[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.  Google Scholar

[8]

Gutkin, Boris Hyperbolic magnetic billiards on surfaces of constant curvature, Comm. Math. Phys., 217 (2001), 33-53.  Google Scholar

[9]

E. Gutkin, Billiard dynamics: A survey with the emphasis on open problems, Regul. Chaotic Dyn., 8 (2003), 1-13.  Google Scholar

[10]

E. Gutkin and S. Tabachnikov, Billiards in Finsler and Minkowski geometries, J. Geom. Phys., 40 (2002), 277-301.  Google Scholar

[11]

A. Knauf, Closed orbits and converse KAM theory, Nonlinearity, 3 (1990), 961-973.  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 "Modern Dynamical Systems and Applications" 233-250, Cambridge Univ. Press, Cambridge, 2004.  Google Scholar

[14]

T. Tasnadi, The behavior of nearby trajectoriies in magnetic billiards, J. Math. Phys., 37 (1996), 5577-5598.  Google Scholar

[15]

A. Veselov, Confocal surfaces and integrable billiards on the sphere and in the Lobachevsky space, J. Geom. Phys., 7 (1990), 81-107.  Google Scholar

[16]

M. Wojtkowski, Two applications of Jacobi fields to the billiard ball problem, J. Differential Geom., 40 (1994), 155-164.  Google Scholar

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