Advertisement

Journal of Mathematical Sciences

, Volume 182, Issue 2, pp 233–245 | Cite as

Billiards, scattering by rough obstacles, and optimal mass transportation

  • A. PlakhovEmail author
Article

Abstract

This article presents a brief exposition of recent results of the author on billiard scattering by rough obstacles. We define the notion of a rough body and give a characterization of scattering by rough bodies. Then we define the resistance of a rough body; it can be interpreted as the aerodynamic resistance of the somersaulting body moving through a rarefied medium. We solve the problems of maximum and minimum resistance for rough bodies (more precisely, for bodies obtained by roughening a prescribed convex set) in arbitrary dimension. Surprisingly, these problems are reduced to special problems of optimal mass transportation on the sphere.

Keywords

Convex Body Incident Particle Solar Sailing Transport Plan Optimal Transportation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    G. A. Chechkin, A. L. Piatnitski, and A. S. Shamaev, Homogenization. Methods and Applications. Transl. Math. Monogr., 234, Am. Math. Soc., Providence, Rhode Island (2007).zbMATHGoogle Scholar
  2. 2.
    N. Chernov, “Entropy, Lyapunov exponents, and mean free path for billiards,” J. Stat. Phys., 88, 1–29 (1997).MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    P. D. Fieseler, “A method for solar sailing in a low Earth orbit,” Acta Astron., 43, 531–541 (1998).CrossRefGoogle Scholar
  4. 4.
    K. Moe and M. M. Moe, “Gas-surface interactions and satellite drag coefficients,” Planet. Space Sci., 53, 793–801 (2005).CrossRefGoogle Scholar
  5. 5.
    J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces, Taylor & Francis (1991).Google Scholar
  6. 6.
    B. N. J. Persson, “Contact mechanics for randomly rough surfaces,” Surf. Sci. Rep., 61, 201–227 (2006).MathSciNetCrossRefGoogle Scholar
  7. 7.
    A. Plakhov, “Newton’s problem of the body of minimum mean resistance,” Sb. Math., 195, 1017–1037 (2004).MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    A. Plakhov, “Billiard scattering on rough sets. Two-dimensional case,” SIAM J. Math. Anal., 40, 2155–2178 (2009).MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    A. Plakhov, “Scattering in billiards and problems of Newtonian aerodynamics,” Russ. Math. Surv., 64, 873-938 (2009).MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of AveiroAveiroPortugal

Personalised recommendations