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November  2011, 30(4): 1211-1235. doi: 10.3934/dcds.2011.30.1211

Mathematical retroreflectors

Alexander Plakhov 1,

1. 

Department of Mathematics, University of Aveiro, Aveiro 3810-193

Received  May 2010 Revised  August 2010 Published  May 2011

Retroreflectors are optical devices that reverse the direction of incident beams of light. Here we present a collection of billiard type retroreflectors consisting of four objects; three of them are asymptotically perfect retroreflectors, and the fourth one is a retroreflector which is very close to perfect. Three objects of the collection have recently been discovered and published or submitted for publication. The fourth object --- notched angle --- is a new one; a proof of its retroreflectivity is given.
Keywords: Billiards, problems of maximum resistance., retroreflectors, shape optimization.
Mathematics Subject Classification: Primary: 37D50, 49Q10; Secondary: 70G6.
Citation: Alexander Plakhov. Mathematical retroreflectors. Discrete and Continuous Dynamical Systems, 2011, 30 (4) : 1211-1235. doi: 10.3934/dcds.2011.30.1211
References:
[1]

P. Bachurin, K. Khanin, J. Marklof and A. Plakhov, Perfect retroreflectors and billiard dynamics, J. Modern Dynam., 5 (2011), 33-48. doi: 10.3934/jmd.2011.5.33.

[2]

K. I. Borg, L. H. Söderholm and H. Essén, Force on a spinning sphere moving in a rarefied gas, Physics of Fluids, 15 (2003), 736-741. doi: 10.1063/1.1541026.

[3]

L. Bunimovich, Mushrooms and other billiards with divided phase space, Chaos, 11 (2001), 802-808. doi: 10.1063/1.1418763.

[4]

J. E. Eaton, On spherically symmetric lenses, Trans. IRE Antennas Propag., 4 (1952), 66-71.

[5]

P. D. F. Gouveia, "Computação de Simetrias Variacionais e Optimização da Resistência Aerodinâmica Newtoniana," (Portuguese) [Computation of Variational Symmetries and Optimization of Newtonian Aerodynamic Resistance], Ph.D thesis, University of Aveiro, Portugal, 2007.

[6]

P. Gouveia, A. Plakhov and D. Torres, Two-dimensional body of maximum mean resistance, Applied Math. and Computation, 215 (2009), 37-52. doi: 10.1016/j.amc.2009.04.030.

[7]

S. G. Ivanov and A. M. Yanshin, Forces and moments acting on bodies rotating around a symmetry axis in a free molecular flow, Fluid Dyn., 15 (1980), 449-453. doi: 10.1007/BF01089985.

[8]

K. Moe and M. M. Moe, Gas-surface interactions and satellite drag coefficients, Planet. Space Sci., 53 (2005), 793-801. doi: 10.1016/j.pss.2005.03.005.

[9]

I. Newton, "Philosophiae Naturalis Principia Mathematica," London: Streater, 1687.

[10]

A. Plakhov, Billiards in unbounded domains reversing the direction of motion of a particle, Russ. Math. Surv., 61 (2006), 179-180. doi: 10.1070/RM2006v061n01ABEH004308.

[11]

A. Plakhov, Billiards and two-dimensional problems of optimal resistance, Arch. Ration. Mech. Anal., 194 (2009), 349-382. doi: 10.1007/s00205-008-0137-1.

[12]

A. Plakhov and P. Gouveia, Problems of maximal mean resistance on the plane, Nonlinearity, 20 (2007), 2271-2287. doi: 10.1088/0951-7715/20/9/013.

[13]

A. Plakhov, Scattering in billiards and problems of Newtonian aerodynamics, Russ. Math. Surv., 64 (2009), 873-938. doi: 10.1070/RM2009v064n05ABEH004642.

[14]

S. Tabachnikov, "Billiards," Paris: Société Mathématique de France, 1995.

[15]

C.-T. Wang, Free molecular flow over a rotating sphere, AIAA J., 10 (1972), 713-714. doi: 10.2514/3.50192.

show all references

References:
[1]

P. Bachurin, K. Khanin, J. Marklof and A. Plakhov, Perfect retroreflectors and billiard dynamics, J. Modern Dynam., 5 (2011), 33-48. doi: 10.3934/jmd.2011.5.33.

[2]

K. I. Borg, L. H. Söderholm and H. Essén, Force on a spinning sphere moving in a rarefied gas, Physics of Fluids, 15 (2003), 736-741. doi: 10.1063/1.1541026.

[3]

L. Bunimovich, Mushrooms and other billiards with divided phase space, Chaos, 11 (2001), 802-808. doi: 10.1063/1.1418763.

[4]

J. E. Eaton, On spherically symmetric lenses, Trans. IRE Antennas Propag., 4 (1952), 66-71.

[5]

P. D. F. Gouveia, "Computação de Simetrias Variacionais e Optimização da Resistência Aerodinâmica Newtoniana," (Portuguese) [Computation of Variational Symmetries and Optimization of Newtonian Aerodynamic Resistance], Ph.D thesis, University of Aveiro, Portugal, 2007.

[6]

P. Gouveia, A. Plakhov and D. Torres, Two-dimensional body of maximum mean resistance, Applied Math. and Computation, 215 (2009), 37-52. doi: 10.1016/j.amc.2009.04.030.

[7]

S. G. Ivanov and A. M. Yanshin, Forces and moments acting on bodies rotating around a symmetry axis in a free molecular flow, Fluid Dyn., 15 (1980), 449-453. doi: 10.1007/BF01089985.

[8]

K. Moe and M. M. Moe, Gas-surface interactions and satellite drag coefficients, Planet. Space Sci., 53 (2005), 793-801. doi: 10.1016/j.pss.2005.03.005.

[9]

I. Newton, "Philosophiae Naturalis Principia Mathematica," London: Streater, 1687.

[10]

A. Plakhov, Billiards in unbounded domains reversing the direction of motion of a particle, Russ. Math. Surv., 61 (2006), 179-180. doi: 10.1070/RM2006v061n01ABEH004308.

[11]

A. Plakhov, Billiards and two-dimensional problems of optimal resistance, Arch. Ration. Mech. Anal., 194 (2009), 349-382. doi: 10.1007/s00205-008-0137-1.

[12]

A. Plakhov and P. Gouveia, Problems of maximal mean resistance on the plane, Nonlinearity, 20 (2007), 2271-2287. doi: 10.1088/0951-7715/20/9/013.

[13]

A. Plakhov, Scattering in billiards and problems of Newtonian aerodynamics, Russ. Math. Surv., 64 (2009), 873-938. doi: 10.1070/RM2009v064n05ABEH004642.

[14]

S. Tabachnikov, "Billiards," Paris: Société Mathématique de France, 1995.

[15]

C.-T. Wang, Free molecular flow over a rotating sphere, AIAA J., 10 (1972), 713-714. doi: 10.2514/3.50192.

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