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.  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]

Timothy Chumley, Renato Feres. Entropy production in random billiards. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1319-1346. doi: 10.3934/dcds.2020319

[2]

Qiao Liu. Local rigidity of certain solvable group actions on tori. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 553-567. doi: 10.3934/dcds.2020269

[3]

Jian Zhang, Tony T. Lee, Tong Ye, Liang Huang. An approximate mean queue length formula for queueing systems with varying service rate. Journal of Industrial & Management Optimization, 2021, 17 (1) : 185-204. doi: 10.3934/jimo.2019106

[4]

Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344

[5]

Wenxiong Chen, Congming Li, Shijie Qi. A Hopf lemma and regularity for fractional $ p $-Laplacians. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3235-3252. doi: 10.3934/dcds.2020034

[6]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276

[7]

Susmita Sadhu. Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020342

[8]

Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084

[9]

Zaizheng Li, Qidi Zhang. Sub-solutions and a point-wise Hopf's lemma for fractional $ p $-Laplacian. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020293

[10]

Xianyong Chen, Weihua Jiang. Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021013

2019 Impact Factor: 0.5

Metrics

  • PDF downloads (45)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]