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), 81-126. 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 planearXiv: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 planearXiv: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.

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