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
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show all references

References:
[1]

J. Statist. Phys., 83 (1996), 81-126. doi: 10.1007/BF02183641.  Google Scholar

[2]

J. Phys. A, 18 (1985), 1361-1378.  Google Scholar

[3]

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]

Ergodic Theory Dynam. Systems, 20 (2000), 1619-1626.  Google Scholar

[6]

A Modern Introduction. Cambridge Studies in Advanced Mathematics, 98. Cambridge University Press, Cambridge, 2006.  Google Scholar

[7]

Comm. Math. Phys., 208 (1999), 65-90. doi: 10.1007/s002200050748.  Google Scholar

[8]

Comm. Math. Phys., 217 (2001), 33-53.  Google Scholar

[9]

Regul. Chaotic Dyn., 8 (2003), 1-13.  Google Scholar

[10]

J. Geom. Phys., 40 (2002), 277-301.  Google Scholar

[11]

Nonlinearity, 3 (1990), 961-973.  Google Scholar

[12]

Panor. Synth., 1, 1995.  Google Scholar

[13]

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]

J. Math. Phys., 37 (1996), 5577-5598.  Google Scholar

[15]

J. Geom. Phys., 7 (1990), 81-107.  Google Scholar

[16]

J. Differential Geom., 40 (1994), 155-164.  Google Scholar

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